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Aupomorphisms of the category of the free nilpotent groups of the fixed class of nilpotency

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arXiv:math/0509541v4 [math.GR] 14 Apr 2006

AUTOMORPHISMS OF THE CATEGORY OF THE FREE NILPOTENT GROUPS OF THE FIXED CLASS OF NILPOTENCY.
A.Tsurkov
Department of Mathematics and Statistics, Bar Ilan University, Ramat Gan, 52900, Israel. Jerusalem College of Technology, 21 Havaad Haleumi, Jerusalem, 91160, Israel. tsurkov@jct.ac.il February 2, 2008
Abstract This research was motivated by universal algebraic geometry. One of the central questions of universal algebraic geometry is: when two algebras have the same algebraic geometry? For answer of this question (see [8],[10]) we must consider the variety Θ, to which our algebras belongs, the category Θ0 of all ?nitely generated free algebras of Θ and research how the group AutΘ0 of all the automorphisms of the category Θ0 are di?erent from the group InnΘ0 of the all inner automorphisms of the category Θ0 . An automorphism Υ of the arbitrary category K is called inner, if it is isomorphic as functor to the identity automorphism of the category K, or, in details, for every A ∈ ObK there exists sΥ : A → A Υ (A) isomorphism of these objects of the category K and for every α ∈ MorK (A, B) the diagram A ↓α B ? → sΥ A sΥ B ? → Υ (A) Υ (α) ↓ Υ (B)

is commutative. In the case when Θ is a variety of all groups we have the classical results which let us resolve this problem by an indirect way. In [3] proved that for every free group Fi the group Aut (AutFi ) coincides with the group Inn (AutFi ), from this result in [4] was concluded that Aut (EndFi ) = Inn (EndFi ) and from this fact by theorem of reduction [2] it can be concluded that AutΘ0 = InnΘ0 . In the case when Θ = Nd

1

is a variety of the all nilpotent class no more then d groups we know by [6], that if the number of generators i of the free nilpotent class d group N Fid are bigger enough than d, then Aut AutN Fid = Inn AutN Fid . But we have no description of Aut EndN Fid and so can not use the theorem of reduction. In this paper the method of verbal operations is used. This method was established in [9]. In [9] by this method was very easily proved that AutΘ0 = InnΘ0 when Θ is the variety of all groups and the variety of all abelian groups. In this paper we will prove, by this method, that AutN0 = InnN0 for every d ≥ 2. d d

1

Introduction and methodology.

This research was motivated by universal algebraic geometry. One of the central questions of universal algebraic geometry is: when two algebras have the same algebraic geometry? For answer of this question (see [8],[10]) we must consider the variety Θ, to which our algebras belongs, the category Θ0 of all ?nitely generated free algebras of Θ and research how the group AutΘ0 of all the automorphisms of the category Θ0 are di?erent from the group InnΘ0 of the all inner automorphisms of the category Θ0 . An automorphism Υ of the arbitrary category K is called inner, if it is isomorphic as functor to the identity automorphism of the category K, or, in details, for every A ∈ ObK there exists sΥ : A → Υ (A) isomorphism of these objects of the category K and for every A α ∈ MorK (A, B) the diagram ? → A sΥ Υ (A) A ↓α Υ (α) ↓ B sΥ Υ (B) B ? → is commutative. In the case when Θ is a variety of all groups we have the classical results which let us resolve this problem by an indirect way. In [3] proved that for every free group Fi the group Aut (AutFi ) coincides with the group Inn (AutFi ), from this result in [4] was concluded that Aut (EndFi ) = Inn (EndFi ) and from this fact by theorem of reduction [2] it can be concluded that AutΘ0 = InnΘ0 . In the case when Θ = Nd is a variety of the all nilpotent class no more then d groups we know by [6], that if the number of generators i of the free nilpotent class d group N Fid are bigger enough than d, then Aut AutN Fid = Inn AutN Fid . But we have no description of Aut EndN Fid and so can not use the theorem of reduction. In this paper the method of verbal operations is used. This method was established in [9]. In [9] by this method was very easily proved that AutΘ0 = InnΘ0 when Θ is the variety of all groups and the variety of all abelian groups. In this paper we will prove, by this method, that AutN0 = InnN0 for d d every d ≥ 2. We start the explanation of the method of verbal operations in the general situation. We consider the variety Θ of one-sorted algebras. The signature of our algebras we denote ?.

2

For the construction of the category Θ0 we must ?x a countable set of symbols X0 = {x1 , x2 , . . . , xn , . . .}. As objects of the category Θ0 we consider all free algebras W (X) of the variety Θ generated by the ?nite subsets X ? X0 . Morphisms of the category Θ0 are homomorphisms of these algebras. De?nition 1.1. We call the automorphism Φ of the category Θ0 strongly stable if it ful?lls these three conditions: A1) Φ preserves all objects of Θ0 , A2) there exists a system of bijections sΦ : B → B | B ∈ ObΘ0 such that Φ B acts on the morphisms of Θ0 by these bijections, i. e., Φ (α) = sΦ α sΦ B A for every α : A → B (A, B ∈ ObΘ0 ); A3) sΦ |X = idX for every B = W (X) ∈ ObΘ0 . B The variety Θ is called an IBN variety if for every free algebras W (X) , W (Y ) ∈ Θ we have W (X) ? W (Y ) if and only if |X| = |Y |. In the [9, Theorem 2] = proved, that if Θ is an IBN variety of one-sorted algebras, then every automorphism Ψ ∈ AutΘ0 can be decomposed: Ψ = ΥΦ, where Υ, Φ ∈ AutΘ0 , Υ is an inner automorphism and Φ is a strongly stable one. So, if we want to know the di?erence of the group AutΘ0 from the group InnΘ0 , we must study the strongly stable automorphisms of the category Θ0 . Before the explanation of the notion of the verbal operation we will introduce the short notation, which will be widely used in this paper. In this notation k-tuple (c1 , . . . , ck ) ∈ C k (C is an arbitrary set) we denote by single letter c and we will even allow ourself to write c ∈ C instead c ∈ C k and to write ”homomorphism α : A ? a → b ∈ B” instead ”homomorphism α : A → B, which transforms ai to the bi , where 1 ≤ i ≤ k”. For every word w (x) ∈ W (X), where X = {x1 , . . . , xk } and every C ∈ Θ we ? can de?ne a k-ary operation wC (c) = w (c) (in full notation c = (c1 , . . . , ck ) ∈ k k ? C , x = (x1 , . . . , xk ) ∈ (W (X)) ) or, more formal, wC (c) = γ c (w (x)), where γ c is a well de?ned homomorphism γ c : W (X) ? x → c ∈ C (in full notation: homomorphism γ c : W (X) → C, which transforms xi to the ci for 1 ≤ i ≤ k). This operation we call the verbal operation induced on the algebra C by the word w (x) ∈ W (X). A little more detailed discussion about de?nition of the verbal operation you can see in [10, Section 2.1]. Now we will consider two kinds of substances, which will be also important for our method: 1. systems of bijections conditions: sB : B → B | B ∈ ObΘ0 which ful?lls these two
?1

(1.1)

B1) for every homomorphism α : A → B (A, B ∈ ObΘ0 ) the mappings sB αs?1 and s?1 αsA is also a homomorphism; A B 3

B2) sB |X = idX for every B = W (X) ∈ ObΘ0 . 2. systems of words {wω (x) ∈ W (Xω ) | ω ∈ ?} which ful?lls these two conditions: Op1) Xω = {x1 , . . . , xk }, where k is an arity of ω, for every ω ∈ ?; Op2) for every B = W (X) ∈ ObΘ0 there exists an isomorphism σ B : B → B ? (algebra B ? has the same domain as the algebra B and its operations ω ? are induced by wω (x) for every ω ∈ ?) such as B σ B |X = idX . The system of bijections sΦ : B → B | B ∈ ObΘ0 , described in A2) and B A3) of the de?nition of the strongly stable automorphism ful?lls conditions B1) and B2) with sB = sΦ . B We take a system of bijections S = sB : B → B | B ∈ ObΘ0 which ful?lls conditions B1) and B2). If arity of ω ∈ ? is k, we take Xω = {x1 , . . . , xk } ? X0 . Aω = W (Xω ) - free algebra in Θ. We have that ω (x) ∈ Aω (x = (x1 , . . . , xk )) so there exists wω (x) ∈ Aω such that wω (x) = sAω (ω (x)) . (1.2)

The system of words {wω (x) ∈ Aω | ω ∈ ?} we denote W (S). This system ful?lls condition Op1) by our construction, condition Op2) with σ B = sB (B ∈ ObΘ0 ) by [9, Theorem 3]. For every C ∈ Θ we denote ω ? the verbal operation C induced on the algebra C by wω (x). C ? will be the algebra, which has the same domain as the algebra C and its operations are {ω? | ω ∈ ?}. By [10, C Proposition 3.1] one can conclude that for every C ∈ Θ the algebra C ? belongs to Θ. Contrariwise, if we have a system of words W = {wω (x) | ω ∈ ?}, which ful?lls conditions Op1) and Op2), then the isomorphisms σ B : B → B ? (B ∈ ObΘ0 ) are bijections. For the system of bijections σ B : B → B | B ∈ ObΘ0 , which we denote S (W ), B1) ful?lls by [5, 1.8, Lemma 8] and [10, Corollary from Proposition 3.2]; B2) ful?lls by construction of these bijections. If we have a system of bijections S = sB : B → B | B ∈ ObΘ0 which ful?lls conditions B1) and B2) then we can de?ne an automorphism Φ (S) = Φ of the category Θ0 : Φ preserves all objects of the category Θ0 and acts on its morphisms according to formula (1.1) with sΦ = sB . Obviously Φ ful?lls B conditions A1) - A3) with sΦ : B → B | B ∈ ObΘ0 = S, i.e. Φ is a strongly B stable automorphism. Actually, the two di?erent systems of bijections S1 = sΦ : B → B | B ∈ ObΘ0 and S2 = sΦ : B → B | B ∈ ObΘ0 can provide 1,B 2,B by formula (1.1) the same action on homomorphisms and, so, the same strongly stable automorphism of the category Θ0 . Proposition 1.1. Every strongly stable automorphism Φ of the category Θ0 can be obtained as Φ (S (W )) where W = {wω (x) ∈ Aω | ω ∈ ?} is a system of words which ful?lls conditions Op1) and Op2).

4

Proof. Let Φ be a strongly stable automorphism of the category Θ0 . The system of bijections S = sΦ : B → B | B ∈ ObΘ0 ful?lls conditions B1) and B B2). Lets consider {wω (x) ∈ Aω | ω ∈ ?} = W (S). By [10, Proposition 3.3], W (S) ful?lls conditions Op1) and Op2). By [10, Proposition 3.4], S (W (S)) = S, so Φ (S (W (S))) = Φ (S). Φ (S) and Φ both preserve all objects of Θ0 and act on the morphisms of Θ0 by the same system of bijections S according to formula (1.1). Therefore they coincide. So, if we describe the strongly stable automorphisms of the category Θ0 , we must concentrate on ?nding out the systems of words, which ful?ll conditions Op1) and Op2). However, in describing this, we must remember, that di?erent systems of words which ful?ll conditions Op1) and Op2) can provide us the same automorphism, because di?erent systems of bijections can provide us the same automorphism. This method we apply to the variety Nd of the all nilpotent class no more then d groups. The variety A2 of the abelian groups with the exponent no more then 2 is a subvariety of Nd for every d ∈ N. So, by the second theorem of Fudzivara [7, III.7.6] the variety Nd is an IBN variety.

2

Applying the method.

The free i-generated group in Nd is denoted as N Fid . The group signature is {1, ?1, ·}, where 1 is a 0-ary operation of the taking of the unit, ?1 is an unary operation of taking an inverse element and · is a binary operation of the multiplication. So, by our method, we must ?nd out the systems of the words {w1 , w?1 , w· } (2.1)

d d d such that w1 ∈ N F0 , w?1 ∈ N F1 , w· ∈ N F2 (condition Op1) ) and condition d 0 Op2) ful?lls for all N Fi ∈ ObNd (i ∈ N). But, as it will be clear above, we d d can consider only the word w (x, y) ∈ N F2 (x, y are generators of N F2 ) which ful?lls this condition: d d is a set N F2 with the binary verbal operation induced by Opd ) If N F2 d d ? are w (x, y) ∈ N F2 (this operation we denote as ”?”), then in N F2 d d ? ful?lled all group axioms and exists an isomorphism σ d : N F2 → N F2 , d such that σ d (x) = x, σ d (y) = y (x, y are the generators of N F2 ). d are ful?lled all group axioms then w (x, y) = Proposition 2.1. If in N F2 d xyg2 (x, y) (g2 (x, y) ∈ γ 2 N F2 , γ i (G) is a i-th group in the lowest central d series of the group G), 1? = 1, a?k = ak , for every a ∈ N F2 , k ∈ Z and ?k d ? d d ? ? γ i N F2 for every i ∈ N (1? and a is the unit of the N F2 γ i N F2 ? ?

and degree of the element a according to the new operation).
d Proof. Every word in N F2 can be written as w (x, y) = xt y s g2 (x, y) (t, s ∈ Z). So we assume that x ? y = xt y s g2 (x, y). Then 1 ? 1 = 1, so 1 = 1? . Therefore

5

d x = x ? 1 = xt , so t = 1. Analogously s = 1. For every a ∈ N F2 and every k2 k1 k1 +k2 k2 k1 , because g2 a , a is calculated in the k1 , k2 ∈ Z we have a ? a = a commutative group a . d So, for every a, b ∈ N F2 holds (a, b)? = a?1 b?1 g2 a?1 , b?1 ? abg2 (a, b) = ?1 ?1 d d a b abl2 = (a, b) l2 ∈ γ 2 N F2 (l2 ∈ γ 2 N F2 , (a, b) is a commutator of a and b and (a, b)? is a commutator of a and b according to the new operation

”?”). We assume that for k < i it is proved that γ k li?1 ∈ γ i?1
d N F2 ? d , b ∈ N F2 , then we have

d N F2

?

d ? γ k N F2 . If

?1 ?1 (li?1 , b)? = li?1 b?1 g2 li?1 , b?1 ? li?1 bg2 (li?1 , b) = ?1 ?1 ?1 ?1 = li?1 b?1 g2 li?1 , b?1 li?1 bg2 (li?1 , b) g2 li?1 b?1 g2 li?1 , b?1 , li?1 bg2 (li?1 , b) . ?1 d d g2 li?1 , b?1 , g2 (li?1 , b) ∈ γ i N F2 , because li?1 ∈ γ i?1 N F2 . So ?1 ?1 (li?1 , b)? ≡ li?1 b?1 li?1 bg2 li?1 b?1 , li?1 b ?1 ?1 li?1 b?1 , li?1 b ≡ li?1 , b d modγ i N F2

. .

d b?1 , li?1 ≡ 1 modγ i N F2

?1 d g2 (x, y) has the weight 2 or more, so g2 li?1 b?1 , li?1 b ∈ γ i N F2 and (li?1 , b)? ∈ d γ i N F2 .

Corollary 1. If the system of words (2.1) ful?lls conditions Op1) and Op2) then w1 = 1, w?1 = x?1 . By this Corollary we can concentrate on a research of the verbal binary d d operations in the N F2 . We will ?nd out the word w (x, y) ∈ N F2 , which ful?ll d condition Op ).
d Corollary 2. In the condition of the Proposition 2.1 the group N F2 a nilpotent class d group. ? ?

is also

d are ful?lled all group axioms then the hoBy this Corollary, if in N F2 x x d d ? momorphism σ d : N F2 ? → ∈ N F2 is well de?ned. y y 2 Proposition 2.2. The verbal operation induced on N F2 by the word w (x, y) = m xy (y, x) ful?lls the group axioms for every m ∈ Z, but the homomorphism 2 2 ? σ 2 : N F2 → N F2 is an isomorphism if and only if m = 0 or m = 1, i.e., w (x, y) = xy or w (x, y) = yx.

Proof. Let x ? y = xy (y, x) . We have 1? = 1, a?(?1) = a?1 . For every 2 a, b, c ∈ N F2 we have (a ? b) ? c = ab (b, a) ? c = = ab (b, a) c (c, ab (b, a))
m m m

m

= abc (b, a) (c, a) (c, b) . 6

m

m

m

a ? (b ? c) = a ? bc (c, b) = abc (c, b) (bc (c, b) , a) So (a ? b) ? c = a ? (b ? c).
m m m

m

=
m

= abc (c, b) (b, a)m (c, a)m .

(y, x)? = y ?1 ? x?1 ? y ? x = y ?1 x?1 x?1 , y ?1 = (y, x) σ 2 (x y
1?2m

m

? yx (x, y)
m

m

=
1?2m α3

yx, y ?1 x?1
α3

m α1

= (y, x)

1?2m

y, x?1
α3

x, y ?1
α1

= (y, x)

. =

α1 α2

(y, x) ) = σ 2 (x )?σ 2 (y )?σ 2 ((y, x) ) = x
α3

α2

?y

α2

?((y, x)? )

= xα1 ?y α2 ?((y, x)? )

= xα1 ?y α2 ?(y, x)

(1?2m)α3

= xα1 y α2 (y, x)

α1 α2 m+(1?2m)α3

2 (α1 , α2 , α3 ∈ Z). If 1 ? 2m = ±1, i.e., m = 0, 1, then (y, x) ∈ σ 2 N F2 . /

Proposition 2.3. γi
d ? N F2

d If N F2

?

d as in the condition Opd ), then γ i N F2

=

for every i ∈ N.

Proof. By Corollary 1 from Proposition 2.1, the operations ”1” and ”?1” of d d ? the group signature in N F2 and in N F2 coincide. By [10, Proposition 3.2], d d original operations in N F2 are the verbal operations induced on N F2 by the words with respect to the operations ”1”, ”?1” and ”?”. So we have a situation d d of Proposition 2.1 in which the group N F2 and N F2 changed they place. Hence,
d γ i N F2 ? γ i d N F2 ?

for every i ∈ N.

Lemma 2.1. If the word w (x, y) ful?lls conditions Opd ), then the word κ (w (x, y)) ∈ d?1 d?1 d N F2 ful?lls conditions Opd?1 ) (κ : N F2 → N F2 is the natural epimorphism).
d?1 d?1 Proof. N F2 ∈ Nd . So on N F2 we can induce the operations of the group d ?1 d d signature by words 1 ∈ N F0 , x ∈ N F1 , w (x, y) ∈ N F2 . This operations we d?1 denote by ”1”, ”?1” and ”?” correspondingly. N F2 with this operations we d?1 ? . denote N F2 d 1 → γ d N F2 ↓ σd d 1 → γ d N F2 ?

?→ ?→

d N F2 ↓ σd d ? N F2

κ ? → ? → κ

d?1 N F2 ↓ σ d?1 d?1 ? N F2

→ 1 (2.2) → 1

d?1 d d are induced by same words; κ : N F2 → N F2 is The operations in N F2 a homomorphism, so, by [5, 1.8, Lemma 8], κ is also a homomorphism from n?1 ? n?1 ? n (N F2 )? to N F2 . Therefore N F2 is an homomorphic image of the ? d?1 d group N F2 , hence it is also a group. For every a, b ∈ N F2 we have a ? b = x a d?1 d w (a, b) = αw (x, y), where α : N F2 ? → ∈ N F2 . We can y b κ (x) a d?1 d?1 also consider a homomorphism α : N F2 ? → ∈ N F2 . κ (y) b

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We have α = ακ, so a ? b = α (κw (x, y)). Therefore the operation ”?” is d?1 d?1 induced in N F2 by the word κw = w (κ (x) , κ (y)) ∈ N F2 (κ (x) and κ (y) d?1 are free generators of N F2 ). This operation ful?lls all group axioms, hence, d?1 ? is also a nilpotent class d ? 1 by Corollary 2 from Proposition 2.1, N F2 κ (x) κ (x) d?1 d?1 ? group. So σ d?1 : N F2 ? → is a well ∈ N F2 κ (y) κ (y) de?ned homomorphism. Our goal is to prove that it is an isomorphism. d d ? d = γ d N F2 . ? γ d N F2 By Proposition 2.3, we have σ d γ d N F2
d d By consideration of the acting of σ ?1 , we achieve γ d N F2 ? σ d γ d N F2 . d d In the diagram (2.2) we have two exact rows, because ker κ = γ d N F2 . The left square of this diagram is commutative. The right square is commutative too, because σ d and σ d?1 preserve the generators of the corresponding free nilpotent groups. σ d is an isomorphism, so σ d?1 too.

Lemma 2.2. For every d ≥ 2 there are only two words: w = xy and w = yx d from N F2 which ful?lls condition Opd ). Proof. For d = 2 it is proved in the Proposition 2.2. We will assume that it is proved for all natural numbers lesser then d. d d Let w (x, y) ∈ N F2 ful?lls condition Opd ) and ? be an operation in N F2 induced by w (x, y). By Lemma 2.1 and assumption of induction κw (x, y) = κ (x) κ (y) or κw (x, y) = κ (y) κ (x). d In the ?rst case we have w (x, y) = xyr (x, y), where r (x, y) ∈ γ d N F2 . For n every a, b, c ∈ N F2 we have (a ? b)?c = abr (a, b)?c = abr (a, b) cr (abr (a, b) , c) = x a d abcr (a, b) r (ab, c) (r (a, b) = α (r (x, y)) where α : N F2 ? → ∈ y b d d d d N F2 for every a, b ∈ N F2 , so r (a, b) ∈ γ d N F2 ? Z N F2 ). Also we have a?(b ? c) = a?bcr (b, c) = abcr (b, c) r (a, bcr (b, c)) = abcr (b, c) r (a, bc). ? ful?lls the axiom of associativity, so r (a, b) r (ab, c) = r (b, c) r (a, bc) . (2.3)

We will prove that for d ≥ 3 there isn’t any nontrivial word r (x, y), which is generated by commutators of the weight d and ful?lls the condition (2.3). d We will shift all our calculation from the free nilpotent group N F2 to the N Ld 2 - free nilpotent class n Lie algebra over Q with generators x and y. By [1, ? d d 8.3.9], N F2 = N Ld , where N F2 is a Maltsev completion of the group 2 d d ? N F2 and N L2 is a group which coincides with N Ld as a set and has a 2 multiplication de?ned by Campbell-Hausdor? formula. By consideration of the Campbell-Hausdor? formula we have that r (a, b) = q (a, b), where q (x, y) is the word in N Ld , which we achieve from r (x, y) by replacement of the circular 2 d brackets of the group commutators by Lie brackets and multiplication in N F2 d by addition in N L2 , q (a, b) is a result of substitution of the a and b instead x d and y correspondingly (a, b ∈ N F2 ) to the word q (x, y). Also by consideration of the Campbell-Hausdor? formula ab ≡ a + b mod N Ld 2 8
2

for every a, b ∈

d N F2 ( N Ld is the i-th ideal of the lowest central series of the algebra N Ld , 2 2 d 1 ≤ i ≤ d) and ab = a + b if a, b ∈ γ d N F2 . Therefore the condition (2.3) we can write as q (a, b) + q (a + b, c) = q (b, c) + q (a, b + c) . (2.4)

i

We shall substitute a = λx, b = x, c = y - (λ ∈ Z) to (2.4) and we will achieve: q ((λ + 1) x, y) = q (x, y) + q (λx, x + y) . (2.5)

N Ld is a direct sum of its polyhomogeneous (homogeneous as according x, as 2 according y separately) components (proof of this fact is similar to the proof
d?1

of [1, 2.2.5]). Let q (x, y) =
i=1

qi (x, y), where qi (x, y) is a polyhomogeneous

component of q (x, y) corresponding to the degree i of x and d ? i of y. By (2.5) we have
d?1 d?1 d?1

(λ + 1) qi (x, y) =
i=1 i=1

i

qi (x, y) +
i=1

λi qi (x, x + y) .

(2.6)

When we develop qi (x, x + y) by additivity, we achieve qi (x, x + y) = qi (x, y)+
d?1

mi,j (x, y) where mi,j (x, y) is a polyhomogeneous component of qi (x, x + y)
j=i+1

with the degree j of x and d ? j of y. In particular q1 (x, x + y) = q1 (x, y) +
d?1

m1,j (x, y). By comparison of the polyhomogeneous components of (2.6)
j=2

with the degree 2 of x and d ? 2 of y we have (λ + 1) ? 1 ? λ2 q2 (x, y) = λ2 m1,2 (x, y) .
k s=1 k 2

(2.7)

Let {e1 (x, y) , . . . , ek (x, y)} be a linear basis of the d-th ideal of the lowest central series of N Ld and q2 (x, y) = 2 (ρs , ?s ∈ Z). From (2.7) we have (λ + 1) ? 1 ? λ2 ρs = λ2 ?s (s ∈ {1, . . . , k}). We take two values of λ such that we will achieve ρs = 0, ?s = 0. Hence q2 (x, y) = 0, q2 (x, x + y) = 0, m1,2 (x, y) = 0. Step by step, analogously we conclude, that qi (x, y) = 0, qi (x, x + y) = 0, m1,i (x, y) = 0 for i ∈ {3, . . . , d ? 1}. Therefore q (x, y) = q1 (x, y). Now we shall substitute a = x, b = y, c = λy - (λ ∈ Z) to (2.4) and we will achieve: q1 (x, y) + q1 (x + y, λy) = q1 (x, (λ + 1) y) or (λ + 1)
d?1 2

ρs es (x, y), m1,2 (x, y) =

s=1

?s es (x, y)

? λd?1 ? 1 q1 (x, y) = 0,

so q1 (x, y) = 0, q (x, y) = 0, r (x, y) = 1 and w (x, y) = xy. Analogously we consider the case w (x, y) = yx. 9

Theorem 2.1. All automorphisms of the category of the nilpotent class d free groups are inner. Proof. Let W = {w1 , w?1 , w· } be the system of words, which ful?lls conditions Op1) and Op2). By Lemma 2.2 there are no more then two opportunities: W = 1, x?1 , xy or may also be W = 1, x?1 , yx . It is easy to check that both these systems actually ful?ll conditions Op1) and Op2). By Proposition 1.1, all strongly stable automorphisms of N0 can by achieved as Φ (S (W )). If d W ful?lls conditions Op1) and Op2), then S (W ) ful?lls conditions B1) and B2) (see Section 1), so, by [9, Lemma 3 and Theorem 2], Φ (S (W )) is an inner automorphism if and only if for every B ∈ ObN0 exists an isomorphism cB d from B to B ? , where B ? is an algebra which has the same domain as algebra B and operations induced by the W (S (W )) = W (see [10, Proposition 3.4]), such that cB αc?1 = α for every homomorphism α : A → B (A, B ∈ ObΘ0 ). If A W = 1, x?1 , xy , then B ? = B and we can take cB = idB for every B ∈ ObN0 . d And if W = 1, x?1 , yx , then for every B ∈ ObN0 we can take cB : B → B ? d such that cB (b) = b?1 for every b ∈ B.

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Acknowledgements.

I dedicate this paper to the 80th birthday of Prof. B.Plotkin. He motivated this research and was very heedful to it. Of course, I must appreciative to Prof. B.Plotkin and Dr. G. Zhitomirski for theirs marvelous paper [9]. This research could not take place without theirs deep results. Prof. S. Margolis also was very heedful to my research. Very useful discussion with Prof. L. Rowen, Dr. R. Lipyanski, Dr. E. Plotkin and Dr. G. Zhitomirski, also help me in the writing of this paper.

References
[1] Ju. Bahturin, Identities in Lie algebras, (Russian), Nauka, Moscow, 1985. [2] A. Berzins, B. Plotkin, E. Plotkin, Algebraic geometry in varieties of algebras with the given algebra of constants, Journal of Math. Sciences, 102:3, (2000), pp. 4039 – 4070. [3] J. Dyer, E.Formanek, The automorphism group of a free group is complete, J. London Math. Soc. (2), 11:2 (1975), pp. 181 – 190. [4] E. Formanek, A question of B.Plotkin about semigroup of endomorphisms of a free group, Proc. Amer. Math.Soc., 130 (2002), pp. 935 – 937. [5] G. Gr¨tzer, Universal algebra. Second edition. Spinger-Verlag, 1979. a [6] M. Kassabov, On the Automorphism Tower of Free Nilpotent Groups, http://arxiv.org/abs/math.GR/0311488.

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[7] A. Kurosh, Lectures in general algebra (Russian), Fizmatgiz, Moscow, 1962. [8] Plotkin B. 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. [9] B. Plotkin, G. Zhitomirski, On automorphisms of categories of free algebras of some varieties, http://arxiv.org/abs/math.RA/0501331. [10] A. Tsurkov Automorphic equivalence http://arxiv.org/abs/math.GM/0509032. of one-sorted algebras,

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