Automorphic Equivalence of Linear Algebras

A UTOMORPHIC EQUIV ALENCE OF LINEAR ALGEBRAS. A.Tsurko v Institute of Mathematics and Statistics. Univ ersit y S˜ ao P aulo. Rua do Mat˜ ao, 1010 Cidade Univ ersit´ aria S˜ ao Paulo - SP - Brasil - CEP 055 0 8-090 ark ady .tsurk ov@gmail.com No v ember 20, 201 8 Abstract This res earch is motiv ated by universal alge braic geometry . W e con- sider in universa l algebraic geometry the some v ariety of u niversal algebras Θ and algebras H ∈ Θ from this v ariety . One of the central question of the theory is th e follo wing: Wh en do t wo algebras ha ve the same geome- try? What does it mean that the tw o algebras ha ve th e same geometry? The notion of geometric equiv alence of algebras gives a sort of answer to this question. Algebras H 1 and H 2 are called geometrically equiv alent if and only if the H 1 -closed sets coincide with the H 2 -closed sets. The notion of automorphic equiv alence is a generalization of the first notion. Algebras H 1 and H 2 are called aut omorphicaly equiva lent if and only if the H 1 -closed sets coincide with t he H 2 -closed sets after some ”c hanging of coordin ates”. W e can detect th e difference b etw een geometric and automorphic equiv- alence of algebras of the v ariety Θ by researc hing of the automorphisms of the category Θ 0 of the finitely generated free algebras of the v ariety Θ. By [3] the aut omorphic equiv alence of algebras provided by inner au t omor- phism d egenerated to the geometric equival ence. So the va rious d ifferences b etw een geometric and automorphic equiv alence of algebras can b e found in the v ariety Θ if the factor group A / Y is big. Hear A is th e group of all automorphisms of the category Θ 0 , Y is a normal subgroup of all inn er automorphisms of the category Θ 0 . In [4] the v ariet y of all Lie algebras and the v ariet y of all associative algebras o ver the infinite fi eld k were stu died. If the field k h as not nontrivial automorphisms th en group A / Y in the first case is trivial and in th e second case h as order 2. W e consider in this pap er the v ariet y of all linear algebras ov er the infinite field k . W e pro ve that group A / Y is isomorphic to the group ( U ( k S 2 ) /U ( k { e } )) ⋋ Aut k , where S 2 is the 1 symmetric group of th e set which has 2 elements , U ( k S 2 ) is the group of all inverti ble elements of the group algebra k S 2 , e ∈ S 2 , U ( k { e } ) is a group of all invertible elemen ts of the subalgebra k { e } , A u t k is th e group of all automorph isms of th e field k . So eve n the field k has not nontrivial automorph isms the group A / Y is infinite. This kind of result is obtained for the first time. The example of tw o linear algebras whic h are automorphically eq uiv- alen t but n ot geometrically equiv alent is p resen ted in the last section of this pap er. This kind of example is also obtained for the first time. 1 In tro duction. In the fir st tw o sections we cons ider some v ariety Θ of one-so r ted algebras of the signature Ω. Denote by X 0 = { x 1 , x 2 , . . . , x n , . . . } a countable set of symbols, and b y F ( X 0 ) the set of all finite subsets of X 0 . W e will consider the categor y Θ 0 , whose ob jects ar e a ll free algebr as F ( X ) of the v ariety Θ generated b y finite subsets X ∈ F ( X 0 ). Mor phis ms of the categ o ry Θ 0 are homomorphisms of free algebras . W e deno te some time F ( X ) = F ( x 1 , x 2 , . . . , x n ) if X = { x 1 , x 2 , . . . , x n } and even F ( X ) = F ( x ) if X has only o ne element. W e assume tha t our v ar iet y Θ p oss esses the IBN prop erty: for fr ee algebra s F ( X ) , F ( Y ) ∈ Θ we hav e F ( X ) ∼ = F ( Y ) if and only if | X | = | Y | . In this case we have [4, Theore m 2 ] this dec o mpo sition A = YS . (1.1) of the group A of a ll automor phisms of the categ ory Θ 0 . Hear Y is a g roup of all inner auto morphisms of the categ ory Θ 0 and S is a gr oup of all strongly stable a utomorphisms of the catego ry Θ 0 . Definition 1.1 An aut omorphism Υ of a c ate gory K is inner , if it is isomor- phic as a functor to the identity automorphism of t he c ate gory K . This mea ns that for every A ∈ O b K there exists an isomorphism s Υ A : A → Υ ( A ) such that for every α ∈ Mo r K ( A, B ) the diagr am A − → s Υ A Υ ( A ) ↓ α Υ ( α ) ↓ B s Υ B − → Υ ( B ) commutes. Definition 1.2 . An aut omorphism Φ of the c ate gory Θ 0 is c al le d str ongly stable if it satisfies the c onditions: A1) Φ pr eserves al l obje cts of Θ 0 , 2 A2) t her e exist s a system of bije ctions  s Φ F : F → F | F ∈ ObΘ 0  such that Φ acts on the morphisms α : D → F of Θ 0 by this way: Φ ( α ) = s Φ F α  s Φ D  − 1 , (1.2) A3) s Φ F | X = id X , for every fr e e algebr a F = F ( X ) . The subgro up Y is a normal in A . W e will calculate the factor gr oup A / Y ∼ = S / S ∩ Y . This calculation is v ery impo r tant for universal algebraic geometry . All definitions of the bas ic no tio ns of the universal alge br aic geometry c a n be found, for example, in [1 ], [2 ] and [3]. In univ ersal algebr aic ge o metry we consider a ”set of equations” T ⊂ F × F in so me finitely genera ted free algebra F of the arbitrar y v ariety of universal algebr as Θ and we ” r esolve” these equatio ns in Hom ( F, H ), where H ∈ Θ . The s et Hom ( F , H ) serves as an ”affine space ov er the a lgebra H ”. Denote by T ′ H the set { µ ∈ Hom ( F , H ) | T ⊂ ker µ } . This is the s et o f a ll solutions o f the set of equations T . F or every set of ”p oints” R of the a ffine space Hom ( F , H ) we consider a cong ruence of e q uations defined by this set: R ′ H = T µ ∈ R ker µ . F or every set of eq uations T we consider its alge br aic closure T ′′ H in re sp e c t to the alg ebra H . A s e t T ⊂ F × F is ca lle d H -closed if T = T ′′ H . An H -closed set is alwa y s a c o ngruence. Definition 1.3 Alge br as H 1 , H 2 ∈ Θ ar e ge ometric al ly e qu ivalent if and only if for every X ∈ F ( X 0 ) and every T ⊂ F ( X ) × F ( X ) fulfil ls T ′′ H 1 = T ′′ H 2 . Denote the family o f all H -clo sed co ngruences in F by C l H ( F ). W e can consider the catego r y C Θ ( H ) of the c o or dinate algebr as connected with the algebra H ∈ Θ. O b jects of this ca teg ory a re q uotient algebra s F ( X ) /T , wher e X ∈ F ( X 0 ), T ∈ C l H ( F ( X )). Mo rphisms of this catego ry are homomor phis ms of alg ebras. Definition 1.4 L et I d ( H, X ) = T ϕ ∈ Hom( F ( X ) ,H ) ker ϕ b e the minimal H -close d c ongruenc e in F ( X ) . A lgebr as H 1 , H 2 ∈ Θ ar e automorphic al ly e quival ent if and only if ther e ex ists a p air (Φ , Ψ) , wher e Φ : Θ 0 → Θ 0 is an aut omorphism, Ψ : C Θ ( H 1 ) → C Θ ( H 2 ) is an isomorphism subje ct to c onditions: A. Ψ ( F ( X ) /I d ( H 1 , X )) = F ( Y ) /I d ( H 2 , Y ) , wher e Φ ( F ( X )) = F ( Y ) , B. Ψ ( F ( X ) /T ) = F ( Y ) / e T , wher e T ∈ C l H 1 ( F ( X )) , e T ∈ C l H 2 ( F ( Y )) , C. Ψ takes t he natur al epimorphi sm τ : F ( X ) /I d ( H 1 , X ) → F ( X ) /T to the natu r al epimorphism Ψ ( τ ) : F ( Y ) / I d ( H 2 , Y ) → F ( Y ) / e T . Note that if such a pair ( Φ , Ψ) exists, then Ψ is uniquely defined b y Φ. W e can say , in certain sens e, that automor phic equiv alence of a lgebras is a coinciding of the structur e o f closed s ets a fter some ”changing o f co o rdinates” provided by a utomorphism Φ. 3 Algebras H 1 and H 2 are geometrica lly equiv alent if and o nly if an inner au- tomorphism Φ : Θ 0 → Θ 0 provides the automorphic e quiv alence of algebra s H 1 and H 2 . So, only stro ngly s table auto morphism Φ can provide us a utomorphic equiv alence of algebras which not coincides with geometric eq uiv alence of a lge- bras. Ther efore, in so me sense, difference fro m the auto morphic e q uiv alence to the g eometric equiv a lence is meas ured by the factor group A / Y ∼ = S / S ∩ Y . 2 V erbal op erations and strongly stable auto- morphisms. F or every word w = w ( x 1 , . . . , x k ) ∈ F ( X ), where F ( X ) ∈ ObΘ 0 , X = { x 1 , . . . , x k } and for every algebra H ∈ Θ we can define a k -ary op er ation w ∗ H on H by w ∗ H ( h 1 , . . . , h k ) = w ( h 1 , . . . , h k ) = γ h ( w ( x 1 , . . . , x k )) , where γ h is a homo morphism F ( X ) ∋ x i → γ h ( x i ) = h i ∈ H , 1 ≤ i ≤ k . This op eration we call the v erbal op eration induced o n the alg ebra H by the word w ( x 1 , . . . , x k ) ∈ F ( X ). A system o f words W = { w i | i ∈ I } such that w i ∈ F ( X i ) , X i = { x 1 , . . . , x k i } , deter mines a system of k i -ary op er a tions ( w i ) ∗ H on H . Deno te the set H w ith the system of these ope ration by H ∗ W . W e hav e a corr esp ondence betw een s trongly stable automorphisms and sys- tems of words which define the v erbal oper ation a nd fulfill so me conditio ns . This cor r esp ondence explained in [4] and [5]: W e denote the signa ture of our v ariety Θ by Ω, by k ω we denote the arity of ω for every ω ∈ Ω. W e supp ose that w e have the system of words W = { w ω | ω ∈ Ω } satisfies the co nditions: Op1) w ω ( x 1 , . . . , x k ω ) ∈ F ( X ω ), wher e X ω = { x 1 , . . . , x k ω } ; Op2) for every F = F ( X ) ∈ ObΘ 0 there exists an isomor phism σ F : F → F ∗ W such that σ F | X = i d X . F ∗ W ∈ Θ so isomorphisms σ F are defined uniquely by the sy stem of words W . The set S =  σ F : F → F | F ∈ O bΘ 0  is a system of bijectio ns whic h sat- isfies the conditions : B1) for every homomor phism α : A → B ∈ MorΘ 0 the mappings σ B ασ − 1 A and σ − 1 B ασ A are homomorphisms; B2) σ F | X = i d X for every free alg ebra F ∈ ObΘ 0 . So we can define the stro ngly stable automor phism by this sys tem of bijec- tions. This auto morphism pr eserves all ob jects o f Θ 0 and acts on morphism of Θ 0 by formula (1.2 ), w her e s Φ F = σ F . 4 Vice versa if we hav e a s tr ongly stable automorphism Φ of the categor y Θ 0 then its sy s tem of bijections S =  s Φ F : F → F | F ∈ ObΘ 0  defined uniquely . Really , if F ∈ ObΘ 0 and f ∈ F then s Φ F ( f ) = s Φ F α ( x ) =  s Φ F α  s Φ D  − 1  ( x ) = (Φ ( α )) ( x ) , (2.1) where D = F ( x ) - 1-gener ated free linear a lgebra - and α : D → F ho- momorphism suc h that α ( x ) = f . Obviously that this system of bijections S =  s Φ F : F → F | F ∈ ObΘ 0  fulfills c o nditions B1 and B2 with σ F = s Φ F . If we hav e a system of bijections S =  σ F : F → F | F ∈ ObΘ 0  which fulfills conditions B1 and B2 than w e can define the system of w ords W = { w ω | ω ∈ Ω } satisfies the co nditions Op1 and Op2 by formula w ω ( x 1 , . . . , x k ω ) = σ F ω ( ω (( x 1 , . . . , x k ω ))) ∈ F ω , (2.2) where F ω = F ( X ω ). By formulas (2.1) and (2.2) we ca n chec k that there a re 1. one to one and onto corresp ondenc e betw een stro ngly stable automor - phisms o f the categor y Θ 0 and systems o f bijections sa tisfied the co nditions B1 a nd B 2 2. one to o ne and ont o corr esp ondence b etw een sys tems of bijections satisfied the conditions B1 a nd B2 a nd systems of words satisfied the co nditio ns Op1 a nd Op2. So we can find a strongly s ta ble auto mo rphism Φ of the category Θ 0 by finding a sys tem of words which fulfills co nditions O p1 and Op2. 3 V erbal op erations in linear algebras. F ro m now on, we co nsider the v ariety Θ of all linear a lg ebras over infinite field k . W e consider linear alg ebras a s one-sorted univ ersal algebra s, i. e., multiplication by scalar we co nsider a s 1-ary op era tio n for every λ ∈ k : H ∋ h → λh ∈ H where H ∈ Θ. Hence the signature Ω of algebras of our v ar iety contains these op erations: 0-ary op eratio n 0; | k | 1-ar y o p er ations of multiplications by sca lars; 1-ary op er ation − : h → − h , where h ∈ H , H ∈ Θ; 2-ar y op era tio n · a nd 2-ary op er ation +. W e will finding the system of w or ds W = { w ω | ω ∈ Ω } satisfies the co nditions Op1 and O p2. W e denote the words corre s po nding to these op er ations by w 0 , w λ for a ll λ ∈ k , w − , w · , w + . F or arbitr a ry F ( X ) ∈ ObΘ 0 we deno te F ( X ) = ∞ L i =1 F i the deco mpo sition to the linear spa ces of elements whic h ar e homog e neous according the sum o f degrees of gener ators from the set X . W e also denote the t wo-sides ideals ∞ L i = j F i = F j . F rom now on, the word ”ideal” mea ns tw o sided ideal o f linear algebra. 5 W e denote the group o f all automorphis ms of the field k b y Aut k . Our v a riety Θ p o ssesses the IBN prop erty , b ecause | X | = dim F /F 2 fulfills for all free alge br as F = F ( X ) ∈ Θ. So we hav e the decomp osition (1.1) for group of all automorphisms of the ca tegory Θ 0 . Now we need to prove one technical fact abo ut 1-gener a ted fr e e linear algebra F ( x ) . Lemma 3.1 L et { u 1 , . . . , u r } is t he set of al l monomials of de gr e e n in F ( x ) (b asis of F n ), { v 1 , . . . , v t } is t he set of al l monomials of de gr e e m in F ( x ) (b asis of F m ), ϕ is an arbitr ary function fr om { 1 , . . . , n } t o { 1 , . . . , t } . Den ote by ϕ ( u l ) the monomial which is a r esults of subst itution into monomial u l ( 1 ≤ l ≤ r ) inste ad j -t h fr om left entry of x the monomial v ϕ ( j ) ( 1 ≤ j ≤ n ). Al l these monomials ar e distinct, i. e., ϕ 1 ( u l 1 ) = ϕ 2 ( u l 2 ) if and only if ϕ 1 = ϕ 2 and u l 1 = u l 2 , wher e ϕ 1 , ϕ 2 : { 1 , . . . , n } → { 1 , . . . , t } , u l 1 , u l 2 ∈ { u 1 , . . . , u r } . Pro of. W e will prove this lemma by induction by n - degr ee of monomia ls from { u 1 , . . . , u r } . The cla im of the lemma is trivial for n = 1. W e assume tha t the claim of the lemma is proved for monomials which ha ve degr ee < n . W e supp o se that ϕ 1 ( u l 1 ) = ϕ 2 ( u l 2 ), where deg u l 1 = deg u l 2 = n > 1, ϕ 1 , ϕ 2 : { 1 , . . . , n } → { 1 , . . . , t } . u l i = u (1) l i · u (2) l i , where i = 1 , 2 . W e de no te deg u (1) l i = c i . 1 ≤ c i < n for i = 1 , 2 . F or i = 1 , 2 we have ϕ i ( u l i ) = ϕ (1) i  u (1) l i  · ϕ (2) i  u (2) l i  , where ϕ (1) i : { 1 , . . . , c i } → { 1 , . . . , t } , ϕ (2) i : { 1 , . . . , n − c i } → { 1 , . . . , t } , ϕ (1) i ( j ) = ϕ i ( j ) for 1 ≤ j ≤ c i , ϕ (2) i ( j ) = ϕ i ( c i + j ) for 1 ≤ j ≤ n − c 1 . ϕ 1 ( u l 1 ) = ϕ 2 ( u l 2 ) if and only if ϕ (1) 1  u (1) l 1  = ϕ (1) 2  u (1) l 2  and ϕ (2) 1  u (2) l 1  = ϕ (2) 2  u (2) l 2  . If c 1 6 = c 2 then deg ϕ (1) 1  u (1) l 1  = c 1 m 6 = deg ϕ (1) 2  u (1) l 2  = c 2 m , hence ϕ (1) 1  u (1) l 1  6 = ϕ (1) 2  u (1) l 2  and ϕ 1 ( u l 1 ) 6 = ϕ 2 ( u l 2 ). So c 1 = c 2 and, by our assumption, ϕ (1) 1 = ϕ (1) 2 , u (1) l 1 = u (1) l 2 , ϕ (2) 1 = ϕ (2) 2 , u (2) l 1 = u (2) l 2 . There fo re ϕ 1 = ϕ 2 and u l 1 = u l 2 . Corollary 1 L et f ( x ) , g ( x ) ∈ F ( X ) . f ( g ( x )) is a r esult of substitution of g ( x ) in f ( x ) inste ad x . f ( g ( x )) ∈ F 1 if and only if f ( x ) , g ( x ) ∈ F 1 . Pro of. W e write f ( x ) a nd g ( x ) as sum of its homogeneous comp onents: f ( x ) = f 1 ( x ) + f 2 ( x ) + . . . + f n ( x ), g ( x ) = g 1 ( x ) + g 2 ( x ) + . . . + g m ( x ), f i ( x ) , g i ( x ) ∈ F i . W e assume tha t n > 1 or m > 1, f n ( x ) 6 = 0 and g m ( x ) 6 = 0 . f ( g ( x )) = f 1 ( g ( x )) + f 2 ( g ( x )) + . . . + f n ( g ( x )). Addenda of the maximal po ssible degr ee of x , which can app ear in f ( g ( x )), i. e., addenda of deg ree nm can app ear in f n ( g ( x )). They coincide with addenda of f n ( g m ( x )). D enote f n ( x ) = λ 1 u 1 + . . . + λ r u r , g m ( x ) = µ 1 v 1 + . . . + µ t v t , where { u 1 , . . . , u r } is the set of all monomials of degree n in F ( x ), { v 1 , . . . , v t } is the set of all mono mia ls of deg ree m in F ( x ), λ i , µ j ∈ k . Not all { λ 1 , . . . , λ r } and not all { µ 1 , . . . , µ t } are equal to 0 by o ur ass umption. f n ( g m ( x )) = λ 1 u 1 ( g m ( x )) + . . . + λ r u r ( g m ( x )). If we op en the brack ets in u l ( g m ( x )) = u l ( µ 1 v 1 + . . . + µ t v t ) (1 ≤ l ≤ r ), w e obtain addenda, which are r esults o f substitution into mo nomial u l instead a ll entry of x some monomial from µ 1 v 1 , . . . , µ t v t in all p ossible options. W e can say mo re 6 formal: for every function ϕ : { 1 , . . . , n } → { 1 , . . . , t } we obtain an addendum which is a res ults of substitution into mono mial u l instead j -th from left e ntry of x the monomia l µ ϕ ( j ) v ϕ ( j ) (1 ≤ j ≤ n ). Therefore all addenda, which we obtain after the op ening of the brack ets in f n ( g m ( x )), distinct from the mono mials discussed in Lemm a 3.1 o nly by co efficients. All these addenda hav e deg ree nm > 1, b ecause n > 1 or m > 1. So addenda o f f n ( g m ( x )) can not cancel one another by Lemma 3.1 . These addenda can not b e canceled b y other addenda f ( g ( x ) ), b ecause all other addenda ha ve degree < nm . Ther efore all these addenda equal to 0, b eca use f ( g ( x )) ∈ F 1 . F or l ∈ { 1 , . . . , r } and j ∈ { 1 , . . . , t } we take the addendum which is a results of substitution into mono mial λ l u l instead a ll e n tries of x the mo nomial µ j v j . The co efficient of this addendum is λ l µ n j = 0 . So λ l µ j = 0 for a ll l ∈ { 1 , . . . , r } and all j ∈ { 1 , . . . , t } . It contradicts the fa c t that f n ( x ) 6 = 0 and g m ( x ) 6 = 0 . Theorem 3.1 The system of wor ds W = { w 0 , w λ ( λ ∈ k ) , w − , w + , w · } (3.1) satisfies the c onditions Op1 and Op2 if and only if w 0 = 0 , w λ = ϕ ( λ ) x 1 , w − = − x 1 , w + = x 1 + x 2 , w · = ax 1 x 2 + bx 2 x 1 , wher e ϕ is an automorphism of the field k , a, b ∈ k , a 6 = ± b . Pro of. Let W (see (3.1) ) sa tis fie s the conditions Op1 a nd Op2. w 0 is an element of the 0- generated free linea r alg ebra. There is o nly one element in this algebra : 0 . This is the only one opp ortunity for w 0 . w λ ∈ F ( x ) for every λ ∈ k . Denote multiplications by s calars in ( F ( x )) ∗ W by ∗ , i. e., λ ∗ f = w λ ( f ) for e very f ∈ F ( x ) and every λ ∈ k . ( F ( x )) ∗ W ∈ Θ , therefore, if λ = 0 then 0 ∗ x = w 0 ( x ) = 0. If λ 6 = 0 then 1 ∗ x =  λ − 1 λ  ∗ x = λ − 1 ∗ ( λ ∗ x ) = w λ − 1 ( w λ ( x )) = x. Hence w λ = ϕ ( λ ) x by Corollary 1 from Lemm a 3.1 , wher e ϕ ( λ ) ∈ k . W e can wr ite ϕ ( 0) = 0. Also we hav e that for a ll λ 1 , λ 2 ∈ k fulfills ( λ 1 λ 2 ) ∗ x = ϕ ( λ 1 λ 2 ) x and ( λ 1 λ 2 ) ∗ x = λ 1 ∗ ( λ 2 ∗ x ) = λ 1 ∗ ( ϕ ( λ 2 ) x ) = ϕ ( λ 1 ) ( ϕ ( λ 2 ) x ) = ( ϕ ( λ 1 ) ϕ ( λ 2 )) x. So ϕ ( λ 1 ) ϕ ( λ 2 ) = ϕ ( λ 1 λ 2 ). If µ ∈ k \ { 0 } , then the 1- ary op eratio n of multipli- cation b y sc a lar µ is a verbal op eration defined by so me word w ∗ µ ( x ) ∈ ( F ( x )) ∗ W , written b e the o pe r ations defined by s y stem of words W - see [5, Pr o p osition 4.2]. Hence, µf = w ∗ µ ( f ) ho lds for every f ∈ F ( x ). Also there is w ∗ µ − 1 ( x ) ∈ ( F ( x )) ∗ W such that µ − 1 f = w ∗ µ − 1 ( f ) for every f ∈ F ( x ). x = µ − 1 ( µx ) = w ∗ µ − 1  w ∗ µ ( x )  . There exists by O p2 a n isomorphism σ F ( x ) : F ( x ) → ( F ( x )) ∗ W such that σ F ( x ) ( x ) = x . So ( F ( x )) ∗ W is a lso 1 -generated free linear a lgebra of Θ with 7 the free g enerator x . Hence there exists a dec o mpo sition ( F ( x )) ∗ W = ∞ L i =1 F ∗ i , where F ∗ i are line a r spaces of elements which are homog eneous acco rding the degree of x but in resp ect of op erations defined by sy stem of words W . There- fore w ∗ µ ( x ) = λ ∗ x , where λ ∈ k , b y Corol lary 1 from Lemma 3.1 . So µx = λ ∗ x = ϕ ( λ ) x a nd µ = ϕ ( λ ), hence ϕ : k → k is a surjection. w + ∈ F ( x 1 , x 2 ) = F . There exis ts n ∈ N , such that w + ( x 1 , x 2 ) = p 1 ( x 1 , x 2 ) + p 2 ( x 1 , x 2 ) + . . . + p n ( x 1 , x 2 ) , where p i ( x 1 , x 2 ) ∈ F i , 1 ≤ i ≤ n . W e have for every λ ∈ k that w + ( λ ∗ x 1 , λ ∗ x 2 ) = λ ∗ w + ( x 1 , x 2 ) = ϕ ( λ ) w + ( x 1 , x 2 ) = ϕ ( λ ) p 1 ( x 1 , x 2 ) + ϕ ( λ ) p 2 ( x 1 , x 2 ) + . . . + ϕ ( λ ) p n ( x 1 , x 2 ) and w + ( λ ∗ x 1 , λ ∗ x 2 ) = p 1 ( λ ∗ x 1 , λ ∗ x 2 )+ p 2 ( λ ∗ x 1 , λ ∗ x 2 )+ . . . + p n ( λ ∗ x 1 , λ ∗ x 2 ) = p 1 ( ϕ ( λ ) x 1 , ϕ ( λ ) x 2 ) + p 2 ( ϕ ( λ ) x 1 , ϕ ( λ ) x 2 ) + . . . + p n ( ϕ ( λ ) x 1 , ϕ ( λ ) x 2 ) = ϕ ( λ ) p 1 ( x 1 , x 2 ) + ( ϕ ( λ )) 2 p 2 ( x 1 , x 2 ) + . . . + ( ϕ ( λ )) n p n ( x 1 , x 2 ) . W e can take λ ∈ k such that ϕ ( λ ) is not a solution of any equation x i = x , where 2 ≤ i ≤ n . So, p i ( x 1 , x 2 ) = 0 for 2 ≤ i ≤ n by equality of the homogeneous comp onents. There fo re w + = αx 1 + β x 2 , where α, β ∈ k . If we denote the op eration defined by w + in ( F ( x 1 , x 2 )) ∗ W by ⊥ , then x 1 ⊥ x 2 = x 2 ⊥ x 1 holds, so αx 1 + β x 2 = αx 2 + β x 1 and α = β . Also x 1 ⊥ 0 = x 1 holds and αx 1 = x 1 , so α = β = 1. Now, by cons ider ation of F ( x ), we can co nclude that for all λ 1 , λ 2 ∈ k fulfills ϕ ( λ 1 + λ 2 ) x = ( λ 1 + λ 2 ) ∗ x = λ 1 ∗ x ⊥ λ 2 ∗ x = λ 1 ∗ x + λ 2 ∗ x = ϕ ( λ 1 ) x + ϕ ( λ 2 ) x = ( ϕ ( λ 1 ) + ϕ ( λ 2 )) x, so ϕ ( λ 1 + λ 2 ) = ϕ ( λ 1 ) + ϕ ( λ 2 ) a nd ϕ is an automorphism o f the field k . Its clear now that w − = − x ∈ F ( x ), be c ause w − ( x ) = − 1 ∗ x = ϕ ( − 1) x = ( − 1 ) x = − x. w · ∈ F ( x 1 , x 2 ). W e wr ite w · as sum o f its homogeneo us comp onents acco rd- ing the degr e e of x 1 : w · ( x 1 , x 2 ) = p 0 ( x 1 , x 2 ) + p 1 ( x 1 , x 2 ) + p 2 ( x 1 , x 2 ) + . . . + p n ( x 1 , x 2 ) . W e denote the op eration defined by w · in ( F ( x 1 , x 2 )) ∗ W by × . So we have for every λ ∈ k that ( λ ∗ x 1 ) × x 2 = λ ∗ ( x 1 × x 2 ) = ϕ ( λ ) w · ( x 1 , x 2 ) = 8 ϕ ( λ ) p 0 ( x 1 , x 2 ) + ϕ ( λ ) p 1 ( x 1 , x 2 ) + ϕ ( λ ) p 2 ( x 1 , x 2 ) + . . . + ϕ ( λ ) p n ( x 1 , x 2 ) . and ( λ ∗ x 1 ) × x 2 = w · ( ϕ ( λ ) x 1 , x 2 ) = p 0 ( ϕ ( λ ) x 1 , x 2 ) + p 1 ( ϕ ( λ ) x 1 , x 2 ) + p 2 ( ϕ ( λ ) x 1 , x 2 ) + . . . + p n ( ϕ ( λ ) x 1 , x 2 ) = p 0 ( x 1 , x 2 ) + ϕ ( λ ) p 1 ( x 1 , x 2 ) + ( ϕ ( λ )) 2 p 2 ( x 1 , x 2 ) + . . . + ( ϕ ( λ )) n p n ( x 1 , x 2 ) . W e can take, as ab ove, λ ∈ k such that b y equality o f the homogeneo us com- po nent s we obtain that w · ( x 1 , x 2 ) = p 1 ( x 1 , x 2 ). Now we write w · ( x 1 , x 2 ) = p 1 ( x 1 , x 2 ) a s sum of its homogeneous co mpo nents a c cording the degre e of x 2 : w · ( x 1 , x 2 ) = r 0 ( x 1 , x 2 ) + r 1 ( x 1 , x 2 ) + r 2 ( x 1 , x 2 ) + . . . + r m ( x 1 , x 2 ) . W e hav e for every λ ∈ k that x 1 × ( λ ∗ x 2 ) = λ ∗ ( x 1 × x 2 ) = ϕ ( λ ) w · ( x 1 , x 2 ) = ϕ ( λ ) r 0 ( x 1 , x 2 ) + ϕ ( λ ) r 1 ( x 1 , x 2 ) + ϕ ( λ ) r 2 ( x 1 , x 2 ) + . . . + ϕ ( λ ) r m ( x 1 , x 2 ) . and x 1 × ( λ ∗ x 2 ) = w · ( x 1 , ϕ ( λ ) x 2 ) = r 0 ( x 1 , ϕ ( λ ) x 2 ) + r 1 ( x 1 , ϕ ( λ ) x 2 ) + r 2 ( x 1 , ϕ ( λ ) x 2 ) + . . . + r m ( x 1 , ϕ ( λ ) x 2 ) = r 0 ( x 1 , x 2 ) + ϕ ( λ ) r 1 ( x 1 , x 2 ) + ϕ ( λ ) 2 r 2 ( x 1 , x 2 ) + . . . + ϕ ( λ ) m r m ( x 1 , x 2 ) . And, as ab ov e, we can conclude that w · ( x 1 , x 2 ) = r 1 ( x 1 , x 2 ) where r 1 ( x 1 , x 2 ) is a homog eneous element of F ( x 1 , x 2 ) suc h that deg x 1 r 1 ( x 1 , x 2 ) = 1 and deg x 2 r 1 ( x 1 , x 2 ) = 1. Ther e fore w · ( x 1 , x 2 ) = ax 1 x 2 + b x 2 x 1 , where a, b ∈ k . If a = b then the o pe r ation defined by w · ( x 1 , x 2 ) is commutativ e . If a = − b then the op eratio n defined by w · ( x 1 , x 2 ) is anticomm utative. The iso morphisms σ F : F → F ∗ W , where F ∈ ObΘ 0 can not exists in b oth these cases if F is not a 0-genera ted free alg ebra. Therefore we prove tha t if the system of words (3.1) satisfies the conditions Op1 and Op2 then w 0 = 0, w λ = ϕ ( λ ) x 1 for a ll λ ∈ k , w − = − x 1 , w + = x 1 + x 2 , w · = a x 1 x 2 + bx 2 x 1 , where ϕ is an a utomorphism of the field k , a, b ∈ k , a 6 = ± b . Now we must prove that fo r a ll ϕ ∈ Aut k and all a, b ∈ k such that a 6 = ± b the system of words (3.1 ) where w 0 = 0 , w λ = ϕ ( λ ) x 1 for all λ ∈ k , w − = − x 1 , w + = x 1 + x 2 , w · = ax 1 x 2 + b x 2 x 1 fulfills co ndition Op2. It mea ns that we m ust build for every F = F ( X ) ∈ ObΘ 0 an isomorphism σ F : F → F ∗ W such that σ F | X = i d X . W e will prov e, first of all, that H ∗ W ∈ Θ for every H ∈ Θ. Op eratio ns defined by w 0 , w − , w + coincide with 0 , − , +. So iden tities of the v ar iety Θ (axioms of the linear alg e bra) relating to these o p erations fulfill in H ∗ W . Hence we o nly need to chec k the axioms that inv o lve the op erations defined by w · and w λ ( λ ∈ k ). As above we deno te these op erations b y × a nd by λ ∗ . λ ∗ ( x + y ) = ϕ ( λ ) ( x + y ) = ϕ ( λ ) x + ϕ ( λ ) y = λ ∗ x + λ ∗ y , 9 ( λµ ) ∗ x = ϕ ( λ µ ) x = ϕ ( λ ) ϕ ( µ ) x = ϕ ( λ ) ( µ ∗ x ) = λ ∗ ( µ ∗ x ) , ( λ + µ ) ∗ x = ϕ ( λ + µ ) x = ( ϕ ( λ ) + ϕ ( µ ) ) x = ϕ ( λ ) x + ϕ ( µ ) x = λ ∗ x + µ ∗ x, 1 ∗ x = ϕ (1) x = 1 x = x, x × ( y + z ) = ax ( y + z ) + b ( y + z ) x = axy + axz + by x + bz x = x × y + x × z , ( y + z ) × x = a ( y + z ) x + bx ( y + z ) = ay x + az x + bxy + bxz = y × x + z × x, λ ∗ ( x × y ) = ϕ ( λ ) ( axy + by x ) = a ( ϕ ( λ ) x ) y + by ( ϕ ( λ ) x ) = ( ϕ ( λ ) x ) × y = ( λ ∗ x ) × y = x × ( λ ∗ y ) fulfills fo r every x, y , z ∈ H , λ, µ ∈ k . Hence there exists a homomo rphism σ F : F → F ∗ W such that σ F | X = id X for every F = F ( X ) ∈ O bΘ 0 . Our g oal is to pr ov e that these homomorphisms are iso mo rphisms. W e will prove by induction b y i that σ F ( F i ) = F i . (3.2) for every i ∈ N . If X = { x 1 , . . . , x n } then every element of F 1 has for m λ 1 x 1 + . . . + λ n x n , where λ 1 , . . . , λ n ∈ k . σ F ( λ 1 x 1 + . . . + λ n x n ) = λ 1 ∗ σ F ( x 1 )+ . . . + λ n ∗ σ F ( x n ) = ϕ ( λ 1 ) x 1 + . . . + ϕ ( λ n ) x n , so σ F ( F 1 ) ⊂ F 1 . σ F  ϕ − 1 ( λ 1 ) x 1 + . . . + ϕ − 1 ( λ n ) x n  = λ 1 x 1 + . . . + λ n x n , so σ F ( F 1 ) = F 1 . Let (3.2) pr ov ed for i such that 1 ≤ i < r . Ev ery element o f F r is a linear combination of the monomials of the for m uv , wher e u ∈ F i , v ∈ F j , i + j = r . σ F ( uv ) = σ F ( u ) × σ F ( v ) = aσ F ( u ) σ F ( v ) + bσ F ( v ) σ F ( u ) , so σ F ( F r ) ⊂ F r , beca use, by our assumption, σ F ( u ) ∈ F i , σ F ( v ) ∈ F j . Also, if u = σ F ( e u ), v = σ F ( e v ), wher e e u ∈ F r , e v ∈ F t , then σ F ( e u e v ) = σ F ( e u ) × σ F ( e v ) = u × v = auv + bv u σ F ( e v e u ) = σ F ( e v ) × σ F ( e u ) = v × u = av u + buv = buv + av u, fulfills. a 6 = ± b , so the matrix  a b b a  is regula r, hence there exis t α, β ∈ k such that uv = ασ F ( e u e v ) + β σ F ( e v e u ) = σ F  ϕ − 1 ( α ) e u e v + ϕ − 1 ( β ) e v e u  . Therefore σ F ( F r ) = F r . W e can conclude that σ F is an epimor phism. Now we will prov e that ker σ F = 0. Let f ∈ ker σ F ⊂ F ( X ). There ex is ts m ∈ N such that f ∈ m L i =1 F i . σ F  m L i =1 F i  = m L i =1 F i by (3.2). σ F is a linear 10 mapping fr om the linear space m L i =1 F i with the original multiplication by sca la rs in F to the  m L i =1 F i  ∗ W - the linear space m L i =1 F i with the m ultiplication by scalars which we deno te by ∗ . F rom for mulas k P i =1 ( λ i ∗ e i ) = k P i =1 ϕ ( λ i ) e i and k P i =1 λ i e i = k P i =1  ϕ − 1 ( λ i ) ∗ e i  we can conclude that if E is a basis of the line a r space m L i =1 F i then E is a basis of the linear space  m L i =1 F i  ∗ W . So dim m L i =1 F i = dim  m L i =1 F i  ∗ W < ∞ , there fore ker  σ F | m L i =1 F i  = 0 and f = 0. 4 Group A / Y . F ro m now on, W is a s ystem of words (3.1) which fulfills conditions Op1 and Op2. The decomp osition (1.1) is no t split in g eneral case, i. e. S ∩ Y 6 = { 1 } in general cas e. The strongly stable automorphism Φ of the categ o ry Θ 0 which corres p o nds to the sy stem o f words W is inner , by [4, Le mma 3 ], if and only if for every F ∈ ObΘ 0 there exists an isomorphism c F : F → F ∗ W such that c F α = αc D fulfills for every ( α : D → F ) ∈ MorΘ 0 (b y [5, Remar k 3.1] α is a lso a homo morphism from D ∗ W to F ∗ W ). Hear we need to prove o ne tec hnical lemma. Lemma 4.1 If F = F ( X ) ∈ ObΘ 0 and c F : F → F ∗ W is an isomorphism then ther e exists an isomorphi sm c i : F /F i → F ∗ W /F i such that χ ∗ i c F = c i χ i , wher e χ i : F → F /F i and χ ∗ i : F ∗ W → F ∗ W /F i ar e n atur al homomorphisms, i ∈ N . Pro of. If H ∈ Θ and I is an idea l of H . If λ ∈ k , y ∈ I , h ∈ H , then λ ∗ y = ϕ ( λ ) y ∈ I , y × h = ay h + b h y ∈ I , analo g ously h × y ∈ I . Therefore I is an ideal of H ∗ W . Hence F i is an ideal of F ∗ W . If σ F : F → F ∗ W is an is omorphism such that σ F | X = i d X , then by (3 .2) we hav e c − 1 F  F i  = c − 1 F σ F  F i  = F i bec ause c − 1 F σ F : F → F is a n isomor phism. So c F  F i  = F i . It finishes the pr o of. Prop ositio n 4. 1 The stro ngly st able automorphism Φ which c orr esp onds t o t he system of wor ds W is inner if and only if ϕ = id k and b = 0 . Pro of. W e s uppo se that s trongly stable auto mo rphism Φ which c o rresp onds to the s y stem of words W is inner. W e assume that ϕ 6 = id k , i., e., there exis ts λ ∈ k such that ϕ ( λ ) 6 = λ . W e denote F = F ( x ). W e take α ∈ End F , such that α ( x ) = λx . W e supp os e that c F : F → F ∗ W is an isomorphism. c 2 is defined as in the Lemma 4.1, and we by this Lemma we have: χ ∗ 2 c F ( x ) = c 2 χ 2 ( x ) = µ ∗ χ ∗ 2 ( x ) = χ ∗ 2 ( µ ∗ x ) = χ ∗ 2 ( ϕ ( µ ) x ) , 11 where op erations in alg ebra F ∗ W /F 2 we denote by s ame symbols as op era tions in algebra F ∗ W and µ ∈ k \ { 0 } . Ther efore c F ( x ) ≡ ϕ ( µ ) x  mo d F 2  . α  F 2  ⊂ F 2 fulfils, s o αc F ( x ) = α ( ϕ ( µ ) x + f 2 ) ≡ α ( ϕ ( µ ) x ) = ϕ ( µ ) α ( x ) = ϕ ( µ ) λx  mo d F 2  , where f 2 ∈ F 2 . c F α ( x ) = c F ( λx ) = λ ∗ c F ( x ) = ϕ ( λ ) c F ( x ) ≡ ϕ ( λ ) ϕ ( µ ) x  mo d F 2  . µ 6 = 0, so ϕ ( µ ) 6 = 0, ϕ ( λ ) 6 = λ hence αc F 6 = c F α . This contradiction proves that ϕ = id k . Now we denote F = F ( x 1 , x 2 ) ∈ ObΘ 0 . By our assumption there exists an isomorphism c F : F → F ∗ W such that c F α = αc F fulfills for every α ∈ End F . c 2 is de fined as in the Lemma 4.1. α  F 2  ⊂ F 2 so we can define the ho momorphism e α : F /F 2 → F /F 2 such that e αχ 2 = χ 2 α . F rom c F α = α c F we ca n conclude c 2 e α = e αc 2 fulfills. By Lemma 4 .1 c 2 is a reg ular linear mapping. W e can take the endomorphisms α such tha t e α will b e an ar bitr ary linea r mapping from k 2 to k 2 . Therefore c 2 m ust b e a re g ular linea r mapping from k 2 to k 2 which commutate with all linear ma ppings from k 2 to k 2 . Hence c 2 m ust b e a scala r mapping, i.e., χ ∗ 2 c F ( x i ) = c 2 χ 2 ( x i ) = λχ ∗ 2 ( x i ) = χ ∗ 2 ( λx i ) , where λ ∈ k \ { 0 } , i = 1 , 2 . Therefor e c F ( x i ) = λx i + f i , where f i ∈ F 2 , i = 1 , 2. W e can rema rk that now we consider the c a se when ϕ = id k , hence we need no t distinguish b etw een m ultiplication by s calar in F and F ∗ W . Now we take α ∈ End F such that α ( x 1 ) = x 1 x 2 , α ( x 2 ) = 0. If u is a monomial which contain only en tries of x 1 , then deg x 1 α ( u ) + deg x 2 α ( u ) = 2 deg x 1 u . If a mo no mial u con ta in at leas t one entry of x 2 , then α ( u ) = 0. Hence α  F 2  ⊂ F 3 . So we have c F α ( x 1 ) = c F ( x 1 x 2 ) = c F ( x 1 ) × c F ( x 2 ) = ac F ( x 1 ) c F ( x 2 ) + bc F ( x 2 ) c F ( x 1 ) = a ( λx 1 + f 1 ) ( λx 2 + f 2 )+ b ( λx 2 + f 2 ) ( λx 1 + f 1 ) ≡ aλ 2 x 1 x 2 + bλ 2 x 2 x 1  mo d F 3  . αc F ( x 1 ) = α ( λ x 1 + f 1 ) ≡ λx 1 x 2  mo d F 3  . Hence we co nclude b = 0 fr om c F α = αc F . If b = 0, i. e., w · = ax 1 x 2 , a 6 = 0 , then we take c F ( f ) = a − 1 f for every F ∈ ObΘ 0 and every f ∈ F . It is obvious that c F is a re gular linear mapping. c F ( f 1 ) × c F ( f 2 ) = ac F ( f 1 ) c F ( f 2 ) = a  a − 1 f 1   a − 1 f 2  = a − 1 f 1 f 2 = c F ( f 1 f 2 ) . for every f 1 , f 2 ∈ F . So c F : F → F ∗ W is an iso morphism. It fulfils c F α ( d ) = a − 1 α ( d ) = α  a − 1 d  = αc F ( d ) for every ( α : D → F ) ∈ Mo rΘ 0 and every d ∈ D . 12 Prop ositio n 4. 2 The gr oup S ∼ = G ⋋ Aut k , wher e G is the gr oup of al l r e gular 2 × 2 m atric es over field k , which have a form  a b b a  and every ϕ ∈ Aut k acts on t he gr ou p G by this way: ϕ  a b b a  =  ϕ ( a ) ϕ ( b ) ϕ ( b ) ϕ ( a )  . Pro of. W e will define the mapping τ : G ⋋ Aut k → S . If g ϕ ∈ G ⋋ Aut k , where g =  a b b a  , then w e define τ ( g ϕ ) = Φ ∈ S , wher e Φ cor r esp onds to the system of words W with w λ = ϕ ( λ ) x 1 for every λ ∈ k and w · = a x 1 x 2 + bx 2 x 1 . By Sectio n 2 and Theorem 3 .1 τ is bijection. W e consider τ ( g 1 ϕ 1 ) = Φ 1 and τ ( g 2 ϕ 2 ) = Φ 2 , where g 1 ϕ 1 , g 2 ϕ 2 ∈ G ⋋ Aut k and g 1 =  a 1 b 1 b 1 a 1  , g 2 =  a 2 b 2 b 2 a 2  . Both these stro ngly stable automo r - phisms preser ves all ob jects o f Θ 0 and acts o n morphisms o f Θ 0 by theirs systems of bijections n s Φ i F : F → F | F ∈ O bΘ 0 o , for i = 1 , 2, accor ding the formula (1.2). W e hav e Φ 2 Φ 1 ( α ) = s Φ 2 F s Φ 1 F α  s Φ 1 D  − 1  s Φ 2 D  − 1 for every ( α : D → F ) ∈ MorΘ 0 . So stro ng ly stable a uto morphism Φ 2 Φ 1 = τ ( g 2 ϕ 2 ) τ ( g 1 ϕ 1 ) preserves all o b jects of Θ 0 and acts o n morphisms of Θ 0 by sys tem o f bijections n s Φ 2 F s Φ 1 F : F → F | F ∈ O bΘ 0 o . This system of bijections satisfies the c o nditions B1 and B2, so we can define the words w Φ 2 Φ 1 λ for every λ ∈ k and w Φ 2 Φ 1 · which corr esp ond to the automor phism Φ 2 Φ 1 by formula (2 .2). T he words w Φ i λ ( λ ∈ k ) and w Φ i · which corresp ond to the automorphism Φ i hav e forms w Φ i λ = ϕ i ( λ ) x 1 ( λ ∈ k ) and w Φ i · = a i x 1 x 2 + b i x 2 x 1 for i = 1 , 2 . So w Φ 2 Φ 1 λ = s Φ 2 F s Φ 1 F ( λx 1 ) = s Φ 2 F  w Φ 1 λ  = s Φ 2 F ( ϕ 1 ( λ ) x 1 ) = ϕ 2 ( ϕ 1 ( λ )) x 1 = ( ϕ 2 ϕ 1 ) ( λ ) x 1 for every λ ∈ k and w Φ 2 Φ 1 · = s Φ 2 F s Φ 1 F ( x 1 x 2 ) = s Φ 2 F  w Φ 1 ·  = s Φ 2 F ( a 1 x 1 x 2 + b 1 x 2 x 1 ) = ϕ 2 ( a 1 ) s Φ 2 F ( x 1 x 2 ) + ϕ 2 ( b 1 ) s Φ 2 F ( x 2 x 1 ) = ϕ 2 ( a 1 ) ( a 2 x 1 x 2 + b 2 x 2 x 1 ) + ϕ 2 ( b 1 ) ( a 2 x 2 x 1 + b 2 x 1 x 2 ) = ( ϕ 2 ( a 1 ) a 2 + ϕ 2 ( b 1 ) b 2 ) x 1 x 2 + ( ϕ 2 ( a 1 ) b 2 + ϕ 2 ( b 1 ) a 2 ) x 2 x 1 . bec ause s Φ i F : F → F ∗ W i is an iso morphism, i = 1 , 2. Hence Φ 2 Φ 1 = τ ( g 2 ϕ 2 ) τ ( g 1 ϕ 1 ) = τ ( g 2 ϕ 2 ( g 1 ) ϕ 2 ϕ 1 ) = τ ( g 2 ϕ 2 · g 1 ϕ 1 ) . Corollary 1 Gr oup S ∩ Y is isomorph ic to the gr oup k ∗ I 2 of the r e gular 2 × 2 sc alar matric es over field k . 13 Pro of. By Prop o s itions 4.1 and 4 .2. Corollary 2 A / Y ∼ = ( G/k ∗ I 2 ) ⋋ Aut k . Pro of. By Pro po sition 4.2 and Corolla ry 1 we hav e that A / Y ∼ = ( G ⋋ Aut k ) /k ∗ I 2 . And we hav e ( G ⋋ Aut k ) /k ∗ I 2 ∼ = ( G/k ∗ I 2 ) ⋋ Aut k b ecause k ∗ I 2 ⊳ G and for ev- ery ϕ ∈ Aut k ϕ ( k ∗ I 2 ) ⊂ k ∗ I 2 fulfills. The symmetric group of the se t whic h ha s 2 elements - S 2 can be em- bedded in the m ultiplicative structure of the algebra M 2 ( k ) of the 2 × 2 ma- trices over field k : S 2 ∋ (12) →  0 1 1 0  ∈ M 2 ( k ), so G ∼ = U ( k S 2 ), where U ( k S 2 ) is the group of all inv e r tible ele ments o f the group alg ebra k S 2 . Also k ∗ I 2 ∼ = U ( k { e } ), where e ∈ S 2 , k { e } is a subalgebra of k S 2 , U ( k { e } ) is a group of all inv e rtible elemen ts of this subalgebra . Therefore A / Y ∼ = ( U ( k S 2 ) /U ( k { e } )) ⋋ Aut k , where every ϕ ∈ Aut k acts on the alg ebra k S 2 by natur al way: ϕ ( ae + b (12)) = ϕ ( a ) e + ϕ ( b ) (12). 5 Example of t w o linear algebras whic h are au- tomorphically equiv alen t but not geometrically equiv alen t. W e take k = Q . Θ will b e the v ariety of all line a r algebras ov er k . H will be the 2-genera ted linear a lgebra, which is free in the v ariety cor resp onding to the ident ity ( x 1 x 1 ) x 2 = 0 . W e cons ide r the stro ngly stable automorphism Φ of the category Θ 0 corres p o nding to the system of words W , wher e b 6 = 0. Algebras H and H ∗ W are automo rphically equiv a lent by [5 , Theor em 5.1]. Prop ositio n 5. 1 Algebr as H and H ∗ W ar e n ot ge ometric al ly e quivalent. Pro of. Let F = F ( x 1 , x 2 ). The ideal I = I d ( H , { x 1 , x 2 } ) of the all t wo- v ariables identities which are fulfill in the algebr a H will b e the sma llest H - closed set in F , b ecause I = (0) ′′ H , wher e 0 ∈ F . I f algebra s H a nd H ∗ W are geometrically equiv alent then the structures of the H -closed sets and of the H ∗ W -closed s ets in F co incide. Hence I m ust b e the smallest H ∗ W -closed set in F . By [5, Remar k 5 .1] T → σ F T (5.1) is a bijection from the str ucture of the H ∗ W -closed sets in F to the s tructure of the H - closed sets in F . Hea r σ F : F → F ∗ W is an isomo r phism fro m condition Op2. It is clear that the bijection (5.1) preser ves inclus io ns of s e ts. So it transforms the sma llest H ∗ W -closed set to the sma llest H -closed set, i. e. I = σ F I must fulfills. It is obviously that I ⊂ F 3 . By (3.2) σ F I ⊂ F 3 . W e will compare the linear subspaces I /F 4 and ( σ F I ) /F 4 . I = h α (( x 1 x 1 ) x 2 ) | α ∈ End F i . Let 14 α ( x i ) ≡ α 1 i x 1 + α 2 i x 2  mo d F 2  , where i = 1 , 2, α j i ∈ k . Then α (( x 1 x 1 ) x 2 ) ≡ (( α 11 x 1 + α 21 x 2 ) ( α 11 x 1 + α 21 x 2 )) ( α 12 x 1 + α 22 x 2 )  mo d F 4  . W e achiev e a fter the extending of br ack ets that I /F 4 is a subspa ce of the linear space spa nned by the elements of F 3 /F 4 which hav e fo r m ( x i x j ) x k + F 4 , where i, j, k = 1 , 2. But σ F I ∋ σ F (( x 1 x 1 ) x 2 ) = aσ F ( x 1 x 1 ) σ F ( x 2 ) + bσ F ( x 2 ) σ F ( x 1 x 1 ) = = a ( a + b ) ( x 1 x 1 ) x 2 + b ( a + b ) x 2 ( x 1 x 1 ) . W e ha ve that a + b 6 = 0, b 6 = 0, so I /F 4 6 = ( σ F I ) /F 4 and I 6 = σ F I . This contradiction prov es tha t alge br as H and H ∗ W are not g eometrically equiv a lent . References [1] B. P lo tkin, V arieties of algebr as and algebr aic varieties. Cate gories of alge- br aic varieties. Sib erian Adv ance d Mathematics, Allerton Pr e ss, 7 :2 (1997), pp. 6 4 – 97. [2] B. Plotkin, Some notions of algebr aic ge ometry in un iversal algebr a, Algebra and Analysis, 9:4 (199 7 ), pp. 224 – 248, St. Petersburg Ma th. J., 9:4 , (19 98) pp. 8 59 – 879. [3] B. P lotkin, Algebr as with the same (algebr aic) ge ometry, Pro ceedings o f the Int ernationa l Co nfer ence o n Mathematical Logic, Algebra and Set Theor y , dedicated to 100 a nniversary of P .S. Novik ov, Pro ceedings o f the Steklov Institute o f Ma thematics, MIAN, v.24 2, (2003 ), pp. 17 – 2 07. [4] B. P lo tkin, G. Zhitomirski On automorphisms of c ate gories of fr e e algebr as of some varieties, J ournal of Algebra , 306 :2 , (2006 ), 344 – 3 67. [5] A. Tsurkov, Automorphi c e quivalenc e of algebr as. International Jo urnal o f Algebra and Computatio n. 17:5 /6 , (2007 ), 126 3–12 7 1. 15