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Automorphisms of the semigroup of endomorphisms of free algebras of homogeneous varieties

We consider homogeneous varieties of linear algebras over an associative-commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F be a free algebra of some variety A of linear algebras over K freely generated by a finite set X, EndF be the semigroup of endomorphisms of F, and AutEndF be the group of automorphisms of the semigroup EndF. We investigate structure of the group AutEndF and its relation to the algebraical and categorical equivalence of algebras from A. We define a wide class of R1MF-domains containing, in particular, Bezout domains, unique factorization domains, and some other domains. We show that every automorphism of semigroup EndF, where F is a free finitely generated Lie algebra over an R1MF-domain, is semi-inner. This solves the Problem 5.1 left open in [21]. As a corollary, semi-innerity of all automorphism of the category of free Lie algebras over R1MF-domains is obtained. Relations between categorical and geometrical equivalence of Lie algebras over R1MF-domains are clarified. The group AutEndF for the variety of m-nilpotent associative algebras over R1MF-domains is described. As a consequence, a complete description of the group of automorphisms of the full matrix semigroup of n x n matrices over R1MF-domains is obtained. We give an example of the variety of linear algebras over a Dedekind domain such that not all automorphisms of AutEndF are quasi-inner. The results obtained generalize the previous studies of various special cases of varieties of linear algebras over infinite fields.


Introduction
Let us recall the main definitions from Universal Algebraic Geometry [23,24]. Let Θ be a variety of algebras over a commutative-associative ring K and F = F (X) be a free algebra from Θ generated by a finite set X. Here X is supposed to be a subset of some infinite universum X 0 . The set Hom(F, G), G ∈ Θ, can be treated as an affine space whose points are homomorphisms. The algebraic set in Hom(F, G) and the category K Θ (G) of algebraic sets over G can be defined. The category K Θ (G) is a geometric invariant of algebra G. Algebras G 1 and G 2 from Θ are categorically equivalent if the categories K Θ (G 1 ) and K Θ (G 2 ) are correctly isomorphic. Algebras G 1 and G 2 are geometrically equivalent if holds for all finite sets X and for all binary relations T on F and ′ is Galois correspondence between sets in Hom(F, G) and the binary relations on F .
It has been shown in [24] that categorical and geometrical equivalences of algebras are related and their relation is determined by the structure of the group Aut Θ 0 , where Θ 0 is the category of free finitely generated algebras of Θ. Note that the category Θ 0 is small. The group Aut Θ 0 is known for the following varieties: the variety of all groups, the variety of F -groups, where F is a free group of constants, the variety of all semigroups, the variety of commutative-associative algebras with unit element over infinite fields, the variety of associative algebras over infinite fields, the variety of all Lie algebras over infinite fields, the variety of modules over IBN -rings [2,3,15,16,20,21,27].
There is a natural connection between a structure of the groups Aut End F, F ∈ Θ, and Aut Θ 0 . However, a problem of description of the group Aut End F is more complicated and was solved only for the following varieties: the variety of inverse semigroup, the variety of semigroups, the variety of groups, the variety of associative-commutative algebras over infinite fields, and the variety of Lie algebras over infinite fields [4,8,19,20,27].
We define a class of R 1 MF-domains, containing Bezout domains, unique factorization domains, and some other domains. Namely, a domain K is called R 1 MFdomain if each n × m matrix A over K of rank 1 can be represented as a product of an n × 1 matrix by an 1 × m matrix over K. Here by the rank of matrix we understand its rank over the quotient fieldK of K.
Our aim here is to describe the group Aut End F and, as a consequence, to obtain a description of the group Aut Θ 0 for the variety of Lie algebras and the variety of nilpotent associative algebras over R 1 MF-domains.
The main theorems are as follows: Theorem A. Let L be the variety of Lie algebras over an R 1 MF-domain K and F = F (x 1 , .., x n ) be a finitely generated free Lie algebra of L. Then any automorphism of the group Aut End F is semi-inner. This Theorem solves Problem 5.1 in [21] formulated there for the variety of Lie algebras over fields. The description of the group AutEnd F (x 1 , x 2 ) for the variety of Lie algebras over infinite fields has been given in [21]. The group Aut End F (x 1 , .., x n ), n ≥ 2, for the variety of Lie algebras over infinite fields was described in [27].
Theorem B. Let N m be the variety of m-nilpotent (m ≥ 2) associative algebras over an R 1 MF-domain K and F m = F m (X), |X| < ∞, be a finitely generated free algebra of the variety N m . The following three statements hold: 1. If either (a) |K| = p k , k ≥ 1, p = 2, and p k |m − 1, and m = 2r, r > 1, or (b) |K| = 2 k , k ≥ 1, (i.e. p = 2), and 2 k | m − 1, then the group Aut End F m is generated by semi-inner, p-semi-inner, mirror and p-mirror automorphisms of End F m .
2. If |K| = ∞ or |K| = p k , k ≥ 1, and either (c) p k |m − 1 and m = 2r + 1, r ≥ 0, or (d) p k ∤ m − 1 and m = 2, then the group Aut End F m is generated by semi-inner and mirror automorphisms of End F m . 3. If K is any R 1 MF-domain and m = 2, i.e., the multiplication in F m is trivial, then any automorphism of the group Aut End F m is semi-inner. This result generalizes [13] on automorphisms of the full matrix semigroups of n × n matrices over principal ideal domains (see also [9], [10], where this result has been proved for the full matrix semigroup over fields).
Using Theorem A we further prove Theorem C. Every automorphism of the category L • of Lie algebras over an R 1 MF-domain is semi-inner.
Earlier, this theorem has been proved for the case of Lie algebras over infinite fields [21]. Using Theorem B, a description of automorphisms of the category N • m can be derived.
We give also an example of variety Θ of linear algebras over a Dedekind domain for which the group AutEnd F contains an automorphism which is not quasi-inner. Note that all automorphisms of End F are quasi-inner in all above-mentioned varieties of algebras over R 1 MF-domains.
The outline of this paper is as follows. We prove that all automorphisms of End F for the varieties B 2 and N m are quasi-inner (see Theorem 5.3). Then we describe the bijections related to these quasi-inner automorphisms. To this end we investigate the structure of derivative algebras associated with quasi-inner automorphisms (see Propositions 6.6 and 6.7). Such a relation between quasi-inner automorphisms and derivative algebras was noted first in [27]. We prove the main statements of our paper, Theorems A, B, C, and discuss the relation between the categorical and geometrical equivalences of Lie algebras over R 1 MF-domains (see Remark 7.1). Finally, we give the description of quasi-inner automorphisms of the semigroup End A(X) for the variety of associative algebras over domains.
2. Automorphisms of the semigroup End F and of the category Θ 0 Throughout this paper "ring" will mean "commutative-associative ring with 1". Let F = F (x 1 , ..., x n ) be a free algebra of a variety Θ of linear algebras over ring K generated by a set X = {x 1 , ..., x n }.
3. s(αa) = ϕ(α)s(a), for all α ∈ K and a, b ∈ F and an automorphism ϕ : K → K. If ϕ is the identity automorphism of K, we say that Φ is an inner.
Let A = A(x 1 , ..., x n ) be a finitely generated free associative algebra of the variety Ass-K of associative algebras over K.
Now we introduce a new class of quasi-inner automorphisms. Let A = A(X) be a free finitely generated associative algebra over a ring K of characteristic p > 0.
Recall the notions of category isomorphism and equivalence [17]. An isomorphism ϕ : C → D of categories is a functor ϕ from C to D which is a bijection both on objects and morphisms. In other words, there exists a functor ψ : D → C such that ψϕ = 1 C and ϕψ = 1 D .
Let ϕ 1 and ϕ 2 be two functors from C 1 to C 2 . A functor isomorphism s : i.e., the following diagram is commutative The isomorphism of functors ϕ 1 and ϕ 2 is denoted by ϕ 1 ∼ = ϕ 2 . An equivalence between categories C and D is a pair of functors ϕ : C → D and ψ : D → C together with natural isomorphisms ψϕ ∼ = 1 C and ϕψ ∼ = 1 D . If C = D, then we get the notions of automiorphism and autoequivalence of the category C.
For every small category C denote the group of all its automorphisms by Aut C. We will distinguish the following classes of automorphisms of C.
Definition 2.5. [15,21] An automorphism ϕ : C → C is equinumerous if ϕ(A) ∼ = A for any object A ∈ Ob C ; ϕ is stable if ϕ(A) = A for any object A ∈ Ob C ; and ϕ is inner if ϕ and 1 C are naturally isomorphic, i.e., ϕ ∼ = 1 C . In other words, an automorphism ϕ is inner if for all A ∈ Ob C there exists an isomorphism s A : A → ϕ(A) such that Let Θ be a variety of linear algebras over K. Denote by Θ 0 the full subcategory of finitely generated free algebras F (X), |X| < ∞, of the variety Θ.
Further, we will need the following Proposition 2.8. [15,21] For any equinumerous automorphism ϕ ∈ Aut C there exists a stable automorphism ϕ S and an inner automorphism ϕ I of the category C such that ϕ = ϕ S ϕ I .

Quasi-inner automorphisms of End F for varieties of linear algebras
Now we introduce standard endomorphisms in free algebra F = F (x 1 , ..., x n ) of a variety Θ.
Consider the following elements u 1 , ..., u n from F : Since σ is an automorphism of F , u 1 , ..., u n is a base of F . Let us show that endomorphisms Φ(e ij ), i, j ∈ 1, .., n are standard endomorphisms in the base U = {u 1 , ..., u n } : Conversely, let Φ(e ij ), i, j ∈ [1n], be standard endomorphisms of F in a base U = {u 1 , ..., u n }. Denote by µ ka , a ∈ F, k ∈ [1n], endomorphisms of F given on generators X by the rules: µ ka (u m ) = δ km a, u m ∈ U . Then µ ka e kk = µ ka . It is clear that if ρe kk = ρ for some k and ρ ∈ End F , then there exists a ∈ F , such that ρ = µ ka .
The following Lemma establishes a useful property of X σ -matrix of an automorphism Φ we need below. Lemma 3.6. Let α 1 , ..., α n be endomorphisms of F = F (x 1 , ..., x n ). Then there exists an endomorphism α of F such that ij ) for all i, j ∈ [1n] and some σ ∈ S n . There exists a unique endomorphism α of F such that (3.1) is fulfilled for all i, j ∈ [1n] and all σ ∈ S n .

R 1 MF-domains
Let K be an integral domain (domain, for short) andK = F rac K be the quotient field of K.
Definition 4.1. Rank of a matrix A ∈ M n×m (K) is the rank of A over the field K, i.e., rank K A := rankK A. Definition 4.2. We say that a domain K satisfies the rank-1-matrix factorization condition (R 1 MF-condition) if each n × m matrix A over K of rank 1 can be presented as a product of an n × 1 matrix c i by an 1 × m matrix d i over K, i.e., 3. An n × m matrix A is equivalent to an n × m matrix B over a domain K if there exist invertible matrices P and Q such that A = P BQ. Recall (see [14]) that an elementary divisor domain (EDD, for short) K is a domain with the property that each matrix A over K is equivalent to a diagonal matrix Example 4.4. A Bezout domain is a domain in which any finitely generated ideal is principal, (see [6,11]). By [6], Proposition 4.4, every Bezout domain is an R 1 MFdomain. Note that every EDD is a Bezout domain. However, inverse inclusion is still an open question (see [11]). Example 4.5. Let K be a unique factorization domain (UFD, for short). Let us show that K is an R 1 MF-domain.
Let A = (a ij ) ∈ M n×m (K) and rank A = 1. Then there exists g ∈ K such that gA = c 1 · d 1 , where c 1 = (c k1 ) and Assume that g = p is a prime element in K. Since K is UFD, we have p|c i1 or p|d 1j . If p|c i1 for all i then our statement is true. Let there exist s ≤ n such that p|c k1 for all k < s but p ∤ c s1 . Then p|d 1j for all j and this yields the assertion. Now, let g be a non-prime element in K. We can represent g = p 1 ...p r , r > 1, where all p i are prime elements in K. Using induction on r we obtain the proof of this statement in the general case.
Note that the group algebra over field P of the additive group of rational numbers (written multiplicatively) is a Bezout domain but not UFD (see [6], exercise 3.5 ). This suggests that the classes of UFDs and of R 1 MF-domains do not coincide.

Now consider an example of domain which is not an R
Example 4.6. A domain K with quotient fieldK is called a Dedekind domain if it satisfies any of the following equivalent conditions: As follows from Corollary 5.6 (or Remark 5.7) the Dedekind domain It is proved in [14] that a domain K is EDD if and only if every 2 × 1 and 2 × 2 matrices over K are equivalent to a diagonal matrix. Our hypothesis is: a domain K is R 1 MF-domain if and only if each 2 × 2 matrix over K of rank 1 can be represented as a column-by-row product.
Note that there is a notion of rank of matrix over any ring (see, for example, [6], [11]). Thus the notion of R 1 MF-domain can be generalized to non-commutative rings. In future we are going to study such a generalization.
Let V be a free module of finite rank n over an R 1 MF-domain K, P be a subsemigroup of End V generated by non-zero elements Proof. Denote by VK = V KK a vector space over the quotient fieldK of the domain K Let dim VK = n. Since P 11 = 0, there exists u ∈ V such that u 1 = From this follows that the rank of every We may assume that , a basis of the module V over K. It is easy to check that the elements P ij can be represented by the elementary matrices in this basis. This completes the proof.

Homogeneous varieties of algebras and quasi-inner automorphisms of End F
Let Θ be a variety of linear algebras over a ring K and T (Θ) be its T-ideal.
It is known that the varieties of associative algebras, of nilpotent associative algebras, of Lie algebras, of alternative algebras, of Jordan algebras (if 1/2 ∈ K) are homogeneous varieties ( [26]).
Further, we consider only homogeneous varieties with free algebras without 1. A free algebra F (X) of such a variety can be naturally decomposed as where F (k) is a K-submodule generated by all monomial of F (X) of degree k.
Denote by F ′ = ∞ k=2 F (k) . It is easy to prove the following be the variety of linear algebras over an R 1 MF-domain defined by the identity x 2 1 = 0, and N m , m ≥ 2, be the variety of nilpotent algebras of class ≤ m over an R 1 MF-domain, i.e., for every algebra G ∈ N m holds G m = 0. We will say, somewhat not rigorously, that N m is the variety of m-nilpotent algebras.
Proposition 5.3. Let Θ be a homogeneous variety of linear algebras over an R 1 MF-domain K and F = F (x 1 , ..., x n ) be its free algebra. If either Θ ⊆ B 2 or Θ ⊆ N m for some m ≥ 2, then all automorphisms of the semigroup End F are quasi-inner.
Proof. Let Φ be an automorphism of End F and σ ∈ S n . Consider the X σ -matrix T X can be written in the following form: ij for some i, j and σ ∈ S n . Assume, on the contrary, that m (σ) ij = 0 for all i, j and σ ∈ S n . Consider the following n endomorphisms of F : α 1 = e 11 , ..., α n = e nn . Note that by our assumption t ij ∈ F ′ . As a consequence, e ii (t (σ) ij ) = 0. By Lemma 3.6, there exists a unique endomorphism α ∈ End F such that However, e 11 (t and all σ ∈ S n . We arrived at a contradiction with the uniqueness of the endomorphism α. Let us fix i, j ∈ [1n] and σ ∈ S n for which m (σ) ij = 0 and write the elements Φ(e ij )x σ(k) , k ∈ [1n], in the ordered base X (σ) = x σ(1) , ..., x σ(n) : Denote by M n (K) the full matrix semigroup of n × n matrices over K. There exists a mapping ψ : km ) is the matrix of the linear part of (5.2). By Lemma 5.2 and the equality Φ(e kl )Φ(e ls ) = Φ(e ks ), we obtain that ψ is a homomorphism from Φ(S 0 ) to M n (K). Thus, A and σ ∈ S n . From this it follows that the matrices A (σ) Let V be a free module over K with a basisX = x 1 , ..., x n . By Lemma 4.7, the semigroup A and E are conjugate, i.e., there exists an automorphism ρ of V such that ρA are elementary matrices over K in a basis Y = y 1 , ..., y n and ρ(x i ) = y i . Since F is a Hopfian algebra, Y is a base of F . The elements Φ(e i1 )y 1 , i ∈ [1n], can be represented in the base Y as , are elements of Y ε -matrix (ε is the identical substitution from S n ) of automorphism Φ in the base Y , by Lemma 3.6, there exists an endomorphism α ∈ End F such that . Since x 2 1 = 0 is the identity in Θ, we have η(y i ) = 0. By Lemma 3.6, there exists an endomorphism α ∈ End F such that   11 = 0 for all σ ∈ S n and, as a consequence, Φ(e 11 ) = 0, i.e., e 11 = 0. Thus, we arrive at a contradiction again. Therefore, there exists m (σ) ij = 0 for some i, j and σ ∈ S n . As in the case 1 we obtain the equality (5.4). From (5.4) follows It is well known (see [7,18]) that every endomorphism τ of a finitely generated nilpotent algebra G which induces an invertible linear transformation on the free K-module G/G ′ is an automorphism of G. Taking this into account we obtain that α is an automorphism of F . Thus, we have α −1 (y i ) = z i (mod F ′ ), i.e., the elements z i , i ∈ [1n] form a basis of K-module F/F ′ . Since F is a finitely generated nilpotent algebra, the elements z i , i ∈ [1n], form a base of F (see [18]). Now in the both cases, Θ ⊆ B 2 or Θ ⊆ N m , we have Φ(e ij )z m = Φ(e ij )Φ(e m1 )y 1 = Φ(e ij e m1 )y 1 = Φ(δ jm e i1 )y 1 = δ jm z i .
By Proposition 3.2, the automorphism Φ is quasi-inner as claimed.
Now we consider the variety Υ of linear algebras with zero multiplication over a Dedekind domain such that the group Aut End F , where F is a free two-generated algebra over K, contains a non-quasi-inner automorphism. This example is a modification of an example by Isaacs [12].
∈ Z} be a Dedekind domain and Υ be a variety of linear algebras over K with zero multiplication. Let V = K 2 be a free module over K. The module V can be considered as a free two-generated algebra over K of variety Υ. Note that a semigroup End V is the full matrix semigroup M 2×2 (K). We wish to construct an automorphism of the semigroup M 2×2 (K) which is not quasi-inner. Let so that is an automorphism of the semigroup M 2×2 (K). We will show that this automorphism is not quasi-inner.
Note that we have also Φ 2 (x) = m −2 xm 2 = a −1 xa, where Since det a = 1, the matrix a −1 belongs to M 2×2 (K) as well. Therefore, Φ 2 is a quasi-inner automorphism of End F , whereas Φ is not.
This example counts in favour of the following problem: Problem 5.5. Let Θ be a variety of linear algebras over a domain K such that Θ ⊆ B 2 or Θ ⊆ N c for some c. Let Φ be an automorphism of the semigroup End F , where F = F (x 1 , ..., x n ) is a free n-generated algebra in Θ. It is true that there exists a natural number k(n) such that Φ k(n) is a quasi-inner automorphism of End F ?
The example 5.4 leads also to the following statement: Corollary 5.6. There exists a Dedekind domain which is not an R 1 MF-domain.
Proof. Let the assumptions of the example 5.4 be fulfilled. Suppose that the Since the variety Υ belongs to B 2 , by Proposition 5.3 all automorphisms of End F , where F is a finitely generated free algebra of Υ, are quasi-inner. This fact contradicts to the example 5.4.
Remark 5.7. Note that Corollary 5.6 can be also proved by direct calculations.
Indeed, assume that K = {a + b √ −5 | a, b ∈ Z} is an R 1 MF-domain. Take the matrix Since rank c = 1, we can represent the matrix c as a column-by-row product: 4]. It can be shown that this system of polynomial equations of the second order has no solutions over Z. We omit the calculations. This contradiction gives us a different proof of Corollary 5.6.

Derivative algebras associated with a quasi-inner automorphisms
Let Φ ∈ AutEnd F, F ∈ Θ, be a quasi-inner automorphism of the semigroup End F with the adjoint bijection s : F → F . Our goal is to describe these bijections of F . For this purpose we reformulate universal algebra notations and results from [27] for the category of linear algebras.
From (6.1) follows that Denote by F * = F ; •, ⊥, * , 0 a derivative algebra with the same support F as the original algebra and with one nullary operation 0 which coincides with 0 of F , one unary operation • and two binary operations, ⊥ and * , determined in the following way: 1. α • a 1 = θ a1 s(α · x 1 ), ∀α ∈ K, ∀a 1 ∈ F and x 1 ∈ X, 2. a 1 ⊥a 2 = θ a1,a2 s(x 1 + x 2 ), ∀a 1 , a 2 ∈ F and x 1 , x 2 ∈ X, 3. a 1 * a 2 = θ a1,a2 s(x 1 · x 2 ), ∀a 1 , a 2 ∈ F and x 1 , x 2 ∈ X, where the operations written on the right side of these formulas are the main operations in F . We shall say that the derivative algebra F * is associated with the automorphism Φ. Now, for completeness of presentation, we give the proof of the following statement.
1. Algebra F * ∈ Θ; 2. The bijection s is an isomorphism of F into F * .
Proof. We will check the compatibility of the bijection s with operation * . We have On the other hand by (6.1) i.e., s(a 1 · a 2 ) = s(a 1 ) * s(a 2 ). In a similar way it is easy to check the compatibility of s with operations ⊥ and •. Finally, since s is a bijection on F , we have F * ∈ Θ and s : F → F * is an isomorphism. Now using these results we investigate the structure of bijections s adjoint to quasi-inner automorphisms Φ of End F for some classes of homogeneous varieties of linear algebras.
Let Θ be a homogeneous variety of linear algebras with or without 1 over a ring K such that for each its free algebra F (X) the following condition holds: there exists a free associative algebra U (F ) (an enveloping algebra) containing the algebra F and freely generated by the same set X. We will denote by " · " an operation of multiplication of elements from U (F ) (for brevity, we will omit this sign if it is clear from context). Note that the variety of Lie algebras, the variety generated by the free special Jordan algebras and, of course, variety of associative algebras over a ring K are such varieties [1,7,26]. We shall call these varieties A-varieties of linear algebras.
Let Ψ be a homogeneous variety of m-nilpotent linear algebras over a ring K such that for each its free algebra F freely generated by X there exists a nilpotent associative algebra U m (F ) containing the algebra F and freely generated by the same set X. We shall call these varieties A m -varieties of linear algebras. It is known [5] that the variety N m of m-nilpotent Lie algebras is an A m -variety.
Let A(X) be a free associative algebra freely generated by X over a ring K. Take x is ∈ X, are monomials in A and a I ∈ K are almost all 0. Note that the empty product of x i ∈ X to represent 1. By the support of f we understand the set of all x I such that a I = 0. Let A = A(x 1 , x 2 , x 3 ) be a 3-generated associative algebra over a ring K and f (x 1 , x 2 ) be a monomial in A. Denote by M f the support of f (x 1 + x 3 , x 2 ) in A. Now we need the following Lemma 6.2. If f (x 1 , x 2 ) and g(x 1 , x 2 ) be two different monomials in A, then Proof. Let where s ≥ 1 and k ij ∈ N 0 = N ∪ {0}. Let r be the minimal natural number such that k 1r = k 2r . We consider separately even and odd natural numbers r. If r = 2k + 1, k ≥ 0, our statement follows from the fact that all monomials in polynomials ( are different and the statement also follows. The proof is complete. Remark 6.3. Clearly, a similar assertion can be formulated for the support of the polynomial f (x 1 , x 2 + x 3 ).
Consider distributive polynomials in a free algebra F (x 1 , ..., x n ) over a ring K for an A-variety Θ.
Proof. It is clear that any distributive polynomial contains no constant term. Write P (x 1 , x 2 ) = i α i f i (x 1 , x 2 ), where α i ∈ K and f i (x 1 , x 2 ) ∈ U n are different monomials in U n . By definition 6.4 we have Since U n is a free associative algebra, we have by Lemma 6.2 for all i. Let for some i x kis 2 , s ≥ 1. We will prove that, in the representation (6.5), every k ij ∈ {0, 1}. Assume, on the contrary, that there exists k i,2m−1 > 1 for some m ≥ 1. Then the monomial . . x kis 2 appears in (6.4 (a)) on the left but does not on the right. This gives a contradiction. If there exists k i,2m > 1, m ≥ 1, in (6.5), then using the equality (6.4 (b)) we achieve a contradiction in a similar way. Therefore, every k ir , r ∈ [1s], is equal to 0 or 1.
Assume that k i1 = 1. Then arguing as above, we obtain k i,2k+1 = 0 for all k ≥ 1. If all k i,2k = 0, k ≥ 1, then f i = x 1 and we arrive at a contradiction with (6.4 (b)). Thus, without loss of generality, it can be assumed that k i2 = 1. As above, we obtain k i,2k = 0 for all k > 1. Therefore, f i (x 1 , x 2 ) = x 1 x 2 . Now, if k i1 = 0, then, without loss of generality, we can assume that k i2 = 1. As above, we obtain f i (x 1 , x 2 ) = x 2 x 1 . Finally, we have our assertion.
We are now ready to prove Proposition 6.6. Let Θ be an A-variety of algebras over a ring K and Φ ∈ AutEnd F be a quasi-inner automorphism of a semigroup End F . Let F * be the derivative algebras associated with Φ. Then there exist α, β ∈ K, α 2 + β 2 = 0, such that the following statements hold (i) a * b = αa · b + βb · a; (ii) a⊥b = a + b; (iii) ξ • a = ϕ(ξ)a, for any a, b ∈ F * and ξ ∈ K and an automorphism ϕ : K → K.
Proof. We begin by showing (i). Let s be a bijection adjoint to Φ. Since sF (x 1 , x 2 ) = F (x 1 , x 2 ) (see the equality (6.2)), s(x 1 x 2 ) is a polynomial P (x 1 , x 2 ) which belongs to F (x 1 , x 2 ). By definition of the operation * in F * we have Since * is a distributive operation in F * , the polynomial P (x 1 , x 2 ) is distributive. By Lemma 6.5 P (x 1 , x 2 ) = αx 1 x 2 + βx 2 x 1 , α, β ∈ K. If α = β = 0, the derivative algebra F * is an algebra where the multiplication is trivial. Since F * ∈ Θ and Θ is an A-variety, we achieve a contradiction and the result follows.
This contradiction shows that γ = 0 and (ii) follows.
(iii) By definition of operation • in F * , we have Since, according to (6.2), sF (x 1 ) = F (x 1 ), and so (6.14) where P α (x 1 ) ∈ F (x 1 ). Consider the equality By (6.13) we get From (6.15) follows Write P α (x 1 ) as an element of Since U (F (x 1 , x 2 )) is a graded algebra, (x 1 + x 2 ) ki = x ki 1 + x ki 2 for every k i . Hence all k i = 1. Thus, P α (x 1 ) = s(αx 1 ) = α 1 x 1 for some α 1 ∈ K. Denote ϕ(α) = α 1 , where ϕ : K → K is a mapping of K into K. Since s : F (x 1 ) → F (x 1 ) is a bijection on F (x 1 ), the mapping ϕ is a bijection on K. From ξ • (µ • x 1 ) = (ξµ) • x 1 and (ξ + µ) • x 1 = ξ • x 1 + µ • x 1 , ξ, µ ∈ K, follows P ξµ (x 1 ) = P ξ P µ (x 1 ) and P ξ+µ (x 1 ) = P ξ (x 1 ) + P µ (x 1 ). Thus ϕ is an automorphism of K as required. This ends the proof. Now we consider a similar assertion for A m -varieties of algebras. Proposition 6.7. Let Ψ be an A m -variety of algebras over a ring K, F m be a finitely generated free algebra of Ψ, Φ be a quasi-inner automorphism of a semigroup End F m and F * m be the derivative algebra associated with Φ. The following statements hold: (a) If |K| = ∞ or |K| = p k , k ≥ 1, where p k ∤ m − 1 and m = 2, then the conclusions (i), (ii), (iii) of Proposition 6.6 are fulfilled for F * m . (b) If |K| = p k , k ≥ 1, where p k |m−1, then the conclusions (i), (iii) of Proposition 6.6 are fulfilled for F * m but instead (ii) the following is true (ii) ′ For any a, b ∈ F * m we have a⊥b = a + b + γg(a, b), γ ∈ K, such that if γ = 0 then g(x 1 , x 2 ) is a nonzero homogeneous symmetric polynomial of degree m − 1 in U (F m ) satisfied the following system of functional equations: where x i ∈ X, i ∈ [13]. (c) If K any domain and m = 2, then the conclusion (ii) and (iii) of Proposition 6.6 are fulfilled for F * 2 but instead (i) holds: a * b = 0 for any a, b ∈ F 2 .
Proof. We consider the natural embedding F m ⊆ U (F m ), where U (F m ) is the associative free m-nilpotent algebra. Recall that U (F m ) is isomorphic to A(X)/I, where A(X) is a free associative algebra generated by X = {x 1 , ..., x n } and I is an ideal generated as a verbal ideal by elements x 1 · ... · x m . It is easy to check that Lemmas 6.2 and 6.5 remain valid for the algebras F and U (F m ). In the proofs of these Lemmas, it is sufficient to restrict our consideration to m-reduced polynomials in the algebra U (F m ). Let assumptions of the part (a) be fulfilled. In the same manner as in Proposition 6.6 we derive the same conclusions (i), (iii) from this Proposition. Let us prove the equality (ii): a⊥b = a + b, a, b ∈ F * m . We will use the notations from Proposition 6.6 and the operation ⊥ as defined in (6.6). We have in F * Since the part (iii) from Proposition 6.6 is fulfilled in our case, we get Thus the conditions of part (a), |K| = ∞ or |K| = p k , k ≥ 1, where p k ∤ m − 1, give us g(x 1 , x 2 ) = 0. This proves (ii) and therefore, the part (a) of our Proposition. Let assumptions of the part (b) be fulfilled. Again the conclusions (i), (iii) hold in this case and it is remains to check the conclusion (ii) ′ . Since |K| = p k , k ≥ 1, where p k | m − 1, it is possible that g(x 1 , x 2 ) = 0. Let us analyze the proof of the part (ii) of Proposition 6.6 in this case. There we have obtained a contradiction comparing the degrees of polynomials in both parts of the equalities (6.10) and (6.9). As is easy to see there is no contradiction if and only if in these equalities either the polynomial g(x 1 , x 2 ) is homogeneous of degree m − 1 or g(x 1 , x 2 ) = 0.
Lemma 6.8. The system (6.18) has a non-trivial solution in the class of homogeneous symmetric polynomials from U (F m ) of degree m − 1 over a ring K iff m = 2k, k ≥ 1, or char K = 2 and m is arbitrary. In these two cases the polynomial is a non-zero particular solution of the system (6.18).

Proofs of the main theorems
Proof of Theorem A. Let Φ ∈ Aut End F (X) be an automorphism of End F . By Proposition 5.3, Φ is a quasi-inner automorphism of End F . Let s : F → F be the bijection adjoint to an automorphism Φ and F * be the derivative algebra associated with Φ. Recall that under our assumptions s(x i ) = x i for all x i ∈ X.
Denote by [ , ] 1 and [ , ] 2 the Lie operations in the Lie algebras F and F * , respectively, i.e., By PBW Theorem the variety of Lie algebras is an A-variety [1,7]. Using the Proposition 6.6, we can write where the operation · is a multiplication in the universal enveloping algebra U (F ). Since [a, a] 2 = 0, we have α = −β, i.e., [a, b] 2 = α[a, b] 1 . Since F * is a free Lie algebra, α = 0. By virtue of Proposition 6.1, In the same way we obtain Let ξ α : F → F be a bijection on F defined by ξ α (a) = αa for any a ∈ F . Denote bys = ξ α −1 s a bijection on F . It is evident thats is a semi-inner automorphism of End F . Finally, we have Φ(ν) = sνs −1 =sνs −1 , ∀ν ∈ End F, wheres is a semi-inner automorphism. This end the proof.
Remark 7.1. Let H 1 and H 2 be Lie algebras over an R 1 MF-domain K. Using the same arguments as in [24] and Theorem A one can prove that the following conditions are equivalent: 1. The algebras H 1 and H 2 are categorically equivalent. 2. The algebras H σ 1 and H 2 are geometrically equivalent for some σ ∈ Aut K. Here the algebras H σ 1 and H 2 coincide as rings and multiplication by a scalar in H σ 1 is defined by the rule: λ • a = λ σ −1 · a, ∀λ ∈ K, ∀a ∈ H σ 1 . Earlier, for the variety of Lie algebras over infinite fields this result was obtained in [24].
Proof of Theorem B. 1. Let us prove the part 1 of Theorem B. Since Φ is an automorphism of End F m , by Proposition 5.3, Φ is quasi-inner. As above, we may consider a derivative algebra F * m = F m ; * , •, ⊥, 0 associated with Φ. By Proposition 6.7, we have in F * Since m > 2, we have α 2 + β 2 = 0. Taking into account the law of associativity in F m : (a * b) * c = a * (b * c), we arrive at αβ = 0. Since K is a domain, α = 0 or β = 0. Since α 2 + β 2 = 0, we have a * b = αa · b, α = 0, or a * b = βb · a, β = 0. By Proposition 6.1, the adjoint bijection s to Φ is an isomorphism of F m into F * m . Since |K| = p k , k ≥ 1, p = 2, and p k |m − 1, by Proposition 6.7, part (b), we have for the bijection s : F m → F m the following equalities: such that if γ = 0 then g(x 1 , x 2 ) is a nonzero homogeneous symmetric polynomial of degree m − 1 in U (F m ) which satisfies the system (6.18).
Let m = 2k, k ≥ 1, and char K = 2. By Lemma 6.8, we get g(x 1 , x 2 ) = 0. If α = 0 and β = 0 in (7.4), we obtain, as above in Theorem A, that the automorphism Φ is semi-inner. If α = 0 and β = 0, the automorphism Φ is a composition of mirror and semi-inner automorphisms. Therefore, the group Aut End F is generated by semi-inner and mirror automorphisms.
Let m = 2k, k ≥ 1, or char K = 2. By Lemma 6.8, there exists a non-zero homogeneous symmetric polynomial g = g(x 1 , x 2 ) of degree m − 1 in U (F m ) which is a solution of (6.18). Now we take g = g(x 1 , x 2 ) and set (7.5) ∀a∀b ∈ F m , ∃α = 0 ∈ K, s(a · b) = αs(a) · s(b) in (7.4). Since s(a + b) = s(a) + s(b) + γg(s(a), s(b)), we obtain s r (a + b) = s r (a) + s r (b) + rγg(s r (a), s r (b)), r ≥ 1. From the last equality follows We obtain, as in Theorem A, that Φ p is a semi-inner automorphism, whereas Φ p−1 is not. Hence, the automorphism Φ is p-semi-inner. Setting in (7.4) ∀a∀b ∈ F m , ∃β = 0 ∈ K, s(a · b) = βs(b) · s(a), and choosing ϕ the identical automorphism of K, in the same way as above we come to a p-mirror automorphism of End F m . Now we have to prove the existence of p-semi-inner and p-mirror automorphisms of End F m . To this end we define a mapping τ : F m → F m such that where the polynomial g = g(x 1 , x 2 ) is a non-zero homogeneous symmetric polynomials of degree m − 1 in U (F m ) which is a solution of (6.18). Since Hence, the mapping τ is defined correctly. As above (see (7.6)), we obtain From (7.8) and the condition τ (x i ) = x i , ∈ [1n] follows τ p = Id Fm , where Id Fm is the identical mapping on F m . Therefore, τ is a bijection on F m . Thus, there exists a bijection τ −1 : F m → F m and, furthermore, it is easy to check that (7.9) τ −1 (a + b) = τ −1 (a) + τ −1 (b) + (p − 1)g(τ −1 (a), τ −1 (b)), ∀a∀b ∈ F m Define a mapping Ψ of End F m such that Ψ(ν) = τ −1 ντ, ∀ν ∈ End F m . Using (7.9), we obtain that Ψ is an automorphism of End F m . Now it is clear that Ψ p is a semi-inner automorphism, whereas Ψ p−1 is not. Hence, the automorphism Ψ is p-semi-inner. Setting τ (x k1 i1 ...x kr ir ) = x kr ir ...x k1 i1 , x i k ∈ X in the definition (7.7) and choosing ϕ the identical automorphism of K, we come to a p-mirror automorphism of End F m . Now it is clear that the group Aut End F m is generated by semi-inner, mirror, p-semi-inner and p-mirror automorphisms. This proves the part 1 of Theorem B.
In the same manner as above we can prove that every automorphism Φ of the semigroup End F m is either a semi-inner or a mirror automorphism, or a composition of them. This proves the part 2 of Theorem B.
As above we obtain that all automorphisms of End F 2 are semi-inner. The proof is complete.
Proof of Theorem C. Let ϕ ∈ Aut L • . It is clear that ϕ is an equinumerous automorphism. By Proposition 2.8, ϕ can be represented as the composition of a stable automorphism ϕ S and an inner automorphism ϕ I . Since a stable automorphism does not change free algebras from Θ 0 , we obtain that ϕ S ∈ Aut End F (x 1 , ..., x n ), where F = F (x 1 , ..., x n ) is a finitely generated free Lie algebra of L. By Theorem A, ϕ S is a semi-inner automorphism of End F (x 1 , ..., x n ). Using this fact and Reduction Theorem [15,21] we obtain that the composition ϕ = ϕ S ϕ I is a semi-inner automorphism of Θ 0 . This ends the proof. Now we provide an example of the variety N 3 of 3-nilpotent associative algebras over the field F 2 with a free algebra F 3 = F 3 (x 1 , ..., x n ) such that the group AutEnd F 3 contains 2-inner and 2-mirror automorphisms.
Example 7.2. Consider the polynomial g(x 1 , x 2 ) = x 1 x 2 + x 2 x 1 in F 3 . The polynomial g(x 1 , x 2 ) is a solution of the system (6.18) in the class of homogeneous symmetric polynomials from F 3 of degree 2 (see also Lemma 6.8). In a similar way as in proof of Theorem B we can construct a bijection s : F → F with the help of the polynomial g(x 1 , x 2 ) so that (7.10) ∀ξ ∈ K, ∀a, b ∈ F m , ∀x i ∈ X, s(0) = 0, s(x i ) = x i , s(x k1 i1 ...x kr ir ) = x k1 i1 ...x kr ir , s(a + b) = s(a) + s(b) + g(s(a), s(b)) = s(a) + s(b) + s(a)s(b) + s(b)s(a).
As in Theorem B, we can prove correctness of definition of the mapping s. It is clear that s 2 is an automorphism of F 3 , whereas s is not. Thus, the automorphism Φ of the semigroup End F 3 , such that Φ(ν) = sνs −1 for any ν ∈ End F 3 , is 2-inner. Setting in (7.10) s(x k1 i1 ...x kr ir ) = x kr ir ...x k1 i1 , x i k ∈ X, we arrive at a 2-mirror automorphism of End F 3 .

Quasi-inner automorphisms of the semigroup End A(X)
Let A = A(x 1 , ..., x n ) be a free associative algebra over a ring K. A description of Aut End A(x 1 , x 2 ), where A(x 1 , x 2 ) is a free two generated associative algebra over an infinite field, has been obtained in [4]. In this connection the following assertion presents interest Proposition 8.1. Let Φ ∈ Aut End A be a quasi-inner automorphism of End A, where A = A(x 1 , ..., x n ), n ≥ 2, be a free finitely associative algebra over a domain K. Then Φ is either a semi-inner or a mirror automorphism, or a composition of them.
In the same manner as in Theorem B, part 2, we can prove that Φ is either a semi-inner or a mirror automorphism, or a composition of them.