# AUTOMORPHIC EQUIVALENCE OF LINEAR ALGEBRAS.

A.Tsurkov
Institute of Mathematics and Statistics.
University São Paulo.
Rua do Matão, 1010
Cidade Universitária
São Paulo - SP - Brasil - CEP 05508-090
<EMAIL_ADDRESS>
###### Abstract

This research is motivated by universal algebraic geometry. We consider in
universal algebraic geometry the some variety of universal algebras $\Theta$
and algebras $H\in\Theta$ from this variety. One of the central question of
the theory is the following: When do two algebras have the same geometry? What
does it mean that the two algebras have the same geometry? The notion of
geometric equivalence of algebras gives a sort of answer to this question.
Algebras $H_{1}$ and $H_{2}$ are called geometrically equivalent if and only
if the $H_{1}$-closed sets coincide with the $H_{2}$-closed sets. The notion
of automorphic equivalence is a generalization of the first notion. Algebras
$H_{1}$ and $H_{2}$ are called automorphicaly equivalent if and only if the
$H_{1}$-closed sets coincide with the $H_{2}$-closed sets after some ”changing
of coordinates”.

We can detect the difference between geometric and automorphic equivalence of
algebras of the variety $\Theta$ by researching of the automorphisms of the
category $\Theta^{0}$ of the finitely generated free algebras of the variety
$\Theta$. By [3] the automorphic equivalence of algebras provided by inner
automorphism degenerated to the geometric equivalence. So the various
differences between geometric and automorphic equivalence of algebras can be
found in the variety $\Theta$ if the factor group $\mathfrak{A/Y}$ is big.
Hear $\mathfrak{A}$ is the group of all automorphisms of the category
$\Theta^{0}$, $\mathfrak{Y}$ is a normal subgroup of all inner automorphisms
of the category $\Theta^{0}$.

In [4] the variety of all Lie algebras and the variety of all associative
algebras over the infinite field $k$ were studied. If the field $k$ has not
nontrivial automorphisms then group $\mathfrak{A/Y}$ in the first case is
trivial and in the second case has order $2$. We consider in this paper the
variety of all linear algebras over the infinite field $k$. We prove that
group $\mathfrak{A/Y}$ is isomorphic to the group
$\left(U\left(k\mathbf{S}_{\mathbf{2}}\right)\mathfrak{/}U\left(k\left\\{e\right\\}\right)\right)\mathfrak{\leftthreetimes}\mathrm{Aut}k$,
where $\mathbf{S}_{\mathbf{2}}$ is the symmetric group of the set which has
$2$ elements, $U\left(k\mathbf{S}_{\mathbf{2}}\right)$ is the group of all
invertible elements of the group algebra $k\mathbf{S}_{\mathbf{2}}$,
$e\in\mathbf{S}_{\mathbf{2}}$, $U\left(k\left\\{e\right\\}\right)$ is a group
of all invertible elements of the subalgebra $k\left\\{e\right\\}$,
$\mathrm{Aut}k$ is the group of all automorphisms of the field $k$.

So even the field $k$ has not nontrivial automorphisms the group
$\mathfrak{A/Y}$ is infinite. This kind of result is obtained for the first
time.

The example of two linear algebras which are automorphically equivalent but
not geometrically equivalent is presented in the last section of this paper.
This kind of example is also obtained for the first time.

##  1 Introduction.

In the first two sections we consider some variety $\Theta$ of one-sorted
algebras of the signature $\Omega$. Denote by
$X_{0}=\left\\{x_{1},x_{2},\ldots,x_{n},\ldots\right\\}$ a countable set of
symbols, and by $\mathfrak{F}\left(X_{0}\right)$ the set of all finite subsets
of $X_{0}$. We will consider the category $\Theta^{0}$, whose objects are all
free algebras $F\left(X\right)$ of the variety $\Theta$ generated by finite
subsets $X\in\mathfrak{F}\left(X_{0}\right)$. Morphisms of the category
$\Theta^{0}$ are homomorphisms of free algebras.

We denote some time $F\left(X\right)=F\left(x_{1},x_{2},\ldots,x_{n}\right)$
if $X=\left\\{x_{1},x_{2},\ldots,x_{n}\right\\}$ and even
$F\left(X\right)=F\left(x\right)$ if $X$ has only one element.

We assume that our variety $\Theta$ possesses the IBN property: for free
algebras $F\left(X\right),F\left(Y\right)\in\Theta$ we have
$F\left(X\right)\cong F\left(Y\right)$ if and only if
$\left|X\right|=\left|Y\right|$. In this case we have [4, Theorem 2] this
decomposition

$\mathfrak{A=YS}.$ (1.1)

of the group $\mathfrak{A}$ of all automorphisms of the category $\Theta^{0}$.
Hear $\mathfrak{Y}$ is a group of all inner automorphisms of the category
$\Theta^{0}$ and $\mathfrak{S}$ is a group of all strongly stable
automorphisms of the category $\Theta^{0}$.

###### Definition 1.1

An automorphism $\Upsilon$ of a category $\mathfrak{K}$ is inner, if it is
isomorphic as a functor to the identity automorphism of the category
$\mathfrak{K}$.

This means that for every $A\in\mathrm{Ob}\mathfrak{K}$ there exists an
isomorphism $s_{A}^{\Upsilon}:A\rightarrow\Upsilon\left(A\right)$ such that
for every $\alpha\in\mathrm{Mor}_{\mathfrak{K}}\left(A,B\right)$ the diagram

$\begin{array}[]{ccc}A&\overrightarrow{s_{A}^{\Upsilon}}&\Upsilon\left(A\right)\\\
\downarrow\alpha&&\Upsilon\left(\alpha\right)\downarrow\\\
B&\underrightarrow{s_{B}^{\Upsilon}}&\Upsilon\left(B\right)\end{array}$

commutes.

###### Definition 1.2

. An automorphism $\Phi$ of the category $\Theta^{0}$ is called strongly
stable if it satisfies the conditions:

  1. A1)

$\Phi$ preserves all objects of $\Theta^{0}$,

  2. A2)

there exists a system of bijections $\left\\{s_{F}^{\Phi}:F\rightarrow F\mid
F\in\mathrm{Ob}\Theta^{0}\right\\}$ such that $\Phi$ acts on the morphisms
$\alpha:D\rightarrow F$ of $\Theta^{0}$ by this way:

$\Phi\left(\alpha\right)=s_{F}^{\Phi}\alpha\left(s_{D}^{\Phi}\right)^{-1},$
(1.2)

  3. A3)

$s_{F}^{\Phi}\mid_{X}=id_{X},$ for every free algebra $F=F\left(X\right)$.

The subgroup $\mathfrak{Y}$ is a normal in $\mathfrak{A}$. We will calculate
the factor group $\mathfrak{A/Y\cong S/S\cap Y}$. This calculation is very
important for universal algebraic geometry.

All definitions of the basic notions of the universal algebraic geometry can
be found, for example, in [1], [2] and [3]. In universal algebraic geometry we
consider a ”set of equations” $T\subset F\times F$ in some finitely generated
free algebra $F$ of the arbitrary variety of universal algebras $\Theta$ and
we ”resolve” these equations in $\mathrm{Hom}\left(F,H\right)$, where
$H\in\Theta$. The set $\mathrm{Hom}\left(F,H\right)$ serves as an ”affine
space over the algebra $H$”. Denote by $T_{H}^{\prime}$ the set
$\left\\{\mu\in\mathrm{Hom}\left(F,H\right)\mid T\subset\ker\mu\right\\}$.
This is the set of all solutions of the set of equations $T$. For every set of
”points” $R$ of the affine space $\mathrm{Hom}\left(F,H\right)$ we consider a
congruence of equations defined by this set:
$R_{H}^{\prime}=\bigcap\limits_{\mu\in R}\ker\mu$. For every set of equations
$T$ we consider its algebraic closure $T_{H}^{\prime\prime}$ in respect to the
algebra $H$. A set $T\subset F\times F$ is called $H$-closed if
$T=T_{H}^{\prime\prime}$. An $H$-closed set is always a congruence.

###### Definition 1.3

Algebras $H_{1},H_{2}\in\Theta$ are geometrically equivalent if and only if
for every $X\in\mathfrak{F}\left(X_{0}\right)$ and every $T\subset
F\left(X\right)\times F\left(X\right)$ fulfills
$T_{H_{1}}^{\prime\prime}=T_{H_{2}}^{\prime\prime}$.

Denote the family of all $H$-closed congruences in $F$ by $Cl_{H}(F)$. We can
consider the category $C_{\Theta}\left(H\right)$ of the coordinate algebras
connected with the algebra $H\in\Theta$. Objects of this category are quotient
algebras $F\left(X\right)/T$, where $X\in\mathfrak{F}\left(X_{0}\right)$,
$T\in Cl_{H}(F\left(X\right))$. Morphisms of this category are homomorphisms
of algebras.

###### Definition 1.4

Let
$Id\left(H,X\right)=\bigcap\limits_{\varphi\in\mathrm{Hom}\left(F\left(X\right),H\right)}\ker\varphi$
be the minimal $H$-closed congruence in $\ F\left(X\right)$. Algebras
$H_{1},H_{2}\in\Theta$ are automorphically equivalent if and only if there
exists a pair $\left(\Phi,\Psi\right),$ where
$\Phi:\Theta^{0}\rightarrow\Theta^{0}$ is an automorphism,
$\Psi:C_{\Theta}\left(H_{1}\right)\rightarrow$ $C_{\Theta}\left(H_{2}\right)$
is an isomorphism subject to conditions:

  1. A.

$\Psi\left(F\left(X\right)/Id\left(H_{1},X\right)\right)=F\left(Y\right)/Id\left(H_{2},Y\right)$,
where $\Phi\left(F\left(X\right)\right)=F\left(Y\right)$,

  2. B.

$\Psi\left(F\left(X\right)/T\right)=F\left(Y\right)/\widetilde{T}$, where
$T\in Cl_{H_{1}}(F\left(X\right))$, $\widetilde{T}\in
Cl_{H_{2}}(F\left(Y\right))$,

  3. C.

$\Psi$ takes the natural epimorphism
$\overline{\tau}:F\left(X\right)/Id\left(H_{1},X\right)\rightarrow
F\left(X\right)/T$ to the natural epimorphism
$\Psi\left(\overline{\tau}\right):F\left(Y\right)/Id\left(H_{2},Y\right)\rightarrow
F\left(Y\right)/\widetilde{T}$.

Note that if such a pair $\left(\Phi,\Psi\right)$ exists, then $\Psi$ is
uniquely defined by $\Phi$.

We can say, in certain sense, that automorphic equivalence of algebras is a
coinciding of the structure of closed sets after some ”changing of
coordinates” provided by automorphism $\Phi$.

Algebras $H_{1}$ and $H_{2}$ are geometrically equivalent if and only if an
inner automorphism $\Phi:\Theta^{0}\rightarrow\Theta^{0}$ provides the
automorphic equivalence of algebras $H_{1}$ and $H_{2}$. So, only strongly
stable automorphism $\Phi$ can provide us automorphic equivalence of algebras
which not coincides with geometric equivalence of algebras. Therefore, in some
sense, difference from the automorphic equivalence to the geometric
equivalence is measured by the factor group $\mathfrak{A/Y\cong S/S\cap Y}$.

##  2 Verbal operations and strongly stable automorphisms.

For every word $w=w\left(x_{1},\ldots,x_{k}\right)\in F\left(X\right)$, where
$F\left(X\right)\in\mathrm{Ob}\Theta^{0}$,
$X=\left\\{x_{1},\ldots,x_{k}\right\\}$ and for every algebra $H\in\Theta$ we
can define a $k$-ary operation $w_{H}^{\ast}$ on $H$ by

$w_{H}^{\ast}\left(h_{1},\ldots,h_{k}\right)=w\left(h_{1},\ldots,h_{k}\right)=\gamma_{h}\left(w\left(x_{1},\ldots,x_{k}\right)\right),$

where $\gamma_{h}$ is a homomorphism $F\left(X\right)\ni
x_{i}\rightarrow\gamma_{h}\left(x_{i}\right)=h_{i}\in H$, $1\leq i\leq k$.
This operation we call the verbal operation induced on the algebra $H$ by the
word $w\left(x_{1},\ldots,x_{k}\right)\in F\left(X\right)$. A system of words
$W=\left\\{w_{i}\mid i\in I\right\\}$ such that $w_{i}\in F\left(X_{i}\right)$
, $X_{i}=\left\\{x_{1},\ldots,x_{k_{i}}\right\\},$ determines a system of
$k_{i}$-ary operations $\left(w_{i}\right)_{H}^{\ast}$ on $H$. Denote the set
$H$ with the system of these operation by $H_{W}^{\ast}$.

We have a correspondence between strongly stable automorphisms and systems of
words which define the verbal operation and fulfill some conditions. This
correspondence explained in [4] and [5]: We denote the signature of our
variety $\Theta$ by $\Omega$, by $k_{\omega}$ we denote the arity of $\omega$
for every $\omega\in\Omega$. We suppose that we have the system of words
$W=\left\\{w_{\omega}\mid\omega\in\Omega\right\\}$ satisfies the conditions:

  1. Op1)

$w_{\omega}\left(x_{1},\ldots,x_{k_{\omega}}\right)\in
F\left(X_{\omega}\right)$, where
$X_{\omega}=\left\\{x_{1},\ldots,x_{k_{\omega}}\right\\}$;

  2. Op2)

for every $F=F\left(X\right)\in\mathrm{Ob}\Theta^{0}$ there exists an
isomorphism $\sigma_{F}:F\rightarrow F_{W}^{\ast}$ such that
$\sigma_{F}\mid_{X}=id_{X}$.

$F_{W}^{\ast}\in\Theta$ so isomorphisms $\sigma_{F}$ are defined uniquely by
the system of words $W$.

The set $S=\left\\{\sigma_{F}:F\rightarrow F\mid
F\in\mathrm{Ob}\Theta^{0}\right\\}$ is a system of bijections which satisfies
the conditions:

  1. B1)

for every homomorphism $\alpha:A\rightarrow B\in\mathrm{Mor}\Theta^{0}$ the
mappings $\sigma_{B}\alpha\sigma_{A}^{-1}$ and
$\sigma_{B}^{-1}\alpha\sigma_{A}$ are homomorphisms;

  2. B2)

$\sigma_{F}\mid_{X}=id_{X}$ for every free algebra
$F\in\mathrm{Ob}\Theta^{0}$.

So we can define the strongly stable automorphism by this system of
bijections. This automorphism preserves all objects of $\Theta^{0}$ and acts
on morphism of $\Theta^{0}$ by formula (1.2), where $s_{F}^{\Phi}=$
$\sigma_{F}$.

Vice versa if we have a strongly stable automorphism $\Phi$ of the category
$\Theta^{0}$ then its system of bijections
$S=\left\\{s_{F}^{\Phi}:F\rightarrow F\mid F\in\mathrm{Ob}\Theta^{0}\right\\}$
defined uniquely. Really, if $F\in\mathrm{Ob}\Theta^{0}$ and $f\in F$ then

$s_{F}^{\Phi}\left(f\right)=s_{F}^{\Phi}\alpha\left(x\right)=\left(s_{F}^{\Phi}\alpha\left(s_{D}^{\Phi}\right)^{-1}\right)\left(x\right)=\left(\Phi\left(\alpha\right)\right)\left(x\right),$
(2.1)

where $D=F\left(x\right)$ \- $1$-generated free linear algebra - and
$\alpha:D\rightarrow F$ homomorphism such that $\alpha\left(x\right)=f$.
Obviously that this system of bijections $S=\left\\{s_{F}^{\Phi}:F\rightarrow
F\mid F\in\mathrm{Ob}\Theta^{0}\right\\}$ fulfills conditions B1 and B2 with
$\sigma_{F}=s_{F}^{\Phi}$.

If we have a system of bijections $S=\left\\{\sigma_{F}:F\rightarrow F\mid
F\in\mathrm{Ob}\Theta^{0}\right\\}$ which fulfills conditions B1 and B2 than
we can define the system of words
$W=\left\\{w_{\omega}\mid\omega\in\Omega\right\\}$ satisfies the conditions
Op1 and Op2 by formula

$w_{\omega}\left(x_{1},\ldots,x_{k_{\omega}}\right)=\sigma_{F_{\omega}}\left(\omega\left(\left(x_{1},\ldots,x_{k_{\omega}}\right)\right)\right)\in
F_{\omega},$ (2.2)

where $F_{\omega}=F\left(X_{\omega}\right)$.

By formulas (2.1) and (2.2) we can check that there are

  1. 1.

one to one and onto correspondence between strongly stable automorphisms of
the category $\Theta^{0}$ and systems of bijections satisfied the conditions
B1 and B2

  2. 2.

one to one and onto correspondence between systems of bijections satisfied the
conditions B1 and B2 and systems of words satisfied the conditions Op1 and
Op2.

So we can find a strongly stable automorphism $\Phi$ of the category
$\Theta^{0}$ by finding a system of words which fulfills conditions Op1 and
Op2.

##  3 Verbal operations in linear algebras.

From now on, we consider the variety $\Theta$ of all linear algebras over
infinite field $k$. We consider linear algebras as one-sorted universal
algebras, i. e., multiplication by scalar we consider as $1$-ary operation for
every $\lambda\in k$: $H\ni h\rightarrow\lambda h\in H$ where $H\in\Theta$.
Hence the signature $\Omega$ of algebras of our variety contains these
operations: $0$-ary operation $0$; $\left|k\right|$ $1$-ary operations of
multiplications by scalars; $1$-ary operation $-:h\rightarrow-h$, where $h\in
H$, $H\in\Theta$; $2$-ary operation $\cdot$ and $2$-ary operation $+$. We will
finding the system of words $W=\left\\{w_{\omega}\mid\omega\in\Omega\right\\}$
satisfies the conditions Op1 and Op2. We denote the words corresponding to
these operations by $w_{0}$, $w_{\lambda}$ for all $\lambda\in k$, $w_{-}$,
$w_{\cdot}$, $w_{+}$.

For arbitrary $F\left(X\right)\in\mathrm{Ob}\Theta^{0}$ we denote
$F\left(X\right)=\bigoplus\limits_{i=1}^{\infty}F_{i}$ the decomposition to
the linear spaces of elements which are homogeneous according the sum of
degrees of generators from the set $X$. We also denote the two-sides ideals
$\bigoplus\limits_{i=j}^{\infty}F_{i}=F^{j}$. From now on, the word ”ideal”
means two sided ideal of linear algebra.

We denote the group of all automorphisms of the field $k$ by $\mathrm{Aut}k$.

Our variety $\Theta$ possesses the IBN property, because $\left|X\right|=\dim
F/F^{2}$ fulfills for all free algebras $F=F\left(X\right)\in\Theta$. So we
have the decomposition (1.1) for group of all automorphisms of the category
$\Theta^{0}$.

Now we need to prove one technical fact about $1$-generated free linear
algebra $F\left(x\right)$.

###### Lemma 3.1

Let $\left\\{u_{1},\ldots,u_{r}\right\\}$ is the set of all monomials of
degree $n$ in $F\left(x\right)$ (basis of $F_{n}$),
$\left\\{v_{1},\ldots,v_{t}\right\\}$ is the set of all monomials of degree
$m$ in $F\left(x\right)$ (basis of $F_{m}$), $\varphi$ is an arbitrary
function from $\left\\{1,\ldots,n\right\\}$ to $\left\\{1,\ldots,t\right\\}$.
Denote by $\varphi\left(u_{l}\right)$ the monomial which is a results of
substitution into monomial $u_{l}$ ($1\leq l\leq r$) instead $j$-th from left
entry of $x$ the monomial $v_{\varphi\left(j\right)}$ ($1\leq j\leq n$). All
these monomials are distinct, i. e.,
$\varphi_{1}\left(u_{l_{1}}\right)=\varphi_{2}\left(u_{l_{2}}\right)$ if and
only if $\varphi_{1}=\varphi_{2}$ and $u_{l_{1}}=u_{l_{2}}$, where
$\varphi_{1},\varphi_{2}:\left\\{1,\ldots,n\right\\}\rightarrow\left\\{1,\ldots,t\right\\}$,
$u_{l_{1}},u_{l_{2}}\in\left\\{u_{1},\ldots,u_{r}\right\\}$.

Proof. We will prove this lemma by induction by $n$ \- degree of monomials
from $\left\\{u_{1},\ldots,u_{r}\right\\}$. The claim of the lemma is trivial
for $n=1$. We assume that the claim of the lemma is proved for monomials which
have degree $<n$. We suppose that
$\varphi_{1}\left(u_{l_{1}}\right)=\varphi_{2}\left(u_{l_{2}}\right)$, where
$\deg u_{l_{1}}=\deg u_{l_{2}}=n>1$,
$\varphi_{1},\varphi_{2}:\left\\{1,\ldots,n\right\\}\rightarrow\left\\{1,\ldots,t\right\\}$.
$u_{l_{i}}=u_{l_{i}}^{(1)}\cdot u_{l_{i}}^{(2)}$, where $i=1,2$. We denote
$\deg u_{l_{i}}^{(1)}=c_{i}$. $1\leq c_{i}<n$ for $i=1,2$. For $i=1,2$ we have
$\varphi_{i}\left(u_{l_{i}}\right)=\varphi_{i}^{(1)}\left(u_{l_{i}}^{(1)}\right)\cdot\varphi_{i}^{(2)}\left(u_{l_{i}}^{(2)}\right)$,
where
$\varphi_{i}^{(1)}:\left\\{1,\ldots,c_{i}\right\\}\rightarrow\left\\{1,\ldots,t\right\\}$,
$\varphi_{i}^{(2)}:\left\\{1,\ldots,n-c_{i}\right\\}\rightarrow\left\\{1,\ldots,t\right\\}$,
$\varphi_{i}^{(1)}\left(j\right)=\varphi_{i}\left(j\right)$ for $1\leq j\leq
c_{i}$, $\varphi_{i}^{(2)}\left(j\right)=\varphi_{i}\left(c_{i}+j\right)$ for
$1\leq j\leq n-c_{1}$.
$\varphi_{1}\left(u_{l_{1}}\right)=\varphi_{2}\left(u_{l_{2}}\right)$ if and
only if
$\varphi_{1}^{(1)}\left(u_{l_{1}}^{(1)}\right)=\varphi_{2}^{(1)}\left(u_{l_{2}}^{(1)}\right)$
and
$\varphi_{1}^{(2)}\left(u_{l_{1}}^{(2)}\right)=\varphi_{2}^{(2)}\left(u_{l_{2}}^{(2)}\right)$.
If $c_{1}\neq c_{2}$ then
$\deg\varphi_{1}^{(1)}\left(u_{l_{1}}^{(1)}\right)=c_{1}m\neq\deg\varphi_{2}^{(1)}\left(u_{l_{2}}^{(1)}\right)=c_{2}m$,
hence
$\varphi_{1}^{(1)}\left(u_{l_{1}}^{(1)}\right)\neq\varphi_{2}^{(1)}\left(u_{l_{2}}^{(1)}\right)$
and $\varphi_{1}\left(u_{l_{1}}\right)\neq\varphi_{2}\left(u_{l_{2}}\right)$.
So $c_{1}=c_{2}$ and, by our assumption,
$\varphi_{1}^{(1)}=\varphi_{2}^{(1)}$, $u_{l_{1}}^{(1)}=u_{l_{2}}^{(1)}$,
$\varphi_{1}^{(2)}=\varphi_{2}^{(2)}$, $u_{l_{1}}^{(2)}=u_{l_{2}}^{(2)}$.
Therefore $\varphi_{1}=\varphi_{2}$ and $u_{l_{1}}=u_{l_{2}}$.

###### Corollary 1

Let $f\left(x\right),g\left(x\right)\in F\left(X\right)$.
$f\left(g\left(x\right)\right)$ is a result of substitution of
$g\left(x\right)$ in $f\left(x\right)$ instead $x$.
$f\left(g\left(x\right)\right)\in F_{1}$ if and only if
$f\left(x\right),g\left(x\right)\in F_{1}$.

Proof. We write $f\left(x\right)$ and $g\left(x\right)$ as sum of its
homogeneous components:
$f\left(x\right)=f_{1}\left(x\right)+f_{2}\left(x\right)+\ldots+f_{n}\left(x\right)$,
$g\left(x\right)=g_{1}\left(x\right)+g_{2}\left(x\right)+\ldots+g_{m}\left(x\right)$,
$f_{i}\left(x\right),g_{i}\left(x\right)\in F_{i}$. We assume that $n>1$ or
$m>1$, $f_{n}\left(x\right)\neq 0$ and $g_{m}\left(x\right)\neq 0$.
$f\left(g\left(x\right)\right)=f_{1}\left(g\left(x\right)\right)+f_{2}\left(g\left(x\right)\right)+\ldots+f_{n}\left(g\left(x\right)\right)$.
Addenda of the maximal possible degree of $x$, which can appear in
$f\left(g\left(x\right)\right)$, i. e., addenda of degree $nm$ can appear in
$f_{n}\left(g\left(x\right)\right)$. They coincide with addenda of
$f_{n}\left(g_{m}\left(x\right)\right)$. Denote
$f_{n}\left(x\right)=\lambda_{1}u_{1}+\ldots+\lambda_{r}u_{r}$,
$g_{m}\left(x\right)=\mu_{1}v_{1}+\ldots+\mu_{t}v_{t}$, where
$\left\\{u_{1},\ldots,u_{r}\right\\}$ is the set of all monomials of degree
$n$ in $F\left(x\right)$, $\left\\{v_{1},\ldots,v_{t}\right\\}$ is the set of
all monomials of degree $m$ in $F\left(x\right)$, $\lambda_{i},\mu_{j}\in k$.
Not all $\left\\{\lambda_{1},\ldots,\lambda_{r}\right\\}$ and not all
$\left\\{\mu_{1},\ldots,\mu_{t}\right\\}$ are equal to $0$ by our assumption.
$f_{n}\left(g_{m}\left(x\right)\right)=\lambda_{1}u_{1}\left(g_{m}\left(x\right)\right)+\ldots+\lambda_{r}u_{r}\left(g_{m}\left(x\right)\right)$.
If we open the brackets in
$u_{l}\left(g_{m}\left(x\right)\right)=u_{l}\left(\mu_{1}v_{1}+\ldots+\mu_{t}v_{t}\right)$
($1\leq l\leq r$), we obtain addenda, which are results of substitution into
monomial $u_{l}$ instead all entry of $x$ some monomial from
$\mu_{1}v_{1},\ldots,\mu_{t}v_{t}$ in all possible options. We can say more
formal: for every function
$\varphi:\left\\{1,\ldots,n\right\\}\rightarrow\left\\{1,\ldots,t\right\\}$ we
obtain an addendum which is a results of substitution into monomial $u_{l}$
instead $j$-th from left entry of $x$ the monomial
$\mu_{\varphi\left(j\right)}v_{\varphi\left(j\right)}$ ($1\leq j\leq n$).
Therefore all addenda, which we obtain after the opening of the brackets in
$f_{n}\left(g_{m}\left(x\right)\right)$, distinct from the monomials discussed
in Lemma 3.1 only by coefficients. All these addenda have degree $nm>1$,
because $n>1$ or $m>1$. So addenda of $f_{n}\left(g_{m}\left(x\right)\right)$
can not cancel one another by Lemma 3.1. These addenda can not be canceled by
other addenda $f\left(g\left(x\right)\right)$, because all other addenda have
degree $<nm$. Therefore all these addenda equal to $0$, because
$f\left(g\left(x\right)\right)\in F_{1}$. For
$l\in\left\\{1,\ldots,r\right\\}$ and $j\in\left\\{1,\ldots,t\right\\}$ we
take the addendum which is a results of substitution into monomial
$\lambda_{l}u_{l}$ instead all entries of $x$ the monomial $\mu_{j}v_{j}$. The
coefficient of this addendum is $\lambda_{l}\mu_{j}^{n}=0$. So
$\lambda_{l}\mu_{j}=0$ for all $l\in\left\\{1,\ldots,r\right\\}$ and all
$j\in\left\\{1,\ldots,t\right\\}$. It contradicts the fact that
$f_{n}\left(x\right)\neq 0$ and $g_{m}\left(x\right)\neq 0$.

###### Theorem 3.1

The system of words

$W=\left\\{w_{0},w_{\lambda}\left(\lambda\in
k\right),w_{-},w_{+},w_{\cdot}\right\\}$ (3.1)

satisfies the conditions Op1 and Op2 if and only if $w_{0}=0$,
$w_{\lambda}=\varphi\left(\lambda\right)x_{1}$, $w_{-}=-x_{1}$,
$w_{+}=x_{1}+x_{2}$, $w_{\cdot}=ax_{1}x_{2}+bx_{2}x_{1}$, where $\varphi$ is
an automorphism of the field $k$, $a,b\in k$, $a\neq\pm b$.

Proof. Let $W$ (see (3.1) ) satisfies the conditions Op1 and Op2.

$w_{0}$ is an element of the $0$-generated free linear algebra. There is only
one element in this algebra: $0$. This is the only one opportunity for
$w_{0}$.

$w_{\lambda}\in F\left(x\right)$ for every $\lambda\in k$. Denote
multiplications by scalars in $\left(F\left(x\right)\right)_{W}^{\ast}$ by
$\ast$, i. e., $\lambda\ast f=w_{\lambda}\left(f\right)$ for every $f\in
F\left(x\right)$ and every $\lambda\in k$.
$\left(F\left(x\right)\right)_{W}^{\ast}\in\Theta$, therefore, if $\lambda=0$
then $0\ast x=w_{0}\left(x\right)=0$. If $\lambda\neq 0$ then

$1\ast x=\left(\lambda^{-1}\lambda\right)\ast
x=\lambda^{-1}\ast\left(\lambda\ast
x\right)=w_{\lambda^{-1}}\left(w_{\lambda}\left(x\right)\right)=x.$

Hence $w_{\lambda}=\varphi\left(\lambda\right)x$ by Corollary 1 from Lemma
3.1, where $\varphi\left(\lambda\right)\in k$. We can write
$\varphi\left(0\right)=0$. Also we have that for all
$\lambda_{1},\lambda_{2}\in k$ fulfills

$\left(\lambda_{1}\lambda_{2}\right)\ast
x=\varphi\left(\lambda_{1}\lambda_{2}\right)x$

and

$\left(\lambda_{1}\lambda_{2}\right)\ast
x=\lambda_{1}\ast\left(\lambda_{2}\ast
x\right)=\lambda_{1}\ast\left(\varphi\left(\lambda_{2}\right)x\right)=$
$\varphi\left(\lambda_{1}\right)\left(\varphi\left(\lambda_{2}\right)x\right)=\left(\varphi\left(\lambda_{1}\right)\varphi\left(\lambda_{2}\right)\right)x.$

So
$\varphi\left(\lambda_{1}\right)\varphi\left(\lambda_{2}\right)=\varphi\left(\lambda_{1}\lambda_{2}\right)$.
If $\mu\in k\setminus\left\\{0\right\\}$, then the $1$-ary operation of
multiplication by scalar $\mu$ is a verbal operation defined by some word
$w_{\mu}^{\ast}\left(x\right)\in\left(F\left(x\right)\right)_{W}^{\ast}$,
written be the operations defined by system of words $W$ \- see [5,
Proposition 4.2]. Hence, $\mu f=w_{\mu}^{\ast}\left(f\right)$ holds for every
$f\in F\left(x\right)$. Also there is
$w_{\mu^{-1}}^{\ast}\left(x\right)\in\left(F\left(x\right)\right)_{W}^{\ast}$
such that $\mu^{-1}f=w_{\mu^{-1}}^{\ast}\left(f\right)$ for every $f\in
F\left(x\right)$. $x=\mu^{-1}\left(\mu
x\right)=w_{\mu^{-1}}^{\ast}\left(w_{\mu}^{\ast}\left(x\right)\right)$. There
exists by Op2 an isomorphism
$\sigma_{F\left(x\right)}:F\left(x\right)\rightarrow\left(F\left(x\right)\right)_{W}^{\ast}$
such that $\sigma_{F\left(x\right)}\left(x\right)=x$. So
$\left(F\left(x\right)\right)_{W}^{\ast}$ is also $1$-generated free linear
algebra of $\Theta$ with the free generator $x$. Hence there exists a
decomposition
$\left(F\left(x\right)\right)_{W}^{\ast}=\bigoplus\limits_{i=1}^{\infty}F_{i}^{\ast}$,
where $F_{i}^{\ast}$ are linear spaces of elements which are homogeneous
according the degree of $x$ but in respect of operations defined by system of
words $W$. Therefore $w_{\mu}^{\ast}\left(x\right)=\lambda\ast x$, where
$\lambda\in k$, by Corollary 1 from Lemma 3.1. So $\mu x=\lambda\ast
x=\varphi\left(\lambda\right)x$ and $\mu=\varphi\left(\lambda\right)$, hence
$\varphi:k\rightarrow k$ is a surjection.

$w_{+}\in F\left(x_{1},x_{2}\right)=F$. There exists $n\in\mathbb{N}$, such
that

$w_{+}\left(x_{1},x_{2}\right)=p_{1}\left(x_{1},x_{2}\right)+p_{2}\left(x_{1},x_{2}\right)+\ldots+p_{n}\left(x_{1},x_{2}\right),$

where $p_{i}\left(x_{1},x_{2}\right)\in F_{i}$, $1\leq i\leq n$. We have for
every $\lambda\in k$ that

$w_{+}\left(\lambda\ast x_{1},\lambda\ast x_{2}\right)=\lambda\ast
w_{+}\left(x_{1},x_{2}\right)=\varphi\left(\lambda\right)w_{+}\left(x_{1},x_{2}\right)=$
$\varphi\left(\lambda\right)p_{1}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)p_{2}\left(x_{1},x_{2}\right)+\ldots+\varphi\left(\lambda\right)p_{n}\left(x_{1},x_{2}\right)$

and

$w_{+}\left(\lambda\ast x_{1},\lambda\ast x_{2}\right)=p_{1}\left(\lambda\ast
x_{1},\lambda\ast x_{2}\right)+p_{2}\left(\lambda\ast x_{1},\lambda\ast
x_{2}\right)+\ldots+p_{n}\left(\lambda\ast x_{1},\lambda\ast x_{2}\right)=$
$p_{1}\left(\varphi\left(\lambda\right)x_{1},\varphi\left(\lambda\right)x_{2}\right)+p_{2}\left(\varphi\left(\lambda\right)x_{1},\varphi\left(\lambda\right)x_{2}\right)+\ldots+p_{n}\left(\varphi\left(\lambda\right)x_{1},\varphi\left(\lambda\right)x_{2}\right)=$
$\varphi\left(\lambda\right)p_{1}\left(x_{1},x_{2}\right)+\left(\varphi\left(\lambda\right)\right)^{2}p_{2}\left(x_{1},x_{2}\right)+\ldots+\left(\varphi\left(\lambda\right)\right)^{n}p_{n}\left(x_{1},x_{2}\right).$

We can take $\lambda\in k$ such that $\varphi\left(\lambda\right)$ is not a
solution of any equation $x^{i}=x$, where $2\leq i\leq n$. So,
$p_{i}\left(x_{1},x_{2}\right)=0$ for $2\leq i\leq n$ by equality of the
homogeneous components. Therefore $w_{+}=\alpha x_{1}+\beta x_{2}$, where
$\alpha,\beta\in k$. If we denote the operation defined by $w_{+}$ in
$\left(F\left(x_{1},x_{2}\right)\right)_{W}^{\ast}$ by $\bot$, then $x_{1}\bot
x_{2}=x_{2}\bot x_{1}$ holds, so $\alpha x_{1}+\beta x_{2}=\alpha x_{2}+\beta
x_{1}$ and $\alpha=\beta$. Also $x_{1}\bot 0=x_{1}$ holds and $\alpha
x_{1}=x_{1}$, so $\alpha=\beta=1$. Now, by consideration of $F\left(x\right)$,
we can conclude that for all $\lambda_{1},\lambda_{2}\in k$ fulfills

$\varphi\left(\lambda_{1}+\lambda_{2}\right)x=\left(\lambda_{1}+\lambda_{2}\right)\ast
x=\lambda_{1}\ast x\bot\lambda_{2}\ast x=$ $\lambda_{1}\ast x+\lambda_{2}\ast
x=\varphi\left(\lambda_{1}\right)x+\varphi\left(\lambda_{2}\right)x=\left(\varphi\left(\lambda_{1}\right)+\varphi\left(\lambda_{2}\right)\right)x,$

so $\varphi\left(\lambda_{1}+\lambda_{2}\right)=$
$\varphi\left(\lambda_{1}\right)+\varphi\left(\lambda_{2}\right)$ and
$\varphi$ is an automorphism of the field $k$.

Its clear now that $w_{-}=-x\in F\left(x\right)$, because

$w_{-}\left(x\right)=-1\ast x=\varphi\left(-1\right)x=\left(-1\right)x=-x.$

$w_{\cdot}\in F\left(x_{1},x_{2}\right)$. We write $w_{\cdot}$ as sum of its
homogeneous components according the degree of $x_{1}$:

$w_{\cdot}\left(x_{1},x_{2}\right)=p_{0}\left(x_{1},x_{2}\right)+p_{1}\left(x_{1},x_{2}\right)+p_{2}\left(x_{1},x_{2}\right)+\ldots+p_{n}\left(x_{1},x_{2}\right).$

We denote the operation defined by $w_{\cdot}$ in
$\left(F\left(x_{1},x_{2}\right)\right)_{W}^{\ast}$ by $\times$. So we have
for every $\lambda\in k$ that

$\left(\lambda\ast x_{1}\right)\times x_{2}=\lambda\ast\left(x_{1}\times
x_{2}\right)=\varphi\left(\lambda\right)w_{\cdot}\left(x_{1},x_{2}\right)=$
$\varphi\left(\lambda\right)p_{0}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)p_{1}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)p_{2}\left(x_{1},x_{2}\right)+\ldots+\varphi\left(\lambda\right)p_{n}\left(x_{1},x_{2}\right).$

and

$\left(\lambda\ast x_{1}\right)\times
x_{2}=w_{\cdot}\left(\varphi\left(\lambda\right)x_{1},x_{2}\right)=$
$p_{0}\left(\varphi\left(\lambda\right)x_{1},x_{2}\right)+p_{1}\left(\varphi\left(\lambda\right)x_{1},x_{2}\right)+p_{2}\left(\varphi\left(\lambda\right)x_{1},x_{2}\right)+\ldots+p_{n}\left(\varphi\left(\lambda\right)x_{1},x_{2}\right)=$
$p_{0}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)p_{1}\left(x_{1},x_{2}\right)+\left(\varphi\left(\lambda\right)\right)^{2}p_{2}\left(x_{1},x_{2}\right)+\ldots+\left(\varphi\left(\lambda\right)\right)^{n}p_{n}\left(x_{1},x_{2}\right).$

We can take, as above, $\lambda\in k$ such that by equality of the homogeneous
components we obtain that
$w_{\cdot}\left(x_{1},x_{2}\right)=p_{1}\left(x_{1},x_{2}\right)$. Now we
write $w_{\cdot}\left(x_{1},x_{2}\right)=p_{1}\left(x_{1},x_{2}\right)$ as sum
of its homogeneous components according the degree of $x_{2}$:

$w_{\cdot}\left(x_{1},x_{2}\right)=r_{0}\left(x_{1},x_{2}\right)+r_{1}\left(x_{1},x_{2}\right)+r_{2}\left(x_{1},x_{2}\right)+\ldots+r_{m}\left(x_{1},x_{2}\right).$

We have for every $\lambda\in k$ that

$x_{1}\times\left(\lambda\ast x_{2}\right)=\lambda\ast\left(x_{1}\times
x_{2}\right)=\varphi\left(\lambda\right)w_{\cdot}\left(x_{1},x_{2}\right)=$
$\varphi\left(\lambda\right)r_{0}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)r_{1}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)r_{2}\left(x_{1},x_{2}\right)+\ldots+\varphi\left(\lambda\right)r_{m}\left(x_{1},x_{2}\right).$

and

$x_{1}\times\left(\lambda\ast
x_{2}\right)=w_{\cdot}\left(x_{1},\varphi\left(\lambda\right)x_{2}\right)=$
$r_{0}\left(x_{1},\varphi\left(\lambda\right)x_{2}\right)+r_{1}\left(x_{1},\varphi\left(\lambda\right)x_{2}\right)+r_{2}\left(x_{1},\varphi\left(\lambda\right)x_{2}\right)+\ldots+r_{m}\left(x_{1},\varphi\left(\lambda\right)x_{2}\right)=$
$r_{0}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)r_{1}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)^{2}r_{2}\left(x_{1},x_{2}\right)+\ldots+\varphi\left(\lambda\right)^{m}r_{m}\left(x_{1},x_{2}\right).$

And, as above, we can conclude that
$w_{\cdot}\left(x_{1},x_{2}\right)=r_{1}\left(x_{1},x_{2}\right)$ where
$r_{1}\left(x_{1},x_{2}\right)$ is a homogeneous element of
$F\left(x_{1},x_{2}\right)$ such that
$\deg_{x_{1}}r_{1}\left(x_{1},x_{2}\right)=1$ and
$\deg_{x_{2}}r_{1}\left(x_{1},x_{2}\right)=1$. Therefore
$w_{\cdot}\left(x_{1},x_{2}\right)=ax_{1}x_{2}+bx_{2}x_{1}$, where $a,b\in k$.
If $a=b$ then the operation defined by $w_{\cdot}\left(x_{1},x_{2}\right)$ is
commutative. If $a=-b$ then the operation defined by
$w_{\cdot}\left(x_{1},x_{2}\right)$ is anticommutative. The isomorphisms
$\sigma_{F}:F\rightarrow F_{W}^{\ast}$, where $F\in\mathrm{Ob}\Theta^{0}$ can
not exists in both these cases if $F$ is not a $0$-generated free algebra.

Therefore we prove that if the system of words (3.1) satisfies the conditions
Op1 and Op2 then $w_{0}=0$, $w_{\lambda}=\varphi\left(\lambda\right)x_{1}$ for
all $\lambda\in k$, $w_{-}=-x_{1}$, $w_{+}=x_{1}+x_{2}$,
$w_{\cdot}=ax_{1}x_{2}+bx_{2}x_{1}$, where $\varphi$ is an automorphism of the
field $k$, $a,b\in k$, $a\neq\pm b$.

Now we must prove that for all $\varphi\in\mathrm{Aut}k$ and all $a,b\in k$
such that $a\neq\pm b$ the system of words (3.1) where $w_{0}=0$,
$w_{\lambda}=\varphi\left(\lambda\right)x_{1}$ for all $\lambda\in k$,
$w_{-}=-x_{1}$, $w_{+}=x_{1}+x_{2}$, $w_{\cdot}=ax_{1}x_{2}+bx_{2}x_{1}$
fulfills condition Op2. It means that we must build for every
$F=F\left(X\right)\in\mathrm{Ob}\Theta^{0}$ an isomorphism
$\sigma_{F}:F\rightarrow F_{W}^{\ast}$ such that $\sigma_{F}\mid_{X}=id_{X}$.

We will prove, first of all, that $H_{W}^{\ast}\in\Theta$ for every
$H\in\Theta$. Operations defined by $w_{0}$, $w_{-}$, $w_{+}$ coincide with
$0$, $-$, $+$. So identities of the variety $\Theta$ (axioms of the linear
algebra) relating to these operations fulfill in $H_{W}^{\ast}$. Hence we only
need to check the axioms that involve the operations defined by $w_{\cdot}$
and $w_{\lambda}$ ($\lambda\in k$). As above we denote these operations by
$\times$ and by $\lambda\ast$.

$\lambda\ast\left(x+y\right)=\varphi\left(\lambda\right)\left(x+y\right)=\varphi\left(\lambda\right)x+\varphi\left(\lambda\right)y=\lambda\ast
x+\lambda\ast y,$ $\left(\lambda\mu\right)\ast
x=\varphi\left(\lambda\mu\right)x=\varphi\left(\lambda\right)\varphi\left(\mu\right)x=\varphi\left(\lambda\right)\left(\mu\ast
x\right)=\lambda\ast\left(\mu\ast x\right),$ $\left(\lambda+\mu\right)\ast
x=\varphi\left(\lambda+\mu\right)x=\left(\varphi\left(\lambda\right)+\varphi\left(\mu\right)\right)x=\varphi\left(\lambda\right)x+\varphi\left(\mu\right)x=\lambda\ast
x+\mu\ast x,$ $1\ast x=\varphi\left(1\right)x=1x=x,$
$x\times\left(y+z\right)=ax\left(y+z\right)+b\left(y+z\right)x=axy+axz+byx+bzx=x\times
y+x\times z,$ $\left(y+z\right)\times
x=a\left(y+z\right)x+bx\left(y+z\right)=ayx+azx+bxy+bxz=y\times x+z\times x,$
$\lambda\ast\left(x\times
y\right)=\varphi\left(\lambda\right)\left(axy+byx\right)=a\left(\varphi\left(\lambda\right)x\right)y+by\left(\varphi\left(\lambda\right)x\right)=\left(\varphi\left(\lambda\right)x\right)\times
y=$ $\left(\lambda\ast x\right)\times y=x\times\left(\lambda\ast y\right)$

fulfills for every $x,y,z\in H$, $\lambda,\mu\in k$.

Hence there exists a homomorphism $\sigma_{F}:F\rightarrow F_{W}^{\ast}$ such
that $\sigma_{F}\mid_{X}=id_{X}$ for every
$F=F\left(X\right)\in\mathrm{Ob}\Theta^{0}$. Our goal is to prove that these
homomorphisms are isomorphisms. We will prove by induction by $i$ that

$\sigma_{F}\left(F_{i}\right)=F_{i}.$ (3.2)

for every $i\in\mathbb{N}$. If $X=\left\\{x_{1},\ldots,x_{n}\right\\}$ then
every element of $F_{1}$ has form $\lambda_{1}x_{1}+\ldots+\lambda_{n}x_{n}$,
where $\lambda_{1},\ldots,\lambda_{n}\in k$.

$\sigma_{F}\left(\lambda_{1}x_{1}+\ldots+\lambda_{n}x_{n}\right)=\lambda_{1}\ast\sigma_{F}\left(x_{1}\right)+\ldots+\lambda_{n}\ast\sigma_{F}\left(x_{n}\right)=\varphi\left(\lambda_{1}\right)x_{1}+\ldots+\varphi\left(\lambda_{n}\right)x_{n},$

so $\sigma_{F}\left(F_{1}\right)\subset F_{1}$.

$\sigma_{F}\left(\varphi^{-1}\left(\lambda_{1}\right)x_{1}+\ldots+\varphi^{-1}\left(\lambda_{n}\right)x_{n}\right)=\lambda_{1}x_{1}+\ldots+\lambda_{n}x_{n},$

so $\sigma_{F}\left(F_{1}\right)=F_{1}$.

Let (3.2) proved for $i$ such that $1\leq i<r$. Every element of $F_{r}$ is a
linear combination of the monomials of the form $uv$, where $u\in F_{i}$,
$v\in F_{j}$, $i+j=r$.

$\sigma_{F}\left(uv\right)=\sigma_{F}\left(u\right)\times\sigma_{F}\left(v\right)=a\sigma_{F}\left(u\right)\sigma_{F}\left(v\right)+b\sigma_{F}\left(v\right)\sigma_{F}\left(u\right),$

so $\sigma_{F}\left(F_{r}\right)\subset F_{r}$, because, by our assumption,
$\sigma_{F}\left(u\right)\in F_{i}$, $\sigma_{F}\left(v\right)\in F_{j}$.
Also, if $u=\sigma_{F}\left(\widetilde{u}\right)$,
$v=\sigma_{F}\left(\widetilde{v}\right)$, where $\widetilde{u}\in F_{r}$,
$\widetilde{v}\in F_{t}$, then

$\sigma_{F}\left(\widetilde{u}\widetilde{v}\right)=\sigma_{F}\left(\widetilde{u}\right)\times\sigma_{F}\left(\widetilde{v}\right)=u\times
v=auv+bvu$
$\sigma_{F}\left(\widetilde{v}\widetilde{u}\right)=\sigma_{F}\left(\widetilde{v}\right)\times\sigma_{F}\left(\widetilde{u}\right)=v\times
u=avu+buv=buv+avu,$

fulfills. $a\neq\pm b$, so the matrix $\left(\begin{array}[]{cc}a&b\\\
b&a\end{array}\right)$ is regular, hence there exist $\alpha,\beta\in k$ such
that

$uv=\alpha\sigma_{F}\left(\widetilde{u}\widetilde{v}\right)+\beta\sigma_{F}\left(\widetilde{v}\widetilde{u}\right)=\sigma_{F}\left(\varphi^{-1}\left(\alpha\right)\widetilde{u}\widetilde{v}+\varphi^{-1}\left(\beta\right)\widetilde{v}\widetilde{u}\right).$

Therefore $\sigma_{F}\left(F_{r}\right)=F_{r}$. We can conclude that
$\sigma_{F}$ is an epimorphism.

Now we will prove that $\ker\sigma_{F}=0$. Let $f\in\ker\sigma_{F}\subset
F\left(X\right)$. There exists $m\in\mathbb{N}$ such that
$f\in\bigoplus\limits_{i=1}^{m}F_{i}$.
$\sigma_{F}\left(\bigoplus\limits_{i=1}^{m}F_{i}\right)=\bigoplus\limits_{i=1}^{m}F_{i}$
by (3.2). $\sigma_{F}$ is a linear mapping from the linear space
$\bigoplus\limits_{i=1}^{m}F_{i}$ with the original multiplication by scalars
in $F$ to the $\left(\bigoplus\limits_{i=1}^{m}F_{i}\right)_{W}^{\ast}$ \- the
linear space $\bigoplus\limits_{i=1}^{m}F_{i}$ with the multiplication by
scalars which we denote by $\ast$. From formulas
$\sum\limits_{i=1}^{k}\left(\lambda_{i}\ast
e_{i}\right)=\sum\limits_{i=1}^{k}\varphi\left(\lambda_{i}\right)e_{i}$ and
$\sum\limits_{i=1}^{k}\lambda_{i}e_{i}=\sum\limits_{i=1}^{k}\left(\varphi^{-1}\left(\lambda_{i}\right)\ast
e_{i}\right)$ we can conclude that if $E$ is a basis of the linear space
$\bigoplus\limits_{i=1}^{m}F_{i}$ then $E$ is a basis of the linear space
$\left(\bigoplus\limits_{i=1}^{m}F_{i}\right)_{W}^{\ast}$. So
$\dim\bigoplus\limits_{i=1}^{m}F_{i}=\dim\left(\bigoplus\limits_{i=1}^{m}F_{i}\right)_{W}^{\ast}<\infty$,
therefore $\ker\left(\sigma_{F}\mid\bigoplus\limits_{i=1}^{m}F_{i}\right)=0$
and $f=0$.

##  4 Group $\mathfrak{A/Y}$.

From now on, $W$ is a system of words (3.1) which fulfills conditions Op1 and
Op2.

The decomposition (1.1) is not split in general case, i. e. $\mathfrak{S\cap
Y\neq}\left\\{1\right\\}$ in general case. The strongly stable automorphism
$\Phi$ of the category $\Theta^{0}$ which corresponds to the system of words
$W$ is inner, by [4, Lemma 3], if and only if for every
$F\in\mathrm{Ob}\Theta^{0}$ there exists an isomorphism $c_{F}:F\rightarrow
F_{W}^{\ast}$ such that $c_{F}\alpha=\alpha c_{D}$ fulfills for every
$\left(\alpha:D\rightarrow F\right)\in\mathrm{Mor}\Theta^{0}$ (by [5, Remark
3.1] $\alpha$ is also a homomorphism from $D_{W}^{\ast}$ to $F_{W}^{\ast}$).

Hear we need to prove one technical lemma.

###### Lemma 4.1

If $F=F\left(X\right)\in\mathrm{Ob}\Theta^{0}$ and $c_{F}:F\rightarrow
F_{W}^{\ast}$ is an isomorphism then there exists an isomorphism
$c_{i}:F/F^{i}\rightarrow F_{W}^{\ast}/F^{i}$ such that
$\chi_{i}^{\ast}c_{F}=c_{i}\chi_{i}$, where $\chi_{i}:F\rightarrow F/F^{i}$
and $\chi_{i}^{\ast}:F_{W}^{\ast}\rightarrow F_{W}^{\ast}/F^{i}$ are natural
homomorphisms, $i\in\mathbb{N}$.

Proof. If $H\in\Theta$ and $I$ is an ideal of $H$. If $\lambda\in k$, $y\in
I$, $h\in H$, then $\lambda\ast y=\varphi\left(\lambda\right)y\in I$, $y\times
h=ayh+bhy\in I$, analogously $h\times y\in I$. Therefore $I$ is an ideal of
$H_{W}^{\ast}$. Hence $F^{i}$ is an ideal of $F_{W}^{\ast}$.

If $\sigma_{F}:F\rightarrow F_{W}^{\ast}$ is an isomorphism such that
$\sigma_{F}\mid_{X}=id_{X}$, then by (3.2) we have
$c_{F}^{-1}\left(F^{i}\right)=c_{F}^{-1}\sigma_{F}\left(F^{i}\right)=F^{i}$
because $c_{F}^{-1}\sigma_{F}:F\rightarrow F$ is an isomorphism. So
$c_{F}\left(F^{i}\right)=F^{i}$. It finishes the proof.

###### Proposition 4.1

The strongly stable automorphism $\Phi$ which corresponds to the system of
words $W$ is inner if and only if $\varphi=id_{k}$ and $b=0$.

Proof. We suppose that strongly stable automorphism $\Phi$ which corresponds
to the system of words $W$ is inner. We assume that $\varphi\neq id_{k}$, i.,
e., there exists $\lambda\in k$ such that
$\varphi\left(\lambda\right)\neq\lambda$. We denote $F=F\left(x\right)$. We
take $\alpha\in\mathrm{End}F$, such that $\alpha\left(x\right)=\lambda x$. We
suppose that $c_{F}:F\rightarrow F_{W}^{\ast}$ is an isomorphism. $c_{2}$ is
defined as in the Lemma 4.1, and we by this Lemma we have:

$\chi_{2}^{\ast}c_{F}\left(x\right)=c_{2}\chi_{2}\left(x\right)=\mu\ast\chi_{2}^{\ast}\left(x\right)=\chi_{2}^{\ast}\left(\mu\ast
x\right)=\chi_{2}^{\ast}\left(\varphi\left(\mu\right)x\right),$

where operations in algebra $F_{W}^{\ast}/F^{2}$ we denote by same symbols as
operations in algebra $F_{W}^{\ast}$ and $\mu\in
k\setminus\left\\{0\right\\}$. Therefore
$c_{F}\left(x\right)\equiv\varphi\left(\mu\right)x\left(\mathop{\mathrm{m}od}F^{2}\right)$.
$\alpha\left(F^{2}\right)\subset F^{2}$ fulfils, so

$\alpha
c_{F}\left(x\right)=\alpha\left(\varphi\left(\mu\right)x+f_{2}\right)\equiv\alpha\left(\varphi\left(\mu\right)x\right)=\varphi\left(\mu\right)\alpha\left(x\right)=\varphi\left(\mu\right)\lambda
x\left(\mathop{\mathrm{m}od}F^{2}\right),$

where $f_{2}\in F^{2}$.

$c_{F}\alpha\left(x\right)=c_{F}\left(\lambda x\right)=\lambda\ast
c_{F}\left(x\right)=\varphi\left(\lambda\right)c_{F}\left(x\right)\equiv\varphi\left(\lambda\right)\varphi\left(\mu\right)x\left(\mathop{\mathrm{m}od}F^{2}\right).$

$\mu\neq 0$, so $\varphi\left(\mu\right)\neq 0$,
$\varphi\left(\lambda\right)\neq\lambda$ hence $\alpha c_{F}\neq c_{F}\alpha$.
This contradiction proves that $\varphi=id_{k}$.

Now we denote $F=F\left(x_{1},x_{2}\right)\in\mathrm{Ob}\Theta^{0}$. By our
assumption there exists an isomorphism $c_{F}:F\rightarrow F_{W}^{\ast}$ such
that $c_{F}\alpha=\alpha c_{F}$ fulfills for every $\alpha\in\mathrm{End}F$.
$c_{2}$ is defined as in the Lemma 4.1. $\alpha\left(F^{2}\right)\subset
F^{2}$ so we can define the homomorphism
$\widetilde{\alpha}:F/F^{2}\rightarrow F/F^{2}$ such that
$\widetilde{\alpha}\chi_{2}=\chi_{2}\alpha$. From $c_{F}\alpha=\alpha c_{F}$
we can conclude $c_{2}\widetilde{\alpha}=\widetilde{\alpha}c_{2}$ fulfills. By
Lemma 4.1 $c_{2}$ is a regular linear mapping. We can take the endomorphisms
$\alpha$ such that $\widetilde{\alpha}$ will be an arbitrary linear mapping
from $k^{2}$ to $k^{2}$. Therefore $c_{2}$ must be a regular linear mapping
from $k^{2}$ to $k^{2}$ which commutate with all linear mappings from $k^{2}$
to $k^{2}$. Hence $c_{2}$ must be a scalar mapping, i.e.,

$\chi_{2}^{\ast}c_{F}\left(x_{i}\right)=c_{2}\chi_{2}\left(x_{i}\right)=\lambda\chi_{2}^{\ast}\left(x_{i}\right)=\chi_{2}^{\ast}\left(\lambda
x_{i}\right),$

where $\lambda\in k\setminus\left\\{0\right\\}$, $i=1,2$. Therefore
$c_{F}\left(x_{i}\right)=\lambda x_{i}+f_{i}$, where $f_{i}\in F^{2}$,
$i=1,2$. We can remark that now we consider the case when $\varphi=id_{k}$,
hence we need not distinguish between multiplication by scalar in $F$ and
$F_{W}^{\ast}$.

Now we take $\alpha\in\mathrm{End}F$ such that
$\alpha\left(x_{1}\right)=x_{1}x_{2}$, $\alpha\left(x_{2}\right)=0$. If $u$ is
a monomial which contain only entries of $x_{1}$, then
$\deg_{x_{1}}\alpha\left(u\right)+\deg_{x_{2}}\alpha\left(u\right)=2\deg_{x_{1}}u$.
If a monomial $u$ contain at least one entry of $x_{2}$, then
$\alpha\left(u\right)=0$. Hence $\alpha\left(F^{2}\right)\subset F^{3}$. So we
have

$c_{F}\alpha\left(x_{1}\right)=c_{F}\left(x_{1}x_{2}\right)=c_{F}\left(x_{1}\right)\times
c_{F}\left(x_{2}\right)=ac_{F}\left(x_{1}\right)c_{F}\left(x_{2}\right)+bc_{F}\left(x_{2}\right)c_{F}\left(x_{1}\right)=$
$a\left(\lambda x_{1}+f_{1}\right)\left(\lambda
x_{2}+f_{2}\right)+b\left(\lambda x_{2}+f_{2}\right)\left(\lambda
x_{1}+f_{1}\right)\equiv
a\lambda^{2}x_{1}x_{2}+b\lambda^{2}x_{2}x_{1}\left(\mathop{\mathrm{m}od}F^{3}\right).$
$\alpha c_{F}\left(x_{1}\right)=\alpha\left(\lambda
x_{1}+f_{1}\right)\equiv\lambda
x_{1}x_{2}\left(\mathop{\mathrm{m}od}F^{3}\right).$

Hence we conclude $b=0$ from $c_{F}\alpha=\alpha c_{F}$.

If $b=0$, i. e., $w_{\cdot}=ax_{1}x_{2}$, $a\neq 0$, then we take
$c_{F}\left(f\right)=a^{-1}f$ for every $F\in\mathrm{Ob}\Theta^{0}$ and every
$f\in F$. It is obvious that $c_{F}$ is a regular linear mapping.

$c_{F}\left(f_{1}\right)\times
c_{F}\left(f_{2}\right)=ac_{F}\left(f_{1}\right)c_{F}\left(f_{2}\right)=a\left(a^{-1}f_{1}\right)\left(a^{-1}f_{2}\right)=a^{-1}f_{1}f_{2}=c_{F}\left(f_{1}f_{2}\right).$

for every $f_{1},f_{2}\in F$. So $c_{F}:F\rightarrow F_{W}^{\ast}$ is an
isomorphism. It fulfils

$c_{F}\alpha\left(d\right)=a^{-1}\alpha\left(d\right)=\alpha\left(a^{-1}d\right)=\alpha
c_{F}\left(d\right)$

for every $\left(\alpha:D\rightarrow F\right)\in\mathrm{Mor}\Theta^{0}$ and
every $d\in D$.

###### Proposition 4.2

The group $\mathfrak{S\cong}G\mathfrak{\leftthreetimes}\mathrm{Aut}k$, where
$G$ is the group of all regular $2\times 2$ matrices over field $k$, which
have a form $\left(\begin{array}[]{cc}a&b\\\ b&a\end{array}\right)$ and every
$\varphi\in\mathrm{Aut}k$ acts on the group $G$ by this way:
$\varphi\left(\begin{array}[]{cc}a&b\\\
b&a\end{array}\right)=\left(\begin{array}[]{cc}\varphi\left(a\right)&\varphi\left(b\right)\\\
\varphi\left(b\right)&\varphi\left(a\right)\end{array}\right)$.

Proof. We will define the mapping
$\tau:G\mathfrak{\leftthreetimes}\mathrm{Aut}k\rightarrow\mathfrak{S}$. If
$g\varphi\in G\mathfrak{\leftthreetimes}\mathrm{Aut}k$, where
$g=\left(\begin{array}[]{cc}a&b\\\ b&a\end{array}\right)$, then we define
$\tau\left(g\varphi\right)=\Phi\in\mathfrak{S}$, where $\Phi$ corresponds to
the system of words $W$ with $w_{\lambda}=\varphi\left(\lambda\right)x_{1}$
for every $\lambda\in k$ and $w_{\cdot}=ax_{1}x_{2}+bx_{2}x_{1}$. By Section 2
and Theorem 3.1 $\tau$ is bijection.

We consider $\tau\left(g_{1}\varphi_{1}\right)=\Phi_{1}$ and
$\tau\left(g_{2}\varphi_{2}\right)=\Phi_{2}$, where
$g_{1}\varphi_{1},g_{2}\varphi_{2}\in
G\mathfrak{\leftthreetimes}\mathrm{Aut}k$ and
$g_{1}=\left(\begin{array}[]{cc}a_{1}&b_{1}\\\ b_{1}&a_{1}\end{array}\right)$,
$g_{2}=\left(\begin{array}[]{cc}a_{2}&b_{2}\\\ b_{2}&a_{2}\end{array}\right)$.
Both these strongly stable automorphisms preserves all objects of $\Theta^{0}$
and acts on morphisms of $\Theta^{0}$ by theirs systems of bijections
$\left\\{s_{F}^{\Phi_{i}}:F\rightarrow F\mid
F\in\mathrm{Ob}\Theta^{0}\right\\}$, for $i=1,2$, according the formula (1.2).
We have
$\Phi_{2}\Phi_{1}\left(\alpha\right)=s_{F}^{\Phi_{2}}s_{F}^{\Phi_{1}}\alpha\left(s_{D}^{\Phi_{1}}\right)^{-1}\left(s_{D}^{\Phi_{2}}\right)^{-1}$
for every $\left(\alpha:D\rightarrow F\right)\in\mathrm{Mor}\Theta^{0}$. So
strongly stable automorphism
$\Phi_{2}\Phi_{1}=\tau\left(g_{2}\varphi_{2}\right)\tau\left(g_{1}\varphi_{1}\right)$
preserves all objects of $\Theta^{0}$ and acts on morphisms of $\Theta^{0}$ by
system of bijections

$\left\\{s_{F}^{\Phi_{2}}s_{F}^{\Phi_{1}}:F\rightarrow F\mid
F\in\mathrm{Ob}\Theta^{0}\right\\}.$

This system of bijections satisfies the conditions B1 and B2, so we can define
the words $w_{\lambda}^{\Phi_{2}\Phi_{1}}$ for every $\lambda\in k$ and
$w_{\cdot}^{\Phi_{2}\Phi_{1}}$ which correspond to the automorphism
$\Phi_{2}\Phi_{1}$ by formula (2.2). The words
$w_{\lambda}^{\Phi_{i}}(\lambda\in k)$ and $w_{\cdot}^{\Phi_{i}}$ which
correspond to the automorphism $\Phi_{i}$ have forms
$w_{\lambda}^{\Phi_{i}}=\varphi_{i}\left(\lambda\right)x_{1}(\lambda\in k)$
and $w_{\cdot}^{\Phi_{i}}=a_{i}x_{1}x_{2}+b_{i}x_{2}x_{1}$ for $i=1,2$. So

$w_{\lambda}^{\Phi_{2}\Phi_{1}}=s_{F}^{\Phi_{2}}s_{F}^{\Phi_{1}}\left(\lambda
x_{1}\right)=s_{F}^{\Phi_{2}}\left(w_{\lambda}^{\Phi_{1}}\right)=s_{F}^{\Phi_{2}}\left(\varphi_{1}\left(\lambda\right)x_{1}\right)=\varphi_{2}\left(\varphi_{1}\left(\lambda\right)\right)x_{1}=\left(\varphi_{2}\varphi_{1}\right)\left(\lambda\right)x_{1}$

for every $\lambda\in k$ and

$w_{\cdot}^{\Phi_{2}\Phi_{1}}=s_{F}^{\Phi_{2}}s_{F}^{\Phi_{1}}\left(x_{1}x_{2}\right)=s_{F}^{\Phi_{2}}\left(w_{\cdot}^{\Phi_{1}}\right)=s_{F}^{\Phi_{2}}\left(a_{1}x_{1}x_{2}+b_{1}x_{2}x_{1}\right)=$
$\varphi_{2}\left(a_{1}\right)s_{F}^{\Phi_{2}}\left(x_{1}x_{2}\right)+\varphi_{2}\left(b_{1}\right)s_{F}^{\Phi_{2}}\left(x_{2}x_{1}\right)=$
$\varphi_{2}\left(a_{1}\right)\left(a_{2}x_{1}x_{2}+b_{2}x_{2}x_{1}\right)+\varphi_{2}\left(b_{1}\right)\left(a_{2}x_{2}x_{1}+b_{2}x_{1}x_{2}\right)=$
$\left(\varphi_{2}\left(a_{1}\right)a_{2}+\varphi_{2}\left(b_{1}\right)b_{2}\right)x_{1}x_{2}+\left(\varphi_{2}\left(a_{1}\right)b_{2}+\varphi_{2}\left(b_{1}\right)a_{2}\right)x_{2}x_{1}.$

because $s_{F}^{\Phi_{i}}:F\rightarrow F_{W_{i}}^{\ast}$ is an isomorphism,
$i=1,2$. Hence

$\Phi_{2}\Phi_{1}=\tau\left(g_{2}\varphi_{2}\right)\tau\left(g_{1}\varphi_{1}\right)=\tau\left(g_{2}\varphi_{2}\left(g_{1}\right)\varphi_{2}\varphi_{1}\right)=\tau\left(g_{2}\varphi_{2}\cdot
g_{1}\varphi_{1}\right).$

###### Corollary 1

Group $\mathfrak{S\cap Y}$ is isomorphic to the group $k^{\ast}I_{2}$ of the
regular $2\times 2$ scalar matrices over field $k$.

Proof. By Propositions 4.1 and 4.2.

###### Corollary 2

$\mathfrak{A/Y\cong}\left(G\mathfrak{/}k^{\ast}I_{2}\right)\mathfrak{\leftthreetimes}\mathrm{Aut}k$.

Proof. By Proposition 4.2 and Corollary 1 we have that
$\mathfrak{A/Y\cong}\left(G\mathfrak{\leftthreetimes}\mathrm{Aut}k\right)\mathfrak{/}k^{\ast}I_{2}$.
And we have
$\left(G\mathfrak{\leftthreetimes}\mathrm{Aut}k\right)\mathfrak{/}k^{\ast}I_{2}\mathfrak{\cong}\left(G\mathfrak{/}k^{\ast}I_{2}\right)\mathfrak{\leftthreetimes}\mathrm{Aut}k$
because $k^{\ast}I_{2}\vartriangleleft G$ and for every
$\varphi\in\mathrm{Aut}k$ $\varphi\left(k^{\ast}I_{2}\right)\subset
k^{\ast}I_{2}$ fulfills.

The symmetric group of the set which has $2$ elements -
$\mathbf{S}_{\mathbf{2}}$ can be embedded in the multiplicative structure of
the algebra $\mathbf{M}_{\mathbf{2}}\left(k\right)$ of the $2\times 2$
matrices over field $k$:
$\mathbf{S}_{\mathbf{2}}\ni\left(12\right)\rightarrow\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\in\mathbf{M}_{\mathbf{2}}\left(k\right)$, so $G\cong
U\left(k\mathbf{S}_{\mathbf{2}}\right)$, where
$U\left(k\mathbf{S}_{\mathbf{2}}\right)$ is the group of all invertible
elements of the group algebra $k\mathbf{S}_{\mathbf{2}}$. Also
$k^{\ast}I_{2}\cong U\left(k\left\\{e\right\\}\right)$, where
$e\in\mathbf{S}_{\mathbf{2}}$, $k\left\\{e\right\\}$ is a subalgebra of
$k\mathbf{S}_{\mathbf{2}}$, $U\left(k\left\\{e\right\\}\right)$ is a group of
all invertible elements of this subalgebra. Therefore
$\mathfrak{A/Y\cong}\left(U\left(k\mathbf{S}_{\mathbf{2}}\right)\mathfrak{/}U\left(k\left\\{e\right\\}\right)\right)\mathfrak{\leftthreetimes}\mathrm{Aut}k$,
where every $\varphi\in\mathrm{Aut}k$ acts on the algebra
$k\mathbf{S}_{\mathbf{2}}$ by natural way:
$\varphi\left(ae+b\left(12\right)\right)=\varphi\left(a\right)e+\varphi\left(b\right)\left(12\right)$.

##  5 Example of two linear algebras which are automorphically equivalent but
not geometrically equivalent.

We take $k=\mathbb{Q}$. $\Theta$ will be the variety of all linear algebras
over $k$. $H$ will be the $2$-generated linear algebra, which is free in the
variety corresponding to the identity $\left(x_{1}x_{1}\right)x_{2}=0$. We
consider the strongly stable automorphism $\Phi$ of the category $\Theta^{0}$
corresponding to the system of words $W$, where $b\neq 0$. Algebras $H$ and
$H_{W}^{\ast}$ are automorphically equivalent by [5, Theorem 5.1].

###### Proposition 5.1

Algebras $H$ and $H_{W}^{\ast}$ are not geometrically equivalent.

Proof. Let $F=F\left(x_{1},x_{2}\right)$. The ideal
$I=Id\left(H,\left\\{x_{1},x_{2}\right\\}\right)$ of the all two-variables
identities which are fulfill in the algebra $H$ will be the smallest
$H$-closed set in $F$, because $I=\left(0\right)_{H}^{\prime\prime}$, where
$0\in F$. If algebras $H$ and $H_{W}^{\ast}$ are geometrically equivalent then
the structures of the $H$-closed sets and of the $H_{W}^{\ast}$-closed sets in
$F$ coincide. Hence $I$ must be the smallest $H_{W}^{\ast}$-closed set in $F$.

By [5, Remark 5.1]

$T\rightarrow\sigma_{F}T$ (5.1)

is a bijection from the structure of the $H_{W}^{\ast}$-closed sets in $F$ to
the structure of the $H$-closed sets in $F$. Hear $\sigma_{F}:F\rightarrow
F_{W}^{\ast}$ is an isomorphism from condition Op2. It is clear that the
bijection (5.1) preserves inclusions of sets. So it transforms the smallest
$H_{W}^{\ast}$-closed set to the smallest $H$-closed set, i. e.
$I=\sigma_{F}I$ must fulfills.

It is obviously that $I\subset F^{3}$. By (3.2) $\sigma_{F}I\subset F^{3}$. We
will compare the linear subspaces $I/F^{4}$ and
$\left(\sigma_{F}I\right)/F^{4}$.
$I=\left\langle\alpha\left(\left(x_{1}x_{1}\right)x_{2}\right)\mid\alpha\in\mathrm{End}F\right\rangle$.
Let
$\alpha\left(x_{i}\right)\equiv\alpha_{1i}x_{1}+\alpha_{2i}x_{2}\left(\mathop{\mathrm{m}od}F^{2}\right)$,
where $i=1,2$, $\alpha_{ji}\in k$. Then

$\alpha\left(\left(x_{1}x_{1}\right)x_{2}\right)\equiv\left(\left(\alpha_{11}x_{1}+\alpha_{21}x_{2}\right)\left(\alpha_{11}x_{1}+\alpha_{21}x_{2}\right)\right)\left(\alpha_{12}x_{1}+\alpha_{22}x_{2}\right)\left(\mathop{\mathrm{m}od}F^{4}\right).$

We achieve after the extending of brackets that $I/F^{4}$ is a subspace of the
linear space spanned by the elements of $F^{3}/F^{4}$ which have form
$\left(x_{i}x_{j}\right)x_{k}+F^{4}$, where $i,j,k=1,2$. But

$\sigma_{F}I\ni\sigma_{F}\left(\left(x_{1}x_{1}\right)x_{2}\right)=a\sigma_{F}\left(x_{1}x_{1}\right)\sigma_{F}\left(x_{2}\right)+b\sigma_{F}\left(x_{2}\right)\sigma_{F}\left(x_{1}x_{1}\right)=$
$=a\left(a+b\right)\left(x_{1}x_{1}\right)x_{2}+b\left(a+b\right)x_{2}\left(x_{1}x_{1}\right).$

We have that $a+b\neq 0$, $b\neq 0$, so
$I/F^{4}\neq\left(\sigma_{F}I\right)/F^{4}$ and $I\neq\sigma_{F}I$. This
contradiction proves that algebras $H$ and $H_{W}^{\ast}$ are not
geometrically equivalent.

## References

  * [1] B. Plotkin, Varieties of algebras and algebraic varieties. Categories of algebraic varieties. Siberian Advanced Mathematics, Allerton Press, 7:2 (1997), pp. 64 – 97.
  * [2] B. Plotkin, Some notions of algebraic geometry in universal algebra, Algebra and Analysis, 9:4 (1997), pp. 224 – 248, St. Petersburg Math. J., 9:4, (1998) pp. 859 – 879.
  * [3] B. Plotkin, Algebras with the same (algebraic) geometry, Proceedings of the International Conference on Mathematical Logic, Algebra and Set Theory, dedicated to 100 anniversary of P.S. Novikov, Proceedings of the Steklov Institute of Mathematics, MIAN, v.242, (2003), pp. 17 – 207.
  * [4] B. Plotkin, G. Zhitomirski On automorphisms of categories of free algebras of some varieties, Journal of Algebra, 306:2, (2006), 344 – 367.
  * [5] A. Tsurkov, Automorphic equivalence of algebras. International Journal of Algebra and Computation. 17:5/6, (2007), 1263–1271.


# Strongly interacting matter in a sphere at nonzero magnetic field

Bing-Jun Zuo<EMAIL_ADDRESS>Department of Physics, Nanjing
University, Nanjing 210093, China  Zheng Zhang  Department of Physics, Nanjing
University, Nanjing 210093, China  Chao Shi<EMAIL_ADDRESS>Department of
Nuclear Science and Technology, Nanjing University of Aeronautics and
Astronautics, Nanjing 210016, China  Yong-Feng Huang<EMAIL_ADDRESS>School of
Astronomy and Space Science, Nanjing University, Nanjing 210023, China  Key
Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry
of Education, Nanjing 210023, China  Xinjiang Astronomical Observatory,
Chinese Academy of Sciences, Urumqi 830011, China

###### Abstract

We investigate the chiral phase transition within a sphere under a uniform
background magnetic field. The Nambu–Jona-Lasinio (NJL) model is employed and
the MIT boundary condition is imposed for the spherical confinement. Using the
wave expansion method, the diagonalizable Hamiltonian and energy spectrum are
derived for the system. By solving the gap equation in the NJL model, the
influence of magnetic field on quark matter in a sphere is studied. It is
found that inverse magnetic catalysis occurs at small radii, while magnetic
catalysis occurs at large radii. Additionally, both magnetic catalysis and
inverse magnetic catalysis are observed at the intermediate radii ($R\approx
4$ fm).

##  I Introduction

It is found that finite size effects can have significant impacts on the state
of dense matter and phase transitions for systems whose scale ranges between 2
fm and 10 fm Chernodub and Gongyo (2017); Chen  _et al._ (2017); Sadooghi  _et
al._ (2021); Palhares  _et al._ (2011); Klein (2017). This help us better
understand the properties of Quantum Chromodynamics (QCD) fireball produced in
high energy heavy-ion collision (HIC) experiments. Furthermore, the magnetic
field strength ($B$) can reach up to $10^{15}$ — $10^{18}G$ in the HIC
Voronyuk  _et al._ (2011); Deng and Huang (2012), which can influence the
energy of the system and even lead to the breaking or restoration of chiral
symmetry Miransky and Shovkovy (2015).

In most conditions, the presence of an external magnetic field can lead to the
breakdown of $U(1)_{A}$ symmetry and dimensional reduction, resulting in an
increase in chiral condensate as the magnetic field increases, which is known
as magnetic catalysis Gusynin  _et al._ (1994); Ebert  _et al._ (2000);
Andersen  _et al._ (2016). However, as argued by the authors of Bali  _et al._
(2012a, b), it is also possible that the increase of the magnetic field at the
critical temperature ($T_{c}$) may lead to a decrease in the chiral
condensate, resulting in a reversal of the magnetic catalysis effect, referred
to as the inverse magnetic catalysis. The underlying reason for inverse
magnetic catalysis is still highly debated Bruckmann  _et al._ (2013); Fraga
_et al._ (2013); Ferrer  _et al._ (2015); Cao (2021). In this study, we will
further investigate the phenomena of (inverse) magnetic catalysis in a finite
size system. In this aspect, some authors have recently studied systems
confined in a cylindrical geometry Chen  _et al._ (2016); Sadooghi  _et al._
(2021). Meanwhile, quark matter confined within a sphere can sometimes be a
realistic condition. Therefore, we will consider a finite size system confined
in a spherical geometry in this study.

To calculate the effects of magnetic field on quark matter, QCD-like effective
models are commonly used. Here we will employ the Nambu–Jona-Lasinio (NJL)
model, which has been widely employed to study the chiral phase transitions
Buballa (2005). It is also important to consider appropriate boundary
conditions for a finite sphere. There are various boundary conditions, such as
the MIT boundary condition Chodos  _et al._ (1974), the periodic boundary
condition Rajaraman and Bell (1982), and the chiral condition Zhang  _et al._
(2020). The MIT boundary condition originates from the MIT bag model, which is
widely used for describing the strong interaction of quarks. Here we will
enforce the MIT boundary condition to describe an impenetrable spherical
cavity. Meanwhile, as an initial attempt, we will consider the one-flavor NJL
model as in Sadooghi  _et al._ (2021) for simplicity, and set the temperature
and baryon chemical potential to zero. Yet, extending the study to two flavor
case is also straightforward.

The results of this study may contribute to a better understanding and
modeling of the behavior of quark-gluon matter in heavy-ion collisions, where
strong magnetic fields and finite size effects could play a role.

This paper is organized as follows. In Section II, the mode solutions of free
fermions in a finite sphere with the MIT boundary condition are introduced.
The solutions are then extended to the case with a strong magnetic field by
using the mode expansion method. In Section III, by solving the gap equation
in the NJL model with the obtained modes, the fermion condensate and the
relationship between the magnetic field and the effective mass of quark matter
is obtained. The occurrence of magnetic catalysis and inverse magnetic
catalysis at different radii is explored. Possible explanations for the
occurrence of inverse magnetic catalysis are suggested. Finally, we present
our conclusions in Section IV.

##  II Effects of magnetic field on fermions in a sphere

###  II.1 Mode solutions in a sphere

Assuming that the magnetic field is uniform and is along the z-axis, the Dirac
equation of fermions is

$\displaystyle[i\gamma^{0}\partial_{t}$
$\displaystyle+i\gamma^{1}\left(\partial_{x}+iqeBy/2\right)+i\gamma^{2}\left(\partial_{y}-iqeBx/2\right)$
(1) $\displaystyle+i\gamma^{3}\partial_{z}-M]\psi=0,$

where $e$ is the charge of electrons (in natural units, $e=1/\sqrt{137}$), $q$
is the charge fraction of the fermions considered, and $qe$ is the charge of
the fermions. Here, we use the Pauli-Dirac representation of the gamma
matrices, i.e.

$\gamma^{0}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),\quad\gamma^{i}=\left(\begin{array}[]{cc}0&\sigma_{i}\\\
-\sigma_{i}&0\end{array}\right),$ (2)

where $\sigma_{i}$ are Pauli matrices. When the magnetic field strength is
zero (i.e. $B=0$), the above equation has a normal solution of
$\psi(x)=Xu(x)e^{-iEt}$, where the constant $X$ can be determined by using the
normalization condition. The free part of the Hamiltonian can be expressed as

$\displaystyle H_{0}=-i\gamma^{0}\gamma^{i}\partial_{i}+\gamma^{0}M.$ (3)

In the presence of a non-zero magnetic field ($B\neq 0$), the total
Hamiltonian is given by $H=H_{0}+H^{\prime}$, where

$\displaystyle
H^{\prime}=\frac{qeB}{2}(\gamma^{0}\gamma^{1}y-\gamma^{0}\gamma^{2}x).$ (4)

In our study, $H^{\prime}$ is expressed in spherical coordinates as

$\displaystyle
H^{\prime}=\frac{iqeBr}{2}\left(\begin{array}[]{cc}0&\bigtriangleup\\\
\bigtriangleup&0\end{array}\right),$ (7)

with

$\displaystyle\bigtriangleup=\sin\theta\left(\begin{array}[]{cc}0&e^{-i\varphi}\\\
-e^{i\varphi}&0\end{array}\right).$ (10)

Following Ref. Zhang  _et al._ (2020), we first consider the mode solutions
for $B=0$. In spherical coordinates, there are four commuting operators:
$\\{H_{0},J^{2},J_{z},K\\}$, where $J^{2}$ represents the total angular
operator and $K$ is defined as

$K=\gamma^{0}(L\cdot\Sigma+1).$ (11)

Here, $L$ represents the orbital angular momentum operator, and $\Sigma$ is
defined as

$\Sigma=\left(\begin{array}[]{cc}\sigma&0\\\ 0&\sigma\end{array}\right).$ (12)

The eigenstate can hence be characterized by the eigenvalues of the four
operators, namely $\\{E,j(j+1),m_{j},\kappa\\}$. For convenience, we will use
$k=(E,j,m_{j},\kappa)$ to denote each eigenstate from now on. The solution of
the Dirac equation in spherical coordinates reads

$\displaystyle
u_{k}(r,\theta,\varphi)=\left\\{\begin{array}[]{cl}&\left[\begin{array}[]{cc}{\sqrt{\frac{E+M}{2E}}j_{j-\frac{1}{2}}(pr)\chi^{+}_{jm_{j}}}\\\
{i\frac{E}{|E|}}\sqrt{\frac{E-M}{2E}}{j_{j+\frac{1}{2}}(pr)\chi^{-}_{jm_{j}}}\end{array}\right],\
\ \ \kappa>0,\\\
&\left[\begin{array}[]{cc}{\sqrt{\frac{E+M}{2E}}j_{j+\frac{1}{2}}(pr)\chi^{-}_{jm_{j}}}\\\
{-i\frac{E}{|E|}}\sqrt{\frac{E-M}{2E}}{j_{j-\frac{1}{2}}(pr)\chi^{+}_{jm_{j}}}\end{array}\right],\
\ \ \kappa<0,\end{array}\right.$ (13)

with $j=\frac{1}{2},\frac{3}{2},\cdots$, $m_{j}=-j,-j+1,\cdots,j$, and
$\kappa=\pm(j+\frac{1}{2})$.

The function $\chi^{\pm}_{jm_{j}}$ in Eq. (13) is the spherical harmonic
function, which is expressed as

$\displaystyle\chi_{jm_{j}}^{+}=\left(\begin{array}[]{l}\sqrt{\frac{j+m_{j}}{2j}}Y_{j-\frac{1}{2}}^{m_{j}-\frac{1}{2}}\\\
\sqrt{\frac{j-m_{j}}{2j}}Y_{j-\frac{1}{2}}^{m_{j}+\frac{1}{2}}\end{array}\right),$
(14)
$\displaystyle\chi_{jm_{j}}^{-}=\left(\begin{array}[]{c}\sqrt{\frac{j-m_{j}+1}{2(j+1)}}Y_{j+\frac{1}{2}}^{m_{j}-\frac{1}{2}}\\\
-\sqrt{\frac{j+m_{j}+1}{2(j+1)}}Y_{j+\frac{1}{2}}^{m_{j}+\frac{1}{2}}\end{array}\right).$

In this work, we adopt the MIT boundary condition, which requires that the
quark current vanishes on the boundary surface. Such a boundary condition can
be equivalently written as Greiner  _et al._ (2007)

$i\not{n}\psi(R)=\psi(R),$ (15)

where $R$ represents the radius of the boundary surface and
$\not{n}=\gamma^{\mu}n_{\mu}$, with $n_{\mu}$ being the normal to the
boundary. Using the boundary condition of Eq. (15), the allowed modes should
satisfy

$j_{l_{\kappa}}(pR)=\operatorname{sgn}(\kappa)\frac{p}{E+M}j_{\overline{l}_{\kappa}}(pR),$
(16)

with

$\displaystyle l_{\kappa}=\left\\{\begin{array}[]{cl}{\kappa-1}&{\text{ for
}\kappa>0}\\\ {-\kappa}&{\text{ for }\kappa<0}\end{array}\right.,$ (17)
$\displaystyle\overline{l}_{\kappa}=\left\\{\begin{array}[]{cc}{\kappa}&{\text{
for }\kappa>0}\\\ {-\kappa-1}&{\text{ for }\kappa<0}\end{array}\right..$

Here $j_{n}$ is the $n$th spherical Bessel function. According to Eq. (16),
the momentum $p$ gets discretized, and we label the i-th solution of Eq. (16)
as $p_{j\kappa,i}$. Using the on-shell condition, the energy $E$ can be
calculated from the momentum $p_{j\kappa,i}$, thus the label $k$ can be
equivalently written as

$k=(i,j,m_{j},\kappa).$ (18)

Imposing the normalization condition, the normalization constant of the Dirac
wave function in Eq. (13) is

$\displaystyle
X_{k}=\left\\{\begin{array}[]{cl}\frac{\sqrt{2}}{R|j_{j+\frac{1}{2}}(p_{j\kappa
i}R)|}\sqrt{\frac{E+M}{2ER-(2j+1)+\frac{M}{E}}},\ \ \kappa>0,\\\
\frac{\sqrt{2}}{R|j_{j-\frac{1}{2}}(p_{j\kappa
i}R)|}\sqrt{\frac{E+M}{2ER+(2j+1)+\frac{M}{E}}},\ \
\kappa<0.\end{array}\right.$ (19)

The mode solutions for a fermion under the MIT boundary condition can be
summarized as

$U_{k}(t,r,\theta,\phi)=X_{k}u_{k}(r,\theta,\phi)e^{-iEt}.$ (20)

Consequently, the solutions for the anti-fermion can be obtained via charge
conjugation

$V_{k}(t,r,\theta,\phi)=i\gamma^{2}U^{*}_{k}(t,r,\theta,\phi).$ (21)

###  II.2 Mode solutions at nonzero magnetic field

We now extend our analysis to take the magnetic field into account. Analytical
solutions of the Dirac equation in the presence of a magnetic field are not
directly available. Although perturbation methods can be used in certain
cases, they are less suitable for strong magnetic fields. Here we go beyond
the perturbative method and directly diagonalize the Hamiltonian. It can
potentially yield a more accurate result for the system with a strong magnetic
field. Such a method has been adopted in other studies, such as the electronic
structures in a magnetic field of a spherical quantum dot Wu and Wan (2012).

In the presence of a magnetic field, only $m_{j}$ and the energy $E$ remain to
be good quantum numbers. We denote the particle eigenstate as $S_{m_{j},n}$,
which is now characterized by $m_{j}$ and a new label $n$ for the energy
eigenvalue. It can be expressed using a complete set of orthonormalized
particle basis functions {$U_{k}$} as

$S_{m_{j},n}=\sum_{ij\kappa}c_{ij\kappa,n}U_{k}.$ (22)

Note $k=(i,j,m_{j},\kappa)$ as defined in Eq. (18). Correspondingly, the anti-
particle eigenstate ($T_{m_{j}\kappa,n}$) is

$T_{m_{j},n}=\sum_{ij\kappa}c^{*}_{ij\kappa,n}V_{k}.$ (23)

Replace $\psi$ in Eq. (1) with $S_{m_{j},n}$ and $T_{m_{j},n}$, we can get the
secular equation for a certain $m_{j}$

$\mid
H_{ij\kappa,i^{\prime}j^{\prime}\kappa^{\prime}}^{m_{j}}-E_{n}^{m_{j}}\delta_{ii^{\prime}}\delta_{jj^{\prime}}\delta_{\kappa\kappa^{\prime}}\mid=0.$
(24)

The matrix elements $H_{ij\kappa,i^{\prime}j^{\prime}\kappa^{\prime}}^{m_{j}}$
can be calculated as

$\displaystyle
H_{ij\kappa,i^{\prime}j^{\prime}\kappa^{\prime}}^{m_{j}}\equiv\langle
j^{\prime},m_{j},\kappa^{\prime},i^{\prime}\mid H\mid j,m_{j},\kappa,i\rangle$
(25)
$\displaystyle=E_{k}\delta_{ii^{\prime}}\delta_{jj^{\prime}}\delta_{kk^{\prime}}+\frac{qeBr}{2}\left[A_{1}\frac{4jm_{j}+2m_{j}}{2j(2j+2)}\delta_{jj^{\prime}}\delta_{kk^{\prime}}+A_{2}\left(\frac{\sqrt{(j+m_{j})(j-m_{j})}}{2j}\delta_{j^{\prime}(j-1)}+\frac{\sqrt{(j+m_{j}+1)(j-m_{j}+1)}}{2j+2}\delta_{j^{\prime}(j+1)}\right)\right],$

where $E_{k}=\sqrt{p_{j\kappa,i}^{2}+M^{2}}$. $A_{1}$ and $A_{2}$ are given by

$\displaystyle A_{1}$
$\displaystyle=\left\\{\begin{array}[]{cl}-\frac{(E_{k}^{\prime}+m)p\displaystyle\int_{0}^{R}r^{2}j_{j^{\prime}-\frac{1}{2}}(p^{\prime}r)j_{j+\frac{1}{2}}(pr)dr+(E_{k}+m)p^{\prime}\int_{0}^{R}r^{2}j_{j^{\prime}+\frac{1}{2}}(p^{\prime}r)j_{j-\frac{1}{2}}(pr)dr}{R^{2}|j_{j^{\prime}+\frac{1}{2}}(p^{\prime}R)j_{j+\frac{1}{2}}(pR)|\sqrt{2E_{k^{\prime}}^{2}R-E_{k^{\prime}}+M}\sqrt{2E_{k}^{2}R-E_{k}+M}}&{\text{
for }\kappa,\kappa^{\prime}>0}\\\
{-\frac{(E_{k}^{\prime}+m)p\displaystyle\int_{0}^{R}r^{2}j_{j^{\prime}+\frac{1}{2}}(p^{\prime}r)j_{j-\frac{1}{2}}(pr)dr+(E_{k}+m)p^{\prime}\int_{0}^{R}r^{2}j_{j^{\prime}-\frac{1}{2}}(p^{\prime}r)j_{j+\frac{1}{2}}(pr)dr}{R^{2}|j_{j^{\prime}-\frac{1}{2}}(p^{\prime}R)j_{j-\frac{1}{2}}(pR)|\sqrt{2E_{k^{\prime}}^{2}R+E_{k^{\prime}}+M}\sqrt{2E_{k}^{2}R+E_{k}+M}}}&{\text{
for }\kappa,\kappa^{\prime}<0}\end{array}\right.,$ (26) $\displaystyle A_{2}$
$\displaystyle=\left\\{\begin{array}[]{cl}-\frac{(E_{k}^{\prime}+m)(E_{k}+m)\displaystyle\int_{0}^{R}r^{2}j_{j^{\prime}-\frac{1}{2}}(p^{\prime}r)j_{j+\frac{1}{2}}(pr)dr+pp^{\prime}\int_{0}^{R}r^{2}j_{j^{\prime}+\frac{1}{2}}(p^{\prime}r)j_{j-\frac{1}{2}}(pr)dr}{R^{2}|j_{j^{\prime}+\frac{1}{2}}(p^{\prime}R)j_{j-\frac{1}{2}}(pR)|\sqrt{2E_{k^{\prime}}^{2}R-E_{k^{\prime}}+M}\sqrt{2E_{k}^{2}R+E_{k}+M}}&{\text{
for }\kappa<0,\kappa^{\prime}>0}\\\
{-\frac{(E_{k}^{\prime}+m)(E_{k}+m)\displaystyle\int_{0}^{R}r^{2}j_{j^{\prime}+\frac{1}{2}}(p^{\prime}r)j_{j-\frac{1}{2}}(pr)dr+p^{\prime}p\int_{0}^{R}r^{2}j_{j^{\prime}-\frac{1}{2}}(p^{\prime}r)j_{j+\frac{1}{2}}(pr)dr}{R^{2}|j_{j^{\prime}-\frac{1}{2}}(p^{\prime}R)j_{j+\frac{1}{2}}(pR)|\sqrt{2E_{k^{\prime}}^{2}R+E_{k^{\prime}}+M}\sqrt{2E_{k}^{2}R-E_{k}+M}}}&{\text{
for }\kappa>0,\kappa^{\prime}<0}\end{array}\right..$

Note we utilize the superscript symbol (′) to indicate the index of the bra
vector.

In deriving Eq. (25), we have used the following formulas

$\displaystyle e^{-i\varphi}\sin\theta Y_{l,m}=$
$\displaystyle\sqrt{\frac{(l+1-m)(l+2-m)}{(2l+1)(2l+3)}}Y_{l+1}^{m-1}$ (27)
$\displaystyle-\sqrt{\frac{(l+m)(l+m-1)}{(2l+1)(2l-1)}}Y_{l-1}^{m-1},$
$\displaystyle e^{i\varphi}\sin\theta Y_{l,m}=$
$\displaystyle-\sqrt{\frac{(l+m+1)(l+m+2)}{(2l+1)(2l+3)}}Y_{l+1}^{m+1}$ (28)
$\displaystyle+\sqrt{\frac{(l-m)(l-m-1)}{(2l+1)(2l-1)}}Y_{l-1}^{m+1}.$

Eq. (27) can be found in Ref. Cohen-Tannoudji  _et al._ (1986), and Eq. (28)
can be derived by applying complex conjugation to Eq. (27).

Finally, we can obtain the numerical solutions of the energy eigenvalues and
eigenstates by diagonalizing the Hamiltonian matrix in Eq. (25).

##  III FERMION CONDENSATE AND CHIRAL PHASE TRANSITION UNDER MAGNETIC FILED

We use the NJL model to describe the interaction between quarks. The
Lagrangian density representing the NJL model is

$\mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-\gamma^{\mu}qeA_{\mu}-m_{0})\psi+\frac{G}{2}[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma^{5}\psi)^{2}],$
(29)

where $G$ is the coupling constant, $m_{0}$ is the current quark mass, and
$qe$ is still the charge of the fermions. Here we take the chiral limit, i.e.
$m_{0}=0$. Although the term of $F_{\mu\nu}F^{\mu\nu}$ in Eq. (29) affects the
pressure, it is not relevant to the chiral phase transition. As a result, it
is neglected in our calculations.

According to the mean-field approximation, the gap equation can be derived by
considering the quark condensate as

$M=-G\langle\bar{\psi}\psi\rangle.$ (30)

In order to calculate the quark condensate, we need to perform a second
quantization. The fermion field operator can be expressed as

$\psi=\sum_{\lambda}\left[S_{\lambda}{\bf b_{\lambda}}+T_{\lambda}{\bf
d_{\lambda}^{\dagger}}\right].$ (31)

Here we use $\lambda=(m_{j},n)$ to identify different states. $S_{\lambda}$
and $T_{\lambda}$ are wave functions given by Eq. (22) and Eq. (23). The
operators ${\bf b_{\lambda}}$ and ${\bf d_{\lambda}^{\dagger}}$ are
annihilation and creation operators, satisfying the canonical anti-commutation
relations,

$\\{{\bf b_{\lambda}},{\bf
b_{\lambda^{\prime}}^{\dagger}}\\}=\delta(\lambda,\lambda^{\prime}),~{}~{}\\{{\bf
d_{\lambda}},{\bf
d_{\lambda^{\prime}}^{\dagger}}\\}=\delta(\lambda,\lambda^{\prime}).$ (32)

All other anti-commutation relations are zero. The vacuum state $\mid
0\rangle$ is defined by

${\bf b_{\lambda}}|0\rangle={\bf d_{\lambda}}|0\rangle=0.$ (33)

According to Ref. Landau and Lifshitz (1999); Zhang  _et al._ (2020), we have
the following relations of

$\displaystyle\langle{\bf b_{\lambda}^{\dagger}}{\bf
b_{\lambda^{\prime}}}\rangle$
$\displaystyle=\frac{1}{e^{\beta(E_{\lambda}-\mu)}+1}\delta(\lambda,\lambda^{\prime}),$
(34) $\displaystyle\langle{\bf d_{\lambda}}{\bf
d_{\lambda^{\prime}}^{\dagger}}\rangle$ $\displaystyle=1-\langle{\bf
d_{\lambda^{\prime}}^{\dagger}}{\bf
d_{\lambda}}\rangle=\left(1-\frac{1}{e^{\beta(E_{\lambda}+\mu)}+1}\right)\delta(\lambda,\lambda^{\prime}),$
$\displaystyle\langle{\bf b_{\lambda}^{\dagger}}{\bf
d_{\lambda^{\prime}}^{\dagger}}\rangle$ $\displaystyle=\langle{\bf
d_{\lambda}}{\bf b_{\lambda^{\prime}}}\rangle=0,$
$\displaystyle\bar{V}_{\lambda}{V}_{\lambda}$
$\displaystyle=-\bar{U}_{\lambda}{U}_{\lambda},$

where $E_{\lambda}$ represents the eigenenergy in the presence of a magnetic
field. Note that in the absence of a magnetic field, the eigenstate is denoted
as $k=(i,j,m_{j},\kappa)$ and
$k^{\prime}=(i^{\prime},j^{\prime},m_{j},\kappa^{\prime})$, whereas in the
presence of a magnetic field, it is denoted as $\lambda=(m_{j},n)$.

Substituting the above equations into $\langle\bar{\psi}\psi\rangle$, we have

$\langle\bar{\psi}\psi\rangle=-\sum_{\lambda}w(E_{\lambda})\sum_{k}\sum_{k^{\prime}}|c_{ij\kappa,n}||c_{i^{\prime}j^{\prime}\kappa^{\prime},n}|\bar{{{U}}}_{k}U_{k^{\prime}},$
(35)

where

$w(E_{\lambda})=1-\frac{1}{e^{\beta\left({E_{\lambda}}+\mu\right)}+1}-\frac{1}{e^{\beta\left({E_{\lambda}}-\mu\right)}+1}.$
(36)

$\bar{{{U}}}_{k}U_{k^{\prime}}$ can be derived from Eqs. (13) and (19), which
depends on the coordinate $r$ inside the sphere. To consider the magnetic
field’s impact on the entire sphere, we can calculate the average value of
$\bar{{{U}}}_{k}U_{k}$ as

$\overline{\bar{{{U}}}_{k}U_{k^{\prime}}}=\frac{1}{V}\int_{V}\bar{{{U}}}_{k}U_{k^{\prime}}~{}dV.$
(37)

In our study, the quark condensate
$\langle\bar{\psi}\psi\rangle_{\text{main}}$ is introduced, which is defined
as

$\displaystyle\langle\bar{\psi}\psi\rangle_{\text{main}}=$
$\displaystyle-\sum_{\lambda}w(E_{\lambda})\sum_{ij}\frac{1}{V}|c_{ij\kappa,n}|^{2}$
(38)
$\displaystyle\frac{-\text{sgn}(\kappa)E_{k}+(2j+1)M+2E_{k}RM}{2E^{2}_{k}R-\text{sgn}(\kappa)(2j+1)E_{k}+M}.$

$\langle\bar{\psi}\psi\rangle_{\text{main}}$ can be calculated by taking terms
in index $k$ and $k^{\prime}$ from $\langle\bar{\psi}\psi\rangle$ in Eq. (35).
It is found that $\langle\bar{\psi}\psi\rangle$ is contributed mainly by
$\langle\bar{\psi}\psi\rangle_{\text{main}}$. With the help of
$\langle\bar{\psi}\psi\rangle_{\text{main}}$, we can analyze the behavior of
$\langle\bar{\psi}\psi\rangle$ in some simple cases. When $R$ is large enough
and $B$ is very small, $\langle\bar{\psi}\psi\rangle$ reduces to the normal
quark condensate (in an infinite space): the coefficients $c_{ij\kappa,n}$
equal 1 and
$\langle\bar{\psi}\psi\rangle=\langle\bar{\psi}\psi\rangle_{\text{main}}$ (for
$B=0$). Meanwhile, when $R$ approaches infinity, the fraction term in
$\langle\bar{\psi}\psi\rangle_{\text{main}}$ approaches $\frac{M}{E_{k}}$.
Additionally, the summation in Eq. (38) will be replaced by an integral over
the momentum $p$. Thus quark condensate is reduced to

$\langle\bar{\psi}\psi\rangle\xlongequal[R\rightarrow\infty]{B\rightarrow
0}-\int\frac{d^{3}p}{(2\pi)^{3}}w(E_{k})\frac{M}{E_{k}}.$ (39)

The non-renormalizability of the NJL model requires that a regularization
scheme should be applied. Here we use the three-momentum cutoff scheme.
Following Chen  _et al._ (2016, 2017); Sadooghi  _et al._ (2021), we take the
cutoff momentum and the charge fraction as

$\Lambda=1000\ \text{MeV},\ \ \ \ \ \ q=1.$ (40)

In case of a zero temperature, the gap equation (30) can be written as

$\displaystyle M=$
$\displaystyle\frac{1}{V}\sum_{\lambda}\varTheta(\Lambda-\bar{p})\sum_{k}\sum_{k^{\prime}}|c_{ij\kappa,n}||c_{i^{\prime}j^{\prime}\kappa^{\prime},n}|\overline{\bar{{{U}}}_{k}U_{k^{\prime}}},$
(41)

where $\varTheta$ is the Heaviside function and $\bar{p}$ is the expected
value of the momentum, $\langle\lambda|p|\lambda\rangle$. The coefficients
$c_{ij\kappa,n}$ can be determined by solving the secular equation Eq.(24).
Since we have adopted a cutoff momentum of $\Lambda$, the dimension of the
Hamiltonian matrix will be reduced from infinite to a finite number, thus
making it numerically solvable.

(a) $G=24/\Lambda^{2}$

(b) $G=32/\Lambda^{2}$

Figure 1: Variation of the effective quark mass in a uniform magnetic field.
The quarks are confined in a sphere with a radius of $R=2,6,10$ fm,
respectively. Panels (a) and (b) correspond to $G=24/\Lambda^{2}$ and
$G=32/\Lambda^{2}$, respectively.

The effects of a strong magnetic field on the effective quark mass for quarks
confined in a sphere are illustrated in Fig. 1. We consider two different
coupling constants, $G=24/\Lambda^{2}$ and $G=32/\Lambda^{2}$ in Fig. 1. In
both cases, we see that the effective mass of quarks increases with the
strength of the magnetic field when $R$ is large. This phenomenon is exactly
the so called chiral magnetic catalysis. In general, such a chiral magnetic
catalysis is also observed in the standard NJL model in an infinite space
Gusynin  _et al._ (1996); Andersen  _et al._ (2016); Menezes  _et al._ (2009).
Moreover, when $R$ becomes small, the inverse magnetic catalysis is observed,
i.e, the effective mass of quarks decreases with the increase of the magnetic
field.

The influence of the radius, $R$, on the occurrence of magnetic catalysis or
inverse magnetic catalysis can be observed more clearly in Fig. 2.
Specifically, magnetic catalysis is observed at $R=5$ fm, inverse magnetic
catalysis occurs at $R=3$ fm, and the case of $R=4$ fm falls between these two
cases. It is well know that, at $T=0$, the standard NJL does not exhibit
inverse magnetic catalysis. To explore the potential causes of the inverse
magnetic catalysis here, we should return to Eq. (38) again. When $R$ is
small, the contribution of the Lowest Orbital Level (LOL), i.e.,
$j=\frac{1}{2}$ and $\kappa>0$, reduces to

$\langle\bar{\psi}\psi\rangle_{\text{main}}\xlongequal[\text{LOL}]{R\rightarrow
0}-\sum_{\lambda}w(E_{\lambda})\sum_{i}\frac{1}{V}|c_{i,n}|^{2}\frac{-E_{k}+2M}{-2E_{k}+M}.$
(42)

Since $E_{k}=\sqrt{p_{j\kappa,i}^{2}+M^{2}}$, the contribution of these
momentum modes to quark condensate will be positive for
$p_{j\kappa,i}^{2}<\frac{M^{2}}{3}$, yielding an anomaly value. With the
increase of the magnetic field, the energy gap between the orbital levels
increases, causing the system to prefer the LOL and resulting in the inverse
magnetic catalysis. Another factor is the intrinsic truncation caused by the
small radius, which prevents the increase of Landau levels and the density of
states. It further contributes to the inverse magnetic catalysis.

Figure 2: Variation of the effective quark mass in a uniform magnetic field
near the critical radius for $G=32/\Lambda^{2}$.

Some oscillations could be seen in Fig. 1 when $R=6,10$ fm. They are quite
similar to the de Haas-van Alphen oscillations Lifshitz and Pitaevskii (2013).
Such an oscillation behavior could be caused by the variation of the density
of states due to the Landau quantization, which has also been observed in the
standard NJL model Chatterjee  _et al._ (2011); Cao and Li (2023). On the
other hand, when the radius is small, the intrinsic truncation imposes a
cutoff on the Landau levels, suppressing the variation of the density of
states. Consequently, the de Haas-van Alphen oscillations cease at small
radii.

##  IV  SUMMARY AND DISCUSSION

In this study, quark matter confined in a sphere with a strong uniform
magnetic field is studied. The wave functions and energy levels are solved by
diagonalizing the Hamiltonian numerically, using eigen solutions of
Hamiltonian with a zero magnetic field as basis. The NJL model is employed,
and by solving its gap equation, the inverse magnetic catalysis effect and
magnetic catalysis effect is studied for a confined sphere with various radii.
It is found that when the radius of the sphere is large, magnetic catalysis
occurs, whereas when the radius is small, inverse magnetic catalysis occurs.
At the intermediate region ($R\approx 4$ fm), both phenomena are present. It
is argued that the inverse magnetic catalysis could be caused by the anomalous
contribution from the LOL. Additionally, the intrinsic truncation due to small
radius prevents the increase of Landau levels and density of states, which may
also contribute to the inverse magnetic catalysis.

For simplicity, we mainly adopt the one-flavor NJL model with $q=1$ to
investigate the chiral phase transition in our study. In fact, we have also
performed calculations for the two-flavor NJL model, in which
$q_{u}=\frac{2}{3}$ and $q_{d}=-\frac{1}{3}$. It is found that the results are
generally similar. In the future, more realistic conditions should be
considered. For example, the study of finite temperature effects and the
impact of nonzero current quark mass in a two-flavor model might be of special
interest, which could provide additional insights into the behavior of QCD
fireballs produced in heavy-ion collisions under realistic conditions.

## Acknowledgements

We thank Yong-Hui Xia for helpful discussions. This study is supported by the
National Natural Science Foundation of China (Grant Nos. 12233002, 12041306),
by the National Key R&D Program of China (2021YFA0718500), by National SKA
Program of China No. 2020SKA0120300, and by the Fundamental Research Funds for
the Central Universities, NO. 1227050553. YFH also acknowledges the support
from the Xinjiang Tianchi Program.

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