Lie algebras and algebras of associative type
In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕αeG
 A
 α
 graded by some group G, over a field of characteristic zero, has a nonzero component A
 1 (where 1 stands for the identity element of G), and A
 1 is a semisimple associative algebra. Let B = ⊕αeG
 B
 α
 be a finite-dimensional semisimple Lie algebra over a prime field F
 p
 , and let B be graded by a commutative group G. If B = F
 p
 ⊗
 ℤ
 A
 L
 , where A
 L
 is the commutator algebra of a ℤ-algebra A = ⊕αeG
 A
 α
 ; if ℚ ◯
 ℤ
 A is an algebra of associative type, then the 1-component of the algebra K
 ◯
 ℤ
 B, where K stands for the algebraic closure of the field F
 p
 , is the sum of some algebras of the form gl(n
 i
 ,K).