diff --git "a/7NE0T4oBgHgl3EQfwAHH/content/tmp_files/load_file.txt" "b/7NE0T4oBgHgl3EQfwAHH/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/7NE0T4oBgHgl3EQfwAHH/content/tmp_files/load_file.txt" @@ -0,0 +1,661 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf,len=660 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='02627v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='RA] 6 Jan 2023 Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms Michela Cerqua and Alberto Facchini Abstract We introduce the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring 푘 with identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' These notions are applied to the study of pre-Lie 푘-algebras and, more generally, Lie-admissible 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Associating with any algebra (퐴, ·) its sub-adjacent anticommutative algebra (퐴, [−, −]) is a functor from the category of 푘-algebras with pre-morphisms to the category of anticommutative 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We describe the commutator of two ideals of a pre-Lie algebra, showing that the condition (Huq=Smith) holds for pre- Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This allows to make use of all the notions concerning multiplicative lattices in the study of the multiplicative lattice of ideals of a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We study idempotent endomorphisms of a pre-Lie algebra 퐿, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', semidirect-product decompositions of 퐿 and bimodules over 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Introduction The aim of this paper is to present pre-Lie algebras from the point of view of their multiplicative lattice of ideals, and to study their idempotent endomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre- Lie algebras were first introduced and studied in [15] by Vinberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' He applied them to the study of convex homogenous cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' He called “left-symmetric algebras” the algebras we call pre-Lie algebras in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We present a notion of pre-morphism and pre-derivation for arbitrary non- associative algebras over a commutative ring 푘 with identity, and apply it to the study of pre-Lie 푘-algebras and, more generally, Lie-admissible 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Asso- ciating with any pre-Lie algebra (퐴, ·) its sub-adjacent Lie algebra (퐴, [−, −]) is a functor from the category PreL푘,푝 of pre-Lie 푘-algebras with pre-morphisms to the category of Lie 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We introduce the notion of module 푀 over a pre-Lie Michela Cerqua e-mail: michela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='cerqua@studenti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='it · Alberto Facchini Dipartimento di Matematica "Tullio Levi Civita", Università di Padova, 35121 Padova, Italy e-mail: facchini@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='it 1 2 Michela Cerqua and Alberto Facchini algebra 퐿 and, like in the case of associative algebras, it is possible to do it in two equivalent ways, via a suitable scalar multiplication 퐿 × 푀 → 푀 or as a 푘-module 푀 with a pre-morphism 휆: (퐿, ·) → (End(푘푀), ◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The category of modules over a pre-Lie 푘-algebra (퐿, ·) is isomorphic to the category of modules over its sub- adjacent Lie 푘-algebra (퐿, [−, −]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We then consider the commutator of two ideals in a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In particular we show that the condition (Huq=Smith) holds for pre-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' With the notion of commutator at our disposal, the lattice of ideals of a pre-Lie algebra becomes a multiplicative lattice [6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' As a consequence we immediately get the notions of abelian pre-Lie algebra, prime ideal, prime spec- trum of a pre-Lie algebra, solvable and nilpotent pre-Lie algebras, metabelian and hyperabelian pre-Lie algebras, centralizer, and center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We then consider idempotent endomorphisms of a pre-Lie algebra, because they immediately show what semi-direct products of pre-Lie algebras are, what the action of a pre-Lie algebra on another pre-Lie algebra is, and lead us to the notion of bimodule over a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We study the “Dorroh extensions” of pre-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Like in the associative case, we get a category equivalence between the category PreL푘 and the category of pre-Lie algebras with identity and with an augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 1 Preliminary notions on non-associative 풌-algebras Let 푘 be a commutative ring with identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In this article, a 푘-algebra is a 푘-module 푘푀 with a further 푘-bilinear operation 푀 × 푀 → 푀, (푥, 푦) ↦→ 푥푦 (equivalently, a 푘-module morphism 푀 ⊗푘 푀 → 푀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A subalgebra (an ideal, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=') of 푀 is a 푘-submodule 푁 of 푀 such that 푥푦 ∈ 푁 for every 푥, 푦 ∈ 푁 (푥푛 ∈ 푁 and 푛푥 ∈ 푁 for every 푥 ∈ 푀 and 푛 ∈ 푁, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=') As usual, if 푁 is an ideal of 푀, the quotient 푘-module 푀/푁 inherits a 푘-algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' There is a one-to-one correspondence between the set of all ideals 푁 of 푀 and the set of all congruences on 푀, that is, all equivalence relations ∼ on 푀 for which 푥 ∼ 푦 and 푧 ∼ 푤 imply 푥 + 푧 ∼ 푦 + 푤, 휆푥 ∼ 휆푦 and 푥푧 ∼ 푦푤 for every 푥, 푦, 푧, 푤 ∈ 푀 and every 휆 ∈ 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The opposite 푀op of an algebra 푀 is defined taking as multiplication in 푀op the mapping (푥, 푦) ↦→ 푦푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If 푀 and 푀′ are two 푘-algebras, a 푘-linear mapping 휑: 푀 → 푀′ is a 푘-algebra homomorphism if 휑(푥푦) = 휑(푥)휑(푦) for every 푥, 푦 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Clearly, 푘-algebras form a variety in the sense of Universal Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Moreover, it is a variety of Ω-groups, that is, a variety which is pointed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', it has exactly one constant) and has amongst its operations and identities those of the variety of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It follows that 푘-algebras form a semiabelian category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Other examples of Ω-groups are abelian groups, non-unital rings, commutative algebras, modules and Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If 푀 is any 푘-algebra, its endomorphisms form a monoid, that is, a semigroup with a two-sided identity, with respect to composition of mappings ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A derivation of a 푘-algebra 푀 is any 푘-linear mapping 퐷 : 푀 → 푀 such that 퐷(푥푦) = (퐷(푥))푦+ 푥(퐷(푦)) for every 푥, 푦 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any 푘-algebra 푀, we can construct the 푘-algebra of derivations Der푘(푀) of the 푘-algebra 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Its elements are all derivations of 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 3 If 푀 is any 푘-algebra and 퐷, 퐷′ are two derivations of 푀, then the composite mapping 퐷퐷′ is not a derivation of 푀 in general, but 퐷퐷′ − 퐷′퐷 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Thus, for any 푘-algebra 푀, we can define the Lie 푘-algebra Der푘 (푀) as the subset of End(푘푀) consisting of all derivations of 푀 with multiplication [퐷, 퐷′] := 퐷퐷′ − 퐷′퐷 for every 퐷, 퐷′ ∈ Der푘(푀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is known that there is not a general notion of representation (or module)over our (non-associative) 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' There is a notion of bimodule over a non-associative ring due to Eillenberg, and this notion works well for Lie algebras, but is not convenient in the study of Jordan algebras and alternative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The situation, as far as modules are concerned, is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1 Modules over an associative 풌-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Given any 푘-algebra 푀, we can consider, for every element 푥 ∈ 푀, the map- ping 휆푥 : 푀 → 푀, defined by 휆푥(푎) = 푥푎 for every 푎 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The mapping 휆: 푀 → End(푘푀) is defined by 휆: 푥 ↦→ 휆푥 for every 푥 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This 휆 is a 푘- algebra morphism if and only if 푀 is associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Thus, for any associative 푘-algebra 푀, it is natural to define a left 푀-module as any 푘-module 푘 퐴 with a 푘-algebra homomorphism 휆: 푀 → End(푘 퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Similarly, we can define right ���-modules as 푘-modules 푘 퐴 with a 푘-algebra antihomomorphism 휌 : 푀 → End(푘 퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Here by 푘- algebra antihomomorphism 휓 : 푀 → 푀′ between two 푘-algebras 푀, 푀′ we mean any 푘-linear mapping 휓 such that 휓(푥푦) = 휓(푦)휓(푥) for every 푥, 푦 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Clearly, a mapping 푀 → 푀′ is a 푘-algebra antihomomorphism if and only if it is a 푘-algebra homomorphism 푀op → 푀′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It follows that right 푀-modules coincide with left 푀op-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' More precisely, when we say that right 푀-modules coincide with left 푀op-modules, we mean that there is a canonical category isomorphism between the category of all right 푀-modules and the category of all left 푀op-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Simi- larly, left 푀-modules coincide with right 푀op-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, if 푀 is commutative, then left 푀-modules and right 푀-modules coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Finally, left modules 퐴 over an associative 푘-algebra 푀 can be equivalently defined using, instead of the 푘- algebra homomorphism 휆: 푀 → End(푘 퐴), a 푘-bilinear mapping 휇: 푀 × 퐴 → 퐴, 휇: (푚, 푎) ↦→ 푚푎, such that (푚푚′)푎 = 푚(푚′푎) for every 푚, 푚′ ∈ 푀 and 푎 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='2 Modules over a Lie 풌-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any 푘-module 퐴 we will denote by 픤픩(퐴) the Lie 푘-algebra End(푘 퐴) of all 푘-endomorphisms of 퐴 with the operation [−, −] defined by [ 푓 , 푔] = 푓 푔 − 푔 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any Lie 푘-algebra 푀 and any element 푥 ∈ 푀, the mapping 휆푥 is an element of the Lie 푘-algebra Der푘 (푀), usually called the adjoint of 푥, or the inner derivation defined by 푥, and usually denoted by ad푀 푥 instead of 휆푥, and the mapping ad: 푀 → 4 Michela Cerqua and Alberto Facchini Der푘(푀) ⊆ 픤픩(푀), defined by ad: 푥 ↦→ ad푀 푥 for every 푥 ∈ 푀, is a Lie 푘-algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Left modules over a Lie 푘-algebra 푀 are defined as 푘-modules 퐴 with a Lie 푘-algebra homomorphism휆: 푀 → 픤픩(퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Similarly, it is possible to define right 푀- modules as 푘-modules 퐴 with a 푘-algebra antihomomorphism 휌 : 푀 → 픤픩(퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' But any Lie 푘-algebra 푀 is isomorphic to its opposite algebra 푀op via the isomorphism 푀 → 푀op, 푥 ↦→ −푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It follows that the category of right 푀-modules is canonically isomorphic to the category of left 푀-modules for any Lie 푘-algebra 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Therefore it is useless to introduce both right and left modules, it is sufficient to introduce left 푀-modules and call them simply “푀-modules”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 2 Pre-Lie 풌-algebras A pre-Lie 푘-algebra is a 푘-algebra (푀, ·) satisfying the identity (푥 · 푦) · 푧 − 푥 · (푦 · 푧) = (푦 · 푥) · 푧 − 푦 · (푥 · 푧) (1) for every 푥, 푦, 푧 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any 푘-algebra (푀, ·), defining the commutator [푥, 푦] = 푥 · 푦 − 푦 · 푥 for every 푥, 푦 ∈ 푀, the algebra (푀, [−, −]) is anticommutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If (푀, ·) is a pre-Lie algebra, one gets that (푀, [−, −]) is a Lie algebra, called the Lie algebra sub-adjacent to the pre-Lie algebra (푀, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras are also called Vinberg algebras or left-symmetric algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This last name refers to the fact that in (1) one exchanges the first two variables on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A right-symmetric algebra is an algebra in which, for every 푥, 푦, 푧 ∈ 푀, (푥 · 푦) · 푧 − 푥 · (푦 · 푧) = (푥 · 푧) · 푦 − 푥 · (푧 · 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is easily seen that the category of left-symmetricalgebras and the category of right-symmetricalgebras are isomorphic (the categorical isomorphism is given by 푀 ↦→ 푀op).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Examples 1 (1) Every associative algebra is clearly a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (2) Derivations on 푘[푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푥푛]푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푘 be a commutative ring with identity, 푛 ≥ 1 be an integer, and 푘[푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푥푛] be the ring of polynomials in the 푛 indeter- minates 푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푥푛 with coefficients in 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐴 be the free 푘[푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푥푛]-module 푘[푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푥푛]푛 with free set {푒1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푒푛} of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' As a 푘-module, 퐴 is the free 푘-module with free set of generators the set { 푥푖1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푥푖푛 푛 푒 푗 | 푖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푖푛 ≥ 0, 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푛}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Consider the usual derivations of the ring 푘[푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푥푛]: 휕 휕푥 푗 (푥푖1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푥푖푛 푛 ) = � 푥푖1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푖 푗푥푖푗−1 푗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푥푖푛 푛 for 푖 푗 > 0, 0 for 푖 푗 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Defineamultiplication on 퐴 setting,forevery 푢 = (푢1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푢푛), 푣 = (푣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푣푛) ∈ 퐴, Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 5 푣 · 푢 = ( 푛 � 푗=1 푣 푗 휕푢1 휕푥 푗 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푛 � 푗=1 푣 푗 휕푢푛 휕푥 푗 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is then possible to see that 퐴 is a pre-Lie 푘-algebra [2, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (3) An example of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푘 be any commutative ring with identity and 퐿 � 푘 ⊕ 푘 a free 푘-module of rank 2 with free set {푒1, 푒2} of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Define a multiplication on 퐿 setting 푒1푒1 = 2푒1, 푒1푒2 = 푒2, 푒2푒1 = 0, 푒2푒2 = 푒1, and extending by 푘-bilinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then 퐿 is a pre-Lie 푘-algebra [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (4) Rooted trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Recall that a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A rooted tree of degree 푛 is a pair (푇, 푟), where 푇 is a tree with 푛 vertices, and its root 푟 is a vertex of 푇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In the following we will label the vertices of 푇 with the numbers 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푛, and the root 푟 with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푘 be a commutative ring with identity and T푛 be the free 푘-module with free set of generators the set of all isomorphism classes of rooted trees of degree 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Set T := � 푛≥1 T푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Define a multiplication on T setting, for every pair 푇1,푇2 of rooted trees, 푇1 · 푇2 = � 푣 ∈푉 (푇2) 푇1 ◦푣 푇2, where 푉(푇2) is the set of vertices of 푇2, and 푇1 ◦푣 푇2 is the rooted tree obtained by adding to the disjoint union of 푇1 and 푇2 a further new edge joining the root vertex of 푇1 with the vertex 푣 of 푇2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The root of 푇1 ◦푣 푇2 is defined to be the same as the root of 푇2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' To get a multiplication on T, extend this multiplication by 푘-bilinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let us give an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Suppose 푇1 = 1 2 3 and 푇2 = 1 2 Then 6 Michela Cerqua and Alberto Facchini 푇1 ◦1 푇2 = 1 2 3 4 5 and 푇1 ◦2 푇2 = 1 2 3 4 5 , where we have relabelled the vertices of푇1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (If푇1 has 푛 vertices and푇2 has 푚 vertices, it is convenient to relabel in 푇1 ◦푣 푇2 the vertices 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푛 of 푇1 with the numbers 푚 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푚 + 푛, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=') Therefore 푇1 · 푇2 = 1 2 3 4 5 + 1 2 3 4 5 In this way, one gets a pre-Lie 푘-algebra T [4, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is a graded 푘-algebra because T푛 · T푚 ⊆ T푛+푚 for every 푛 and 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It can be proved that this is the free pre-Lie 푘-algebra on one generator [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (The free generator of T is the rooted tree with one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=') (5) Upper triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This is an interesting example taken from [13], where all the details can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푘 be a commutative ring with identity in which 2 is invertible, and 푛 be a fixed positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푀 be the 푘-algebra of all 푛 × 푛 matrices, and 푈 be the its subalgebra of upper triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 휑: 푀 → 푈 be the the 푘-linear mapping that associates with any matrix 퐴 = (푎푖 푗) ∈ 푀 the matrix 퐵 = (푏푖 푗) ∈ 푈, where 푏푖 푗 = 푎푖 푗 if 푎푖 푗 is above the main diagonal, 푏푖 푗 = 0 if 푎푖 푗 is below the main diagonal, and 푏푖푖 = 푎푖푖/2 if 푎푖 푗 = 푎푖푖 is on the main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, for every 퐴 ∈ 푀, let 퐴tr be the transpose of the matrix 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Define an operation on 푈 setting, for every 푋,푌 ∈ 푈, 푋 · 푌 := 푋푌 + 휑(푋푌tr + 푌 푋tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then (푈, ·) is a pre-Lie 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 7 As we have defined in Section 1, a 푘-algebra homomorphism 휑: 푀 → 푀′ is a 푘-module morphism such that 휑(푥푦) = 휑(푥)휑(푦) for every 푥, 푦 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' But we also need another notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We say that a 푘-module morphism 휑: 푀 → 푀′, where 푀, 푀′ are arbitrary (not-necessarily associative) 푘-algebras, is a pre-morphism if 휑(푥푦) − 휑(푥)휑(푦) = 휑(푦푥) − 휑(푦)휑(푥) for every 푥, 푦 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A mapping 휑: 푀 → 푀′, where (푀, ·), (푀′, ·) are arbitrary 푘-algebras, is a pre-morphism (푀, ·) → (푀′, ·) if and only if it is a 푘-algebra morphism (푀, [−, −]) → (푀′, [−, −]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If (푀, ·), (푀′, ·) are 푘-algebras and 휑: 푀 → 푀′ is a mapping, then 휑: (푀, ·) → (푀′, ·) is a pre-morphism if and only if 휑(푎푏) − 휑(푎)휑(푏) = 휑(푏푎) − 휑(푏)휑(푎) for every 푎, 푏 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This equality can be re-written as 휑(푎푏)−휑(푏푎) = 휑(푎)휑(푏)−휑(푏)휑(푎), that is, 휑([푎, 푏]) = [휑(푎), 휑(푏)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From this lemma and the definition of pre-morphism, we immediately get that: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (a) Every 푘-algebra morphism is a pre-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) The composite mapping of two pre-morphisms is a pre-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (c) The inverse mapping of a bijective pre-morphism is a pre-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In Section 1, we already considered, for any (not-necessarily associative) 푘- algebra 푀, the mapping 휆: 푀 → End(푘푀), where 휆: 푥 ↦→ 휆푥, 휆푥 : 푀 → 푀, and 휆푥(푎) = 푥푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, we had already remarked that this mapping 휆 is a 푘-algebra morphism if and only if 푀 is associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The mapping 휆 is a pre-morphism if and only if 푀 is a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' There is a category of 푘-algebras with pre-morphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', a category in which objects are 푘-algebras and the Hom-set of all morphisms 푀 → 푀′ consists of all pre-morphisms 푀 → 푀′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This category contains as a full subcategory the category PreL푘,푝 of pre-Lie 푘-algebras (with pre-morphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The category PreL푘,푝 contains as a subcategory the category PreL푘 of pre-Lie algebras with 푘-algebra morphisms, hence a fortiori the category of associative algebras with their morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From lemma 1, we get Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Associating with any 푘-algebra (퐴, ·) its sub-adjacent anticommuta- tive algebra (퐴, [−, −]) is a functor 푈 from the category of 푘-algebras with pre- morphisms to the category of anticommutative 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that the functor 푈, viewed as a functor from the category PreL푘,푝 to the category of Lie 푘-algebras, is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Two pre-Lie algebras 퐴, 퐴′ are iso- morphic in PreL푘,푝 if and only if their sub-adjacent Lie algebras are isomorphic Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Two pre-Lie algebras isomorphic in PreL푘,푝 are not necessarily isomorphic as pre-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The simplest example is, over the field R of real numbers, the example of the two R-algebras R × R and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' They are non-isomorphic associative 8 Michela Cerqua and Alberto Facchini commutative 2-dimensional R-algebras, so that their sub-adjacent Lie algebras are both the 2-dimensional abelian Lie R-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence R × R and C are isomorphic objects in PreLR,푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' All R-linear mappings R × R → C are pre-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' More generally, a 푘-algebra 퐴 is said to be Lie-admissible if, setting [푥, 푦] = 푥푦−푦푥, one gets a Lie algebra (퐴, [−, −]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If the associator of a 푘-algebra 퐴 is defined as (푥, 푦, 푧) = (푥푦)푧 −푥(푦푧) for all 푥, 푦, 푧 in 퐴, then being a pre-Lie algebra is equivalent to (푥, 푦, 푧) = (푦, 푥, 푧) for all 푥, 푦, 푧 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Being a Lie-admissible algebra is equivalent to (푥, 푦, 푧) + (푦, 푧, 푥) + (푧, 푥, 푦) = (푦, 푥, 푧) + (푥, 푧, 푦) + (푧, 푦, 푥) (2) for every 푥, 푦, 푧 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras are Lie-admissible algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' By lemma 1, the functor 푈 : (퐴, ·) ↦→ (퐴, [−, −]) is a fully faithful functor from the category of Lie-admissible 푘-algebras with pre-morphisms to the category of Lie 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Corresponding to the notion of pre-morphism, there is a notion of pre-derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We say that a 푘-module endomorphism 훿: 푀 → 푀, where 푀 is an arbitrary (not- necessarily associative) 푘-algebra, is a pre-derivation if 훿(푥푦) − 훿(푥)푦 − 푥훿(푦) = 훿(푦푥) − 훿(푦)푥 − 푦훿(푥) for every 푥, 푦 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푘 be a commutativering with identity, (퐴, ·) a 푘-algebra, and [−, −] : 퐴 × 퐴 → 퐴 the operation on 퐴 defined by [푥, 푦] := 푥푦 − 푦푥 for every 푥, 푦 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then a 푘-module endomorphism 훿 of 퐴 is a pre-derivation of (퐴, ·) if and only if it is a derivation of the 푘-algebra (퐴, [−, −]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The 푘-module endomorphism 훿 of 퐴 is a pre-derivation of (퐴, ·) if and only if 훿(푥푦) − 훿(푥)푦 − 푥훿(푦) = 훿(푦푥) − 훿(푦)푥 − 푦훿(푥), that is, 훿([푥, 푦]) = [훿(푥), 푦] + [푥, 훿(푦)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (a) Every derivation of a 푘-algebra is a pre-derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) If 훿 and 훿′ are two pre-derivationsof a 푘-algebra 퐴, then [훿, 훿′] := 훿◦훿′−훿′◦훿 is a pre-derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (a) is trivial, and (b) follows from lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any 푘-algebra 퐴, the set PreDer푘 (퐴) of all pre-derivations of 퐴 is a Lie 푘-algebra with the operation [−, −] defined by [훿, 훿′] := 훿 ◦ 훿′ − 훿′ ◦ 훿 for every 훿, 훿′ ∈ PreDer푘 (퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The 푘-algebra (PreDer푘(퐴), [−, −]) is the Lie algebra of all derivations of the 푘-algebra (퐴, [−, −]) (lemma 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let (퐴, ·) be any 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For every 푥 ∈ 퐴 define a 푘-module morphism 푑푥 : 퐴 → 퐴 setting 푑푥(푦) := 푥푦 − 푦푥 for every 푦 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The following conditions are equivalent: Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 9 (a) 푑푥 is a pre-derivation for all 푥 ∈ 퐴, that is, the image 푑(퐴) of the mapping 푑 : 퐴 → End(푘 퐴) is contained in PreDer푘(퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) The mapping 푑 is a pre-morphism of the 푘-algebra (퐴, ·) into the associative 푘-algebra (End(푘 퐴), ◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (c) The 푘-algebra (퐴, ·) is Lie-admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (a) ⇔ (c) The mapping 푑푥 : (퐴, ·) → (퐴, ·) is a pre-derivation if and only if the mapping 푑푥 : (퐴, [−, −]) → (퐴, [−, −]) is a derivation by lemma 5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', if and only if 푑푥([푦, 푧]) = [푑푥(푦), 푧] + [푦, 푑푥(푧)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Since the mapping 푑푥 is defined by 푑푥(푦) = [푥, 푦], this is equivalent to [푥, [푦, 푧]] = [[푥, 푦], 푧] + [푦, [푥, 푧]], for every 푥, 푦, 푧 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This proves that 푑푥 is a pre-derivation for every 푥 ∈ 퐴 if and only if (퐴, [−, −]) is a Lie algebra, that is, if and only if (퐴, ·) is Lie-admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) ⇔ (c) The mapping 푑 is a pre-morphism if and only if 푑푥푦−푑푥◦푑푦 = 푑푦푥−푑푦◦ 푑푥 for every 푥, 푦 ∈ 퐴, that is, if and only if 푑푥푦(푧) −푑푥(푑푦(푧)) = 푑푦푥(푧) −푑푦(푑푥(푧)) for every 푥, 푦, 푧 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This is equivalent to (푥푦)푧 − 푧(푥푦) − 푑푥(푦푧 − 푧푦) = (푦푥)푧 − 푧(푦푥) − 푑푦(푥푧 − 푧푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An easy calculation shows that this is exactly Condition (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', it is equivalent to the fact that 퐴 is Lie-admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If 퐴 is a Lie-admissible 푘-algebra, the mapping 푑푥 is the inner pre-derivation of 퐴 induced by ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 3 Pre-Lie algebras are modules over the sub-adjacent Lie algebra Now we want to give another presentation of pre-Lie algebras, helpful to understand their structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푘 be a commutative ring with identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Given a pre-Lie 푘-algebra (퐴, ·), we have already seen in the paragraph after Lemma 2 that the mapping 휆: (퐴, ·) → End(푘 퐴) is a pre-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Apply to it the functor 푈, getting a Lie 푘-algebra morphism 퐿 := 푈(휆) : (퐴, [−, −]) → 픤픩(퐴) defined by 퐿 : 푎 ↦→ 휆푎 for every 푎 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This mapping 퐿 is set-theoretically equal to the mapping 휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In other words, 퐿 defines a module structure on the 푘-module 푘 퐴, giving it the structure of a module over the sub-adjacent Lie 푘-algebra (퐴, [−, −]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Moreover, [푥, 푦] = 퐿(푥)(푦) − 퐿(푦)(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This construction can be inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let (퐴, [−, −]) be a Lie 푘-algebra, and suppose that its sub-adjacent 푘-module 푘 퐴 has a module structure over the Lie algebra (퐴, [−, −]) via the Lie algebra morphism 퐿 : (퐴, [−, −]) → 픤픩(퐴) and that, for every 푥, 푦 ∈ 퐴, the condition 퐿(푥)(푦) − 퐿(푦)(푥) = [푥, 푦] holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Define a new multiplication · on 퐴 setting 푥 · 푦 = 퐿(푥)(푦) for every 푥, 푦 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then (퐴, ·) turns out to be a pre-Lie 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' These two constructions are one the inverse of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' More precisely, fix a Lie 푘-algebra 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then there is a category isomorphism between the following two categories S퐴 and M퐴, where: (1) S퐴 is the category whose objects are all pre-Lie 푘-algebras (퐴, ·) whose sub-adjacent Lie algebra is the fixed Lie algebra (퐴, [−, −]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The morphisms are all pre-Lie algebra homomorphisms between such pre-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 10 Michela Cerqua and Alberto Facchini (2) M퐴 is the category whose objects are all pre-Lie 푘-algebra morphisms 퐿 : (퐴, [−, −]) → 픤픩(퐴) such that 퐿(푥)(푦) − 퐿(푦)(푥) = [푥, 푦] for every 푥, 푦 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The morphisms 휑 : 퐿 → 퐿′ between two objects 퐿, 퐿′ of M퐴 are the 푘-module morphisms 휑: 퐴 → 퐴 for which all diagrams 푀 푀 푀 푀 휑 퐿(푎) 퐿′(휑(푎)) 휑 commute, for every 푎 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' See [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1 Modules over a pre-Lie 풌-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Modules cannot be defined over arbitrary non-associative algebras, but the definition of pre-Lie algebra immediately suggests us how it is possible to define modules over a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A module 푀 over a pre-Lie 푘-algebra 퐴 is any 푘-module 푀 with a 푘-bilinear mapping ·: 퐴 × 푀 → 푀 such that (푥 · 푦) · 푚 − 푥 · (푦 · 푚) = (푦 · 푥) · 푚 − 푦 · (푥 · 푚) (3) for every 푥, 푦 ∈ 퐴 and 푚 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Like in the case of associative algebras, it is possible to equivalently define a module 푀 over a pre-Lie 푘-algebra (퐴, ·) as any 푘-module 푀 with a pre-morphism 휆: (퐴, ·) → (End(푘푀), ◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For instance, if 퐴 is any pre-Lie 푘-algebra and 퐼 is an ideal of 퐼, taking as 푘- bilinear mapping ·: 퐴 × 퐼 → 퐼 the restriction of the multiplication on 퐴, one sees immediately that 퐼 is a module over 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The category of modules over a pre-Lie 푘-algebra (퐴, ·) and the cate- gory of modules over its sub-adjacent Lie 푘-algebra (퐴, [−, −]) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Modules over the pre-Lie algebra (퐴, ·) are pairs (푘푀, 휆) with 푘 푀 a 푘- module and 휆: 퐴 → End(푘푀) a pre-morphism, and modules over the Lie algebra (퐴, [−, −]) are pairs (푘 푀, 휆) with 푘푀 a 푘-module and 휆: (퐴, [−, −]) → 픤픩(푀) a Lie 푘-algebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' By Lemma 1, they are the same pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that we could have obtained the results in Section 3 in a different way: every pre-Lie algebra is clearly a module over itself, hence, applying Theorem 9, to every pre-Lie algebra (퐴, ·) there corresponds a module 퐴푘 over the sub-adjacent Lie algebra (퐴, [−, −]), that is, a Lie algebra morphism 퐿 : (퐴, [−, −]) → 픤픩(퐴), and [푥, 푦] = 퐿(푥)(푦) − 퐿(푦)(푥) for every 푥, 푦 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 11 Also notice that the modules we have defined in this section over a pre-Lie algebra are left modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We don’t consider right modules because the definition of pre-Lie algebra is not right/left symmetric, that is, the opposite of a pre-Lie algebra is not a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 4 Commutator of two ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (Huq=Smith) for pre-Lie algebras The sum of two ideals of a pre-Lie 푘-algebra 퐴, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', their sum as 푘-submodules of 퐴, is an ideal of 퐴, and any intersection of a family of ideals of 퐴 is an ideal of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It follows that the set I(퐴) of all ideals of a pre-Lie algebra 퐴 is a complete lattice with respect to ⊆, and it is a sublattice of the lattice of all 푘-submodules of 퐴푘, hence I(퐴) is a modular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Moreover, the ideal of 퐴 generated by a subset 푋 of 퐴 is the intersection of all the ideals of 퐴 that contain 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We now need a notion of commutator of two ideals of a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The variety V of pre-Lie 푘-algebras is a Barr-exact category, is a variety of Ω-groups, is protomodular and is semi-abelian [12, Example (2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' More precisely, pre-Lie algebras have an underlying group structure with respect to their addition, so that they have the Mal’tsev term 푝(푥, 푦, 푧) = 푥 − 푦 + 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' See [5, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that 푝(푝(푥, 푦, 0), 푥, 푦)) = 0 for every 푥, 푦 ∈ 퐴, hence the variety V of pre-Lie algebras is protomodular by [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Moreover, 푝 has the property that 푝(푝(푥, 푦, 푡), 푡, 푧) = 푝(푥, 푦, 푧) for all 푥, 푦, 푧, 푡 ∈ 퐴 (semi-associativity), so V is semi-abelian by [5, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We want to show that the Huq and the Smith commutators of two ideals of a pre-Lie 푘-algebra coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Recall that in the case of the semi-abelian variety V of pre-Lie algebras, the Huq commutator of two ideals 퐼 and 퐽 of a pre-Lie algebra 퐴 is the smallest ideal [퐼, 퐽]퐻 of 퐴 for which there is a well-defined canonical morphism 퐼 × 퐽 → 퐴/[퐼, 퐽]퐻 such that (푖, 0) ↦→ 푖 + [퐼, 퐽]퐻 and (0, 푗) ↦→ 푗 + [퐼, 퐽]퐻 for every 푖 ∈ 퐼 and 푗 ∈ 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' That is, [퐼, 퐽]퐻 is the smallest ideal of 퐴 for which the mapping 퐼 × 퐽 → 퐴/[퐼, 퐽]퐻, defined by (푖, 푗) ↦→ 푖 + 푗 + [퐼, 퐽]퐻 for every 푖 ∈ 퐼 and 푗 ∈ 퐽, is a pre-Lie algebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The Huq commutator [퐼, 퐽]퐻 of two ideals 퐼 and 퐽 of a pre-Lie algebra 퐴 is the ideal of 퐴 generated by the subset { 푖푗, 푗푖 | 푖 ∈ 퐼, 푗 ∈ 퐽 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The mapping ¯휎 : 퐼 × 퐽 → 퐴/[퐼, 퐽]퐻, defined by (푖, 푗) ↦→ 푖 + 푗 + [퐼, 퐽]퐻, is a pre-Lie 푘-algebra morphism if and only if it respects multiplication, that is, if and only if ¯휎((푖, 푗) · (푖′, 푗′)) ≡ ¯휎(푖, 푗) ¯휎(푖′, 푗′) for every (푖, 푗), (푖′, 푗′) ∈ 퐼 × 퐽, that is, if and only if 푖푖′ + 푗 푗′ ≡ (푖 + 푗)(푖′ + 푗′) modulo [퐼, 퐽]퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence ¯휎 is a pre-Lie algebra morphism if and only if 푖푗′ + 푗푖′ ≡ 0 modulo [퐼, 퐽]퐻, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', if and only if 푖푗′ + 푗푖′ ∈ [퐼, 퐽]퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The conclusion follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The Smith commutator in the Mal’tsev variety V (see [11]) can be defined, for a pre-Lie 푘-algebra 퐴 with Mal’tsev term 푝(푥, 푦, 푧) and two ideals 퐼, 퐽 of 퐴, as the smallest ideal [퐼, 퐽]푆 of 퐴 for which the function 12 Michela Cerqua and Alberto Facchini 푝 : {(푥, 푦, 푧) | 푥 ≡ 푦 (mod 퐼), 푦 ≡ 푧 (mod 퐽)} → 퐴/[퐼, 퐽]푆, defined by 푝(푥, 푦, 푧) = 푥 − 푦 + 푧 + [퐼, 퐽]푆 is a pre-Lie algebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The Smith commutator [퐼, 퐽]푆 of two ideals 퐼 and 퐽 of a pre-Lie algebra 퐴 is the ideal of 퐴 generated by the subset { 푖푗, 푗푖 | 푖 ∈ 퐼, 푗 ∈ 퐽 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence Huq=Smith for pre-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The mapping 푝 : { (푏 + 푖, 푏, 푏 + 푗) | 푏 ∈ 퐴, 푖 ∈ 퐼, 푗 ∈ 퐽 } → 퐴/[퐼, 퐽]푆 is a pre-Lie algebra morphism if and only if for every 푏, 푏′ ∈ 퐴, 푖, 푖′ ∈ 퐼, 푗, 푗′ ∈ 퐽, one has 푝((푏+푖, 푏, 푏+푗)(푏′+푖′, 푏′, 푏′+푗′)) ≡ 푝(푏+푖, 푏, 푏+푗)푝(푏′+푖′, 푏′, 푏′+푗′) (mod [퐼, 퐽]푆), that is, 푝((푏 +푖)(푏′ +푖′), 푏푏′, (푏 + 푗)(푏′ + 푗′)) ≡ (푏 +푖 + 푗)(푏′ +푖′ + 푗′) mod[퐼, 퐽]푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Equivalently,if and only if 0 ≡ 푖푗′+ 푗푖′ mod[퐼, 퐽]푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Thereforethe Smith commutator [퐼, 퐽]푆 of the two ideals 퐼 and 퐽 is the ideal of 퐴 generated by the subset { 푖푗, 푗푖 | 푖 ∈ 퐼, 푗 ∈ 퐽 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In particolar, [퐼, 퐽]퐻 = [퐼, 퐽]푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From now on we will not distinguish between the Huq commutator [퐼, 퐽]퐻 and the Smith commutator [퐼, 퐽]푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We will simply call it the commutatorof the two ideals 퐼 and 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that the commutator is commutative, in the sense that [퐼, 퐽] = [퐽, 퐼].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let us briefly discuss the structure of this ideal [퐼, 퐽].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is clear that if 푋 is any subset of a pre-Lie 푘-algebra 퐴, the ideal ⟨푋⟩ of 퐴 generated by 푋, that is, the intersection of all the ideals of 퐴 that contain 푋, can be also described as the union ⟨푋⟩ = � 푛≥0 푋푛 of the following ascending chain 푋0 ⊆ 푋1 ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' of 푘-submodules of 퐴: 푋0 is the 푘-submodule of 퐴 generated by 푋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' given 푋푛, set 푋푛+1 = 푋푛+퐴푋푛+푋푛퐴, where 퐴푋푛 denotes the set of all finite sums of products 푎푥 with 푎 ∈ 퐴 and 푥 ∈ 푋푛, and similarly for 푋푛퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In the case of the ideal [퐼, 퐽] this specializes as follows: Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐼 and 퐽 be ideals of a pre-Lie 푘-algebra 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then [퐼, 퐽] = 퐼퐽 + � 푛≥0 푆푛, where 푆푛 = ((.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (((퐽퐼)퐴)퐴) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' )퐴)퐴 and in 푆푛 there are 푛 factors equal to 퐴 on the right of the factor J퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Step 1: 퐴(퐼퐽) ⊆ 퐼퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' By Property (1), we have that 퐴(퐼퐽) ⊆ (퐴퐼)퐽 + (퐼퐴)퐽 + 퐼(퐴퐽) ⊆ 퐼퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Step 2: 퐴(퐽퐼) ⊆ 퐽퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From Step 1, by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Step 3: 퐴푆푛 ⊆ 푆푛 + 푆푛+1 for every 푛 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Induction on 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Step 2 gives the case 푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Suppose that 퐴푆푛 ⊆ 푆푛 + 푆푛+1 for some 푛 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then 퐴푆푛+1 = 퐴(푆푛퐴) ⊆ (퐴푆푛)퐴 + (푆푛퐴)퐴 + 푆푛(퐴퐴) ⊆ (푆푛 + 푆푛+1)퐴 + 푆푛+2 + 푆푛+1 = 푆푛+1 + 푆푛+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 13 Step 4: 푆푛퐴 = 푆푛+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' By definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Step 5: (퐼퐽)퐴 ⊆ 퐼퐽 + 푆0 + 푆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In fact, (퐼퐽)퐴 ⊆ 퐼(퐽퐴) + (퐽퐼)퐴 + 퐽(퐼퐴) ⊆ 퐼퐽 + 푆1 + 푆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Final Step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Clearly, 퐼퐽 + � 푛≥0 푆푛 is a 푘-submodule of 퐴 that contains 퐼퐽 and 퐽퐼 and is contained in the ideal generated by 퐼퐽 ∪퐽퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence it remains to show that it is closed by left and right multiplication by elements of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This is proved in Steps 1, 3, 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Now that we have a good notion of commutator of two ideals 퐼 and 퐽 of a pre-Lie 푘-algebra 퐴, we can introduce the multiplicative lattice of all ideals of 퐴: it is the complete modular lattice I(퐴) of all ideals of 퐴 endowed with the commutator of ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that, trivially, [퐼, 퐽] ⊆ 퐼 ∩ 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' As a consequence of looking at pre-Lie algebras from the point of view of multiplicative lattices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' we immediately get the notions of prime ideal of a pre-Lie 푘-algebra 퐴,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (Zariski) prime spectrum of 퐴,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' semiprime ideal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' abelian pre-Lie algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' idempotent (=perfect) pre- Lie algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' derived series,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' solvable pre-Lie algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' lower central series,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' nilpotent pre-Lie algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푚-system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푛-system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' hyperabelian pre-Lie algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' metabelian pre- Lie algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Jacobson radical,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' centralizer of an ideal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' center of a pre-Lie 푘-algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' hypercenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' See the next Section 5 and [8, 9, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that the monotonicity condition holds for our commutator of ideals of a pre-Lie algebra 퐴, in the sense that if 퐼 ≤ 퐼′ and 퐽 ≤ 퐽′ are ideals of 퐴, then [퐼, 퐽] ≤ [퐼′, 퐽′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also notice that the description of the commutator in Proposition 12 reduces, in the case of 퐼 = 퐽 = 퐴, to the equality [퐴, 퐴] = 퐴2 = 퐴퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Here 퐴2 is the image of the 푘-module morphism 휇: 퐴 ⊗푘 퐴 → 퐴 induced by the 푘-bilinear multiplication of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 5 The commutator is not associative In this section we will show that the commutator of ideals in a pre-Lie algebra 퐴 is not associative in general, that is, if 퐼, 퐽, 퐾 are ideals of 퐴, it is not necessarily true that [퐼, [퐽, 퐾]] = [[퐼, 퐽], 퐾].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In our example, the algebra 퐴 will be factor algebra 퐴 := T/푃, where T is the pre-Lie algebra of rooted trees of Example 4 in Section 2, and 푃 is the ideal of T generated by all rooted trees with at least 5 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Such 푃 is the 푘-submodule of T generated by all rooted trees with at least 5 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The rooted trees with at most 4 vertices up to isomorphism are 14 Michela Cerqua and Alberto Facchini 푣 = 1 , 푒 = 1 2 , 푎 = 1 2 3 , 푏 = 1 2 3 , 푐 = 1 2 3 4 , 푑 = 1 2 3 4 , 푓 = 1 2 3 4 , 푔 = 1 2 3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence our pre-Lie 푘-algebra 퐴 is eight dimensional, and we will denote by 푣, 푒, 푎, 푏, 푐, 푑, 푓 , 푔 the images in 퐴 of the corresponding rooted trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' That is, we will say that {푣, 푒, 푎, 푏, 푐, 푑, 푓 , 푔} is a free set of generators for the free 푘-module 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From the multiplication in T defined in Example 4 of Section 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' we get that the multiplication table in 퐴 is 푣 푒 푎 푏 푐 푑 푓 푔 푣 푒 푎 + 푏 푐 + 2 푓 푓 + 푑 + 푔 0 0 0 0 푒 푏 푓 + 푔 0 0 0 0 0 0 푎 푑 0 0 0 0 0 0 0 푏 푔 0 0 0 0 0 0 0 푐 0 0 0 0 0 0 0 0 푑 0 0 0 0 0 0 0 0 푓 0 0 0 0 0 0 0 0 푔 0 0 0 0 0 0 0 0 Pre-Lie algebras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' their multiplicative lattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' and idempotent endomorphisms 15 From the multiplication table we see that 퐴2 = 퐴퐴 has {푒,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푐} as a set of generators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' and is a seven dimensional free 푘-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Now [퐴2, 퐴2] = � 푛≥0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' ((퐴2 · 퐴2) · 퐴) · · · · · 퐴) · 퐴, where there are 푛 factors equal to 퐴 on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' But, always from the multiplication table, one sees that 퐴2·퐴2 is generated by 푓 + 푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Moreover ( 푓 + 푔)퐴 = 0 and 퐴( 푓 + 푔) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Therefore [퐴2, 퐴2] is one dimension as a free 푘-module, and its free set of generators is { 푓 + 푔}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Similarly, [퐴2, 퐴] = 퐴 · 퐴2 + � 푛≥1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' ((퐴2 · 퐴) · 퐴) · · · · · 퐴) · 퐴, where there are 푛 + 1 factors equal to 퐴 on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From the multiplication table, we see that 퐴 · 퐴2 is generated by 푎 + 푏, 푓 + 푔, 푐 + 2 푓 , 푓 + 푑 + 푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, 퐴2 · 퐴 is generated by {푏, 푑, 푔, 푓 + 푔}, (퐴2 · 퐴) · 퐴 is generated by 푔, and ((퐴2 · 퐴) · 퐴) · 퐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Therefore [퐴2, 퐴] is the 푘-module generated by 푏, 푑, 푔, 푓 , 푎, 푐 and is six dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It follows that [퐴2, 퐴]· 퐴 is generated by {푑, 푔}, 퐴·([퐴2, 퐴]) is generated by {푐+2 푓 , 푓 +푑+푔}, and ([퐴2, 퐴] · 퐴) · 퐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From these equalities we get that [[퐴2, 퐴], 퐴] is generated by {푑, 푔, 푐 + 2 푓 , 푓 + 푑 + 푔}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Equivalently, [[퐴2, 퐴], 퐴] is generated by {푑, 푔, 푓 , 푐} and is four dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In particular [퐴2, 퐴2] ≠ [[퐴2, 퐴], 퐴].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let’s illustrate in detail some of the notions that immediately derive from the commutative multiplication [−, −] (the commutator) in the multiplicative lattice I(퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' First of all, a pre-Lie 푘-algebra 퐴 is abelian if the commutator of 퐴 and itself is zero: [퐴, 퐴] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This is equivalent to saying that 푖푗 = 0 for every 푖, 푗 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' That is, a pre-Lie algebra (퐴, ·) is abelian if and only if 푥 · 푦 = 0 for every 푥, 푦 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (This is equivalent to requiring that the addition +: 퐴 × 퐴 → 퐴 is a pre-Lie algebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=') By definition,an ideal 퐼 of a pre-Lie 푘-algebra 퐴 is prime if it is properly contained in 퐴 and, for every ideal 퐽, 퐾 of 퐴, [퐽, 퐾] ⊆ 퐼 implies 퐽 ⊆ 퐼 or 퐾 ⊆ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An ideal 퐼 of a pre-Lie 푘-algebra 퐴 is semiprime if, for every ideal 퐽 of 퐴, [퐽, 퐽] ⊆ 퐼 implies that 퐽 ⊆ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An ideal of 퐴 is semiprime if and only if it is the intersection of a family of prime ideals (if and only if it is the intersection of all the ideals of 퐴 that contain it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An ideal 푃 of a pre-Lie 푘-algebra 퐴 is prime if and only if the lattice I(퐴/푃) is uniform and 퐴/푃 has no non-zero abelian ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Instead of the commutator [퐼, 퐽] of two ideals 퐼 and 퐽, we could have taken two other “product of ideals” in a pre-Lie 푘-algebra: we could consider the product 퐼퐽, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', the 푘-submodule of 퐴 generated by all products 푖푗, which is a 푘-submodule but not an ideal of 퐴 in general, or the ideal ⟨퐼퐽⟩ generated by the submodule 퐼퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that 퐼퐽 ⊆ ⟨퐼퐽⟩ ⊆ [퐼, 퐽] = ⟨퐼퐽⟩ + ⟨퐽퐼⟩, where the last equality follows from Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Correspondingly, we would have had three different notions of “prime ideal”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In the next proposition (essentially contained in [8, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='7]) we prove that these three notions of “prime ideal” coincide: Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The following conditions are equivalent for an ideal 푃 of a pre-Lie algebra 퐴: (a) If 퐼, 퐽 are ideals of 퐴 and 퐼퐽 ⊆ 푃, then either 퐼 ⊆ 푃 or 퐽 ⊆ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) If 퐼, 퐽 are ideals of 퐴 and ⟨퐼퐽⟩ ⊆ 푃, then either 퐼 ⊆ 푃 or 퐽 ⊆ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (c) If 퐼, 퐽 are ideals of 퐴 and [퐼, 퐽] ⊆ 푃, then either 퐼 ⊆ 푃 or 퐽 ⊆ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 16 Michela Cerqua and Alberto Facchini Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The implications (a) ⇒ (b) ⇒ (c) follow immediately from the fact that 퐼퐽 ⊆ ⟨퐼퐽⟩ ⊆ [퐼, 퐽].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (c) ⇒ (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푃 satisfy condition (c) and fix two ideals 퐼, 퐽 of 퐴 such that 퐼퐽 ⊆ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Since 푃 is an ideal, it follows that ⟨퐼퐽⟩ ⊆ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, [⟨퐽퐼⟩, ⟨퐽퐼⟩] = ⟨⟨퐽퐼⟩⟨퐽퐼⟩⟩ ≤ ⟨퐼퐽⟩ ≤ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From (c), we get that ⟨퐽퐼⟩ ≤ 푃, so that [퐼, 퐽] = ⟨퐼퐽⟩ + ⟨퐽퐼⟩ ≤ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From (c) again, we get that either 퐼 ⊆ 푃 or 퐽 ⊆ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proposition 14 shows that if the pre-Lie algebra 퐴 is an associative algebra, then this notion of prime ideal coincide with the notion of prime ideal in an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proposition 12 shows that, for every pair (퐼, 퐽) of ideals of a pre-Lie algebra 퐴, one has [퐼, 퐽] = 퐼퐽 +⟨퐽퐼⟩ = 퐽퐼 +⟨퐼퐽⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, Step 5 in the proof of that Proposition shows that one always has that 퐼퐽 + 퐽퐼 + (퐼퐽)퐴 = 퐼퐽 + 퐽�� + (퐽퐼)퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A pre-Lie 푘-algebra 퐴 is idempotent (or perfect) if [퐴, 퐴] = 퐴, that is, if 퐴2 = 퐴 (last paragraph of Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Given any pre-Lie algebra 퐴, let Spec(퐴) be the set of all its prime ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For every 퐼 ∈ I(퐴), set 푉(퐼) = { 푃 ∈ Spec(퐴) | 푃 ⊇ 퐼 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then the family of all subsets 푉(퐼) of Spec(퐴), 퐼 ∈ I(퐴), is the family of all the closed sets for a topology on Spec(퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' With this topology, the topological space Spec(퐴) is the (Zariski) prime spectrum of 퐴, and is a sober space [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is not a spectral space in the sense of Hochster in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For instance, if 퐵 is a Boolean ring without identity, then 퐵 is a pre-Lie algebra, but its prime spectrum is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If the pre-Lie algebra 퐴 is an associative algebra, then this notion of prime spectrum coincide with the “standard notion” of prime spectrum of an associative algebra 퐴, where the points of the spectrum are the prime ideals of 퐴 and the closed sets are the subsets 푉(퐼) of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' To tell the truth, there is not a “standard notion” of prime spectrum of an associative algebra that extends the classical notion of prime spectrum for commutative associative algebras with identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' There are several such notions as it is shown in [1] and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For instance, the points of the spectrum could be the completely prime ideals of 퐴, or the spectrum of 퐴 could be defined to be the Zariski spectrum of the commutative ring 퐴/[퐴, 퐴], where [퐴, 퐴] now denotes the ideal of 퐴 generated by all elements 푎푏 − 푏푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A pre-Lie 푘-algebra 퐴 is hyperabelian if it has no prime ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For instance, abelian pre-Lie algebras are hyperabelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐴 be a pre-Lie 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The lower central series (or descending central series) of 퐴 is the descending series 퐴 = 퐴1 ≥ 퐴2 ≥ 퐴3 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , where 퐴푛+1 := [퐴푛, 퐴] for every 푛 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If 퐴푛 = 0 for some 푛 ≥ 1, then 퐴 is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (Notice that it is not necessary to distinguish between left nilpotency and rightnilpotency,becausethecommutatoriscommutative,thatis,[퐴푛, 퐴] = [퐴, 퐴푛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=') The derived series of 퐴 [8, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1] is the descending series 퐴 := 퐴(0) ≥ 퐴(1) ≥ 퐴(2) ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 17 where 퐴(푛+1) := [퐴(푛), 퐴(푛)] for every 푛 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The pre-Lie algebra 퐴 is solvable if 퐴(푛) = 0 for some integer 푛 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is metabelian if 퐴(2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In a multiplicative lattice an element is semisimple if it is the join of a set of minimal idempotent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (An element 푚 of a lattice 퐿 is minimal if, for every 푥 ∈ 퐿, 푥 ≤ 푚 implies 푥 = 푚 or 푥 = 0, that is, if it is minimal in the partially ordered set 퐿 \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An element 푒 of a multiplicative lattice 퐿 is idempotent if 푒 · 푒 = 푒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' “Minimal idempotent element” of 퐿 means minimal element of 퐿 \\ {0} that is also an idempotent element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that for a minimal element 푥 ∈ 퐿 either 푥 · 푥 = 푥 or 푥 · 푥 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', minimal elements are either idempotent or abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The Jacobson radical of 퐿 is the meet of the set of all maximal elements 푎 of 퐿 \\ {1} with 1 · 1 ̸≤ 푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The radical is the join of the set of all solvable elements of 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 6 Idempotent endomorphisms, semidirect products of pre-Lie algebras, and actions Let 푒 be an idempotent endomorphism of a pre-Lie 푘-algebra 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then 퐴 = ker(푒) ⊕ 푒(퐴) (direct sum as 푘-modules), where the kernel ker(푒) of 푒 is an ideal of 퐴 and its image 푒(퐴) is a pre-Lie sub-푘-algebra of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If there is a direct-sum decomposition 퐴 = 퐼 ⊕ 퐵 as 푘-module of a pre-Lie 푘-algebra 퐴, where 퐼 is an ideal of 퐴 and 퐵 is a pre-Lie sub-푘-algebra of 퐴, we will say that 퐴is the semidirect product of 퐼 and 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We are interested in semidirect products because, for any algebraic structure, idempotent endomorphisms are in one-to-one correspondence with semidirect products and are related to the notion of action of the structure on another structure, and bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The proof of the following proposition is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐴 be a pre-Lie 푘-algebra, 퐼 an ideal of 퐴 and 퐵 a pre-Lie sub-푘-algebra of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The following conditions are equivalent: (1) 퐴 = 퐼 ⊕ 퐵 as a 푘-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (2) For every 푎 ∈ 퐴, there are a unique 푖 ∈ 퐼 and a unique 푏 ∈ 퐵 such that 푎 = 푖 + 푏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (3) There exists a pre-Lie 푘-algebra morphism 퐴 → 퐵 whose restriction to 퐵 is the identity and whose kernel is 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (4) There is an idempotent pre-Lie 푘-algebra endomorphism of 퐴 whose image is 퐵 and whose kernel is 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is now clear that there is a one-to-one correspondence between the set of all idempotent endomorphisms of a pre-Lie 푘-algebra 퐴 and the set of all pairs (퐼, 퐵), where 퐼 is an ideal of 퐴, 퐵 is a pre-Lie sub-푘-algebra of 퐴, and 퐴 is the direct sum of 퐼 and 퐵 as a 푘-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let us first consider inner semidirect product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Suppose that (퐴, ·) is a pre-Lie 푘-algebra that is a semidirect product of its ideal 퐼 and its pre-Lie sub-푘-algebra 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then there is a pre-morphism 휆: (퐵, ·) → (End(퐼푘), ◦), given by multiplying on the 18 Michela Cerqua and Alberto Facchini left by elements of 퐵 (this follows from the fact that every ideal is a module, as we have already remarked in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, there is a 푘-module morphism 휌 : 퐵 → End(퐼푘), given by multiplying on the right by elements of 퐵, that is, 휌 : 푏 ↦→ 휌푏, where 휌푏(푖) = 푖 · 푏 for every 푖 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Moreover, Identity (1), applied to elements 푥, 푧 in 퐵 and 푦 ∈ 퐼, can be re-written as 휌푎(휆푏(푖)) − 휆푏(휌푎(푖)) = (휌푎 ◦ 휌푏 − 휌푏·푎)(푖) for every 푎, 푏 ∈ 퐵 and 푖 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Identity (1), applied to elements 푥 in 퐵 and 푦, 푧 ∈ 퐼, can be re-written as 휆푎(푖) · 푗 − 휆푎(푖 · 푗) = 휌푎(푖) · 푗 − 푖 · 휆푎( 푗) for every 푎 ∈ 퐵 and 푖, 푗 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Finally, the same identity (1), applied to elements 푧 in 퐵 and 푥, 푦 ∈ 퐼, can be re-written as 휌푎(푥 · 푦) − 푥 · 휌푎(푦) = 휌푎(푦 · 푥) − 푦 · 휌푎(푥) for every 푎 ∈ 퐵 and 푖, 푗 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Conversely, for outer semidirect product: Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐼 and 퐵 be pre-Lie 푘-algebras and (휆, 휌) a pair of 푘-linear map- pings 퐵 → End(퐼푘) such that: (a) 휆: (퐵, ·) → (End(퐼푘), ◦) is a pre-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) 휌푎 ◦ 휆푏 − 휆푏 ◦ 휌푎 = 휌푎 ◦ 휌푏 − 휌푏·푎 for every 푎, 푏 ∈ 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (c) 휆푎(푖) · 푗 − 휆푎(푖 · 푗) = 휌푎(푖) · 푗 − 푖 · 휆푎( 푗) for every 푎 ∈ 퐵 and 푖, 푗 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (d) 휌푎(푖 · 푗) − 푖 · 휌푎( 푗) = 휌푎( 푗 · 푖) − 푗 · 휌푎(푖) for every 푎 ∈ 퐵 and 푖, 푗 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' On the 푘-module direct sum 퐼 ⊕ 퐵 define a multiplication ∗ setting (푖, 푏) ∗ ( 푗, 푐) = (푖 · 푗 + 휆푏( 푗) + 휌푐(푖), 푏 · 푐) for every (푖, 푏), ( 푗, 푐) ∈ 퐼 ⊕ 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then (퐼 ⊕ 퐵, ∗) is a pre-Lie 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For every 푎, 푏, 푐 ∈ 퐵 and 푥, 푦, 푧 ∈ 퐼 we have that ((푥, 푎) ∗ (푦, 푏)) ∗ (푧, 푐) = (푥 · 푦 + 휆푎(푦) + 휌푏(푥), 푎 · 푏) ∗ (푧, 푐) = = ((푥 · 푦) · 푧 + 휆푎(푦) · 푧 + 휌푏(푥) · 푧 + 휆푎·푏(푧)+ +휌푐(푥 · 푦 + 휆푎(푦) + 휌푏(푥)), (푎 · 푏) · 푐) (4) and (푥, 푎) ∗ ((푦, 푏) ∗ (푧, 푐)) = (푥, 푎) ∗ (푦 · 푧 + 휆푏(푧) + 휌푐(푦), 푏 · 푐) = = (푥 · (푦 · 푧) + 푥 · 휆푏(푧) + 푥 · 휌푐(푦)+ +휆푎(푦 · 푧 + 휆푏(푧) + 휌푐(푦)) + 휌푏·푐(푥), 푎 · (푏 · 푐)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (5) The difference of (4) and (5) is ((푥 · 푦) · 푧 − 푥 · (푦 · 푧) + 휆푎(푦) · 푧 − 휆푎(푦 · 푧)+ +휌푏(푥) · 푧 − 푥 · 휆푏(푧) + 휆푎·푏(푧) − (휆푎 ◦ 휆푏)(푧)+ +휌푐(푥 · 푦) − 푥 · 휌푐(푦) + 휌푐(휆푎(푦)) − 휆푎(휌푐(푦)) + 휌푐(휌푏(푥)) − 휌푏·푐(푥)), (푎 · 푏) · 푐 − 푎 · (푏 · 푐)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Similarly, ((푦, 푏) ∗ (푥, 푎)) ∗ (푧, 푐) − (푦, 푏) ∗ ((푥, 푎) ∗ (푧, 푐)) = = ((푦 · 푥) · 푧 − 푦 · (푥 · 푧) + 휆푏(푥) · 푧 − 휆푏(푥 · 푧) + 휌푎(푦) · 푧 − 푦 · 휆푎(푧)+ +휆푏·푎(푧) − (휆푏 ◦ 휆푎)(푧) + 휌푐(푦 · 푥) − 푦 · 휌푐(푥) + 휌푐(휆푏(푥)) − 휆푏(휌푐(푥))+ +휌푐(휌푎(푦)) − 휌푎·푐(푦)), (푏 · 푎) · 푐 − 푏 · (푎 · 푐)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 19 Hence, for the proof, it suffices to show that 휆푎(푦) · 푧 − 휆푎(푦 · 푧) + 휌푏(푥) · 푧 − 푥 · 휆푏(푧) + 휆푎·푏(푧) − (휆푎 ◦ 휆푏)(푧)+ +휌푐(푥 · 푦) − 푥 · 휌푐(푦) + 휌푐(휆푎(푦)) − 휆푎(휌푐(푦)) + 휌푐(휌푏(푥)) − 휌푏·푐(푥)) = = 휆푏(푥) · 푧 − 휆푏(푥 · 푧) + 휌푎(푦) · 푧 − 푦 · 휆푎(푧)+ +휆푏·푎(푧) − (휆푏 ◦ 휆푎)(푧)+ +휌푐(푦 · 푥) − 푦 · 휌푐(푥) + 휌푐(휆푏(푥)) − 휆푏(휌푐(푥))+ +휌푐(휌푎(푦)) − 휌푎·푐(푦)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (6) Now 휆푎(푦) · 푧 − 휆푎(푦 · 푧) = 휌푎(푦) · 푧 − 푦 · 휆푎(푧) by hypotheses (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 휌푏(푥) · 푧 − 푥 · 휆푏(푧) = 휆푏(푥) · 푧 − 휆푏(푥 · 푧) by hypotheses (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 휆푎·푏(푧) − (휆푎 ◦ 휆푏)(푧) = 휆푏·푎(푧) − (휆푏 ◦ 휆푎)(푧) by hypotheses (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 휌푐(푥 · 푦) − 푥 · 휌푐(푦) = 휌푐(푦 · 푥) − 푦 · 휌푐(푥) by hypotheses (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 휌푐(휆푎(푦)) − 휆푎(휌푐(푦)) = 휌푐(휌푎(푦)) − 휌푎·푐(푦)) by hypotheses (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 휌푐(휌푏(푥)) − 휌푏·푐(푥)) = 휌푐(휆푏(푥)) − 휆푏(휌푐(푥)) by hypotheses (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Summing up these equalities one gets Equality (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence the theorem characterises the four properties that an action (휆, 휌), that is, a pair of 푘-linear mappings 퐵 → End(퐼푘), must have in order to construct the semidirect product of a pre-Lie 푘-algebra 퐵 acting on a pre-Lie 푘-algebra 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1 Bimodules over a pre-Lie algebra The most important case of semidirect product is probably when the pre-Lie algebra 퐼 is abelian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', the case where the action, that is, the pair (휆, 휌) of 푘-linear mappings 퐵 → End(퐼푘), is an action of the pre-Lie 푘-algebra 퐵 on a 푘-module 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In other words, when 퐼 is a 퐵-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let us be more precise, giving the precise definition of what a bimodule over a pre-Lie algebra must be: Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐴 be a pre-Lie 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A bimodule over 퐴 is a 푘-module 푀푘 with a pair (휆, 휌) of 푘-linear mappings 퐴 → End(푀푘) such that: (a) 휆: (퐴, ·) → (End(푀푘), ◦) is a pre-morphism (that is, 푀 is a module over 퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) 휌푎 ◦ 휆푏 − 휆푏 ◦ 휌푎 = 휌푎 ◦ 휌푏 − 휌푏·푎 for every 푎, 푏 ∈ 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that Conditions (c) and (d) of Theorem 16 are always trivially satisfied because in this case the 푘-module 푀 is viewed as an abelian pre-Lie algebra, that is, with null multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This definition already appears, for instance, in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice the nice interpretation of condition (b) given in that paper: In condition (b) the left hand side 휌푎 ◦ 휆푏 − 휆푏 ◦ 휌푎 describes how far the action is from associativity (for bimodules over an associative algebra, it is always required to be zero);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' the right hand side 휌푎◦휌푏−휌푏·푎 describes how far 휌 is from being a 푘-algebra antihomomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 20 Michela Cerqua and Alberto Facchini 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='2 Adjoining the identity to a pre-Lie algebra The class of pre-Lie algebras contains the class of associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For asso- ciative algebras, it is very natural to consider associative algebras with an identity, and when there is not an identity, to adjoin one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This construction is often called the “Dorroh extension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let’s show that this is possible for pre-Lie algebras as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We will see in fact that a more appropriate name for our class of algebras, instead of “pre-Lie algebras”, would have been “pre-associative algebras”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Adjoining an identity to a pre-Lie 푘-algebra 퐴 is exactly our semidirect product of the pre-Lie 푘-algebra 푘 acting on the pre-Lie 푘-algebra 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let’s be more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An identity in a pre-Lie 푘-algebra 퐴 is an element, which we will denote by 1퐴, such that 푎 · 1퐴 = 1퐴 · 푎 = 푎 for every 푎 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If 퐴 has an identity, we will say that 퐴 is unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An element 푒 of 퐴 is idempotent if 푒2 := 푒 · 푒 = 푒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The zero of 퐴 is always an idempotent element of 퐴, and the identity, when it exists, is also an idempotent element of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐴 be any fixed pre-Lie 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then the associative commutative ring 푘 is a pre-Lie 푘-algebra, and there is a one-to-one correspondence between the set of all the pre-Lie 푘-algebra morphisms 푘 → 퐴 and the set of all idempotent elements of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any idempotent element 푒 of 퐴 the corresponding morphism 휑푒 : 푘 → 퐴 is defined by 휑푒(휆) = 휆푒 for every 휆 ∈ 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Conversely, for any morphism 휑: 푘 → 퐴 the corresponding idempotent element of 퐴 is 휑(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any fixed pre-Lie 푘-algebra 퐴 it is possible to construct the semidirect product of 푘 acting on 퐴 via the pair (휆, 휌) of 푘-module morphisms 푘 → End(퐴푘) for which 휆훼 = 휌훼 is multiplication by 훼 for all 훼 ∈ 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then the four conditions (a), (b), (c), (d) of Theorem 16 are all automatically satisfied, and the corresponding semidirect product is the 푘-module direct sum 퐴 ⊕ 푘 with the multiplication defined by (푥, 훼)(푦, 훽) = (푥 · 푦 + 훽푥 + 훼푦, 훼훽) for every (푥, 훼), (푦, 훽) ∈ 퐴 ⊕ 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence 퐴 ⊕ 푘 becomes a pre-Lie 푘-algebra with identity (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The Lie algebra sub-adjacent this pre-Lie algebra 퐴 ⊕ 푘 is the direct sum of the Lie algebra (퐴, [−, −]) and the abelian Lie algebra 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We will denote this semidirect product by 퐴#푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Now let PreL푘,1 be the category of all unital pre-Lie 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Its objects are the pre-Lie 푘-algebras 퐴 with an identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Its morphisms 푓 : 퐴 → 퐵 are the 푘-algebra morphisms 푓 such that 푓 (1퐴) = 1퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' There is also a further category involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='It is the category PreL푘,1,푎 of all unital pre-Lie 푘-algebras with an augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Its objects are all the pairs (퐴, 휀퐴), where 퐴 is a unital pre-Lie 푘-algebra and 휀퐴: 퐴 → 푘 is a morphism in PreL푘,1 that is a left inverse for 휑1퐴: 푘 휑1퐴 � 퐴 휀퐴 �푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The morphisms 푓 : (퐴, 휀퐴) → (퐵, 휀퐵) are the morphisms 푓 : 퐴 → 퐵 in PreL푘,1 such that 휀퐵 푓 = 휀퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For instance, the 푘-algebra 퐴#푘 is clearly a unital 푘-algebra with Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 21 augmentation: the augmentation is the canonical projection 휋2 : 퐴#푘 = 퐴 ⊕ 푘 → 푘 onto the second summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is easy to see that: Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' There is a category equivalence 퐹: PreL푘 → PreL푘,1,푎 that associates with any object 퐴 of PreL푘 the 푘-algebra with augmentation 퐹(퐴) := (퐴#푘, 휋2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The quasi-inverse of 퐹 is the functor PreL푘,1,푎 → PreL푘, that associates with each unital pre-Lie 푘-algebra with augmentation (퐴, 휀퐴) the kernel ker(휀퐴) of the augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Ben-Zvi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Ma and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Reyes, A Kochen-Specker theorem for integer matrices and noncommutative spectrum functors, J.' metadata={'source': 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