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UDC 512.552.4 DOI 10.24147/1812-3996.2024.2.9-20
IDENTITIES OF SPACES OF LINEAR TRANSFORMATIONS AND NONASSOCIATIVE LINEAR ALGEBRAS
A. V. Kislitsin
Postgraduent Student, e-mail: avkislitsin@gmail.com
Dostoevsky Omsk State University, Omsk, Russia Altai State Pedagogical University, Barnaul, Russia
Abstract. The template shows the design rules of articles submitted for publication in the journal "Mathematical Structures and Modelling".
Keywords: template, magazine, rules, pictures, drawings, tables, formulas, quote, tire, references, bibliography, literature.
Dedicated to the 85th anniversary of Prof. V.N. Remeslennikov
1. Introduction
Let F be a field, A be a linear associative F-algebra and E is a subspace in A (but E not a necessary subalgebra of A) which generates A as a linear F-algebra. In this case, we call E a multiplicative vector space (in short, an L-space) over the field F. The algebra A will be called enveloping for the space E, and the space E will be called embedded in the algebra A.
The identity of an L-space E over a field F (embedded in an F-algebra A) is an associative polynomial f (xl,x2,..., xn) which equals zero in A if, instead of its variables xl,x2,... ,xn we substitute any elements from E. In this form, the concept of a multiplicative vector space and its identity was introduced in 2010 by I. M. Isaev and the author [1]. However, (direct) analogs of this concept have been studied earlier.
In 1978, I. V. L'vov considered algebras of the form V = V © E, where V is a vector space and E C End^ V. Nonzero products of elements of this algebra are given by the rule: ^e^ = Vj. It's clear that V E Vax{x(yz) = 0). The nonassociative polynomial zf (RX1, RX2,..., RXn) is an identity of the algebra V iff the associative polynomial f (xl,x2,..., xn) is equal to zero when substituting instead of variables linear combinations of elements from E [2]. In fact, this construction uses the concept of a multiplicative vector space.
In 1973, Yu. P. Razmyslov introduced the concept of a weak identity of an associative Lie pair (A,L), where L is a Lie algebra and A is its associative enveloping. A weak identity of a pair (A,L) is an associative polynomial f (xl , X2, ... , ) that is equal to zero in the algebra A when substituted instead of variable Xi, X2, ... , elements of the algebra L [3].
Following the above construction, the identity of the multiplicative vector space E (with the enveloping algebra A) can be considered (if necessary) as a weak identity of the pair (A, E). The pair (A, E) in this case will be called a multiplicative vector pair.
For example, consider the set
Eq =
№ e4
This set is a vector space over the field F, but it is not an F-algebra. The algebra T2(F) of upper triangular matrices is the enveloping algebra for E0. Consider the polynomial St3(x^x2,x3) = (—1)^Since dimFE0 = 2, the polynomial
a£S 3
St3(x^ x2, x3) is the identity of the F-space E0. But this polynomial is not an identity of the algebra T2(F), because St3(en, e12, e22) = e12 = 0.
2. Identities of Vector Spaces
Let F{X) be a free associative algebra, and 0 = G C F{X). By T(G) we denote the T-ideal of the algebra F{X) generated by the set G, and by L(G) we denote the ideal of F{X) generated by the polynomials of the set G (as an ideal) and closed with respect to substitutions instead of variables of linear combinations of variables. These ideals will be called L-ideals. It's clear that L(G) C T(G).
The set of all identities of a vector space E is a L-ideal of F {X). We will denote such a L-ideal by L(E). The converse is also true: every L-ideal of the algebra F{X) is the set of identities of some L-space.
The identity f of the L-space E follows from the identities /1; /2,... if / G L( f1} f2,...). Thus, for obtaining a corollary from the identity /(x1 , X2, . . . , Xn ), instead of variables X1, X2,..., xn, only linear combinations of variables can be substituted. If we substitute the product of variables instead of variables x1; x2,..., xn in /(x1, x2,..., xra), the resulting polynomial may not be an identity of the L-space.
For example, the space E0 = {e 11 + e12, e22)^ satisfies the identity St3(x,y, z) = 0. However, E0 does not satisfy the identity St3(xi,y, z) = 0. Indeed, if x = e11 + e12, t = e 22, y = en + e12, z = e 22 then St3(xi, y, z) = St3(e 12, en + e12, e22) = -e 12 = 0.
If there exists a finite set G of identities of a multiplicative vector space E, from which all the identities of this space follow, then the space E is called a finitely based L-space (FB-space) with a basis of identities G. If such a finite set does not exist for the L-space E, then we say that the L-space E is infinitely based or not finitely based (NFB-space).
FB-algebras and NFB-algebras are similarly defined in other classes of algebras, which may be related to Specht's problem [?]: does every associative algebra over a field of characteristic zero have a finite basis of identities? In 1987, A. R. Kemer gave a positive answer to Specht's problem [5]. However, there are examples of nonfinitely based associative algebras over an infinite field of characteristic p > 0, alternative solvable algebras over a field of characteristic two, alternative commutative (and hence Jordan) algebras over a field of characteristic three. A detailed review of these problems can be found in [6]. The construction of examples of NFB-algebras is an important direction in the study of varieties of algebras.
There are known examples of NFB-algebras in various classes of algebraic systems:
• in the class of groupoids, an example was constructed by R. C. Lyndon [7];
• in the class of semigroups — P. Perkins [8];
• in the class of rings and linear algebras — S. V. Polin [9];
• in the class of loops — M. R. Vaughan-Lee [10].
Let us give examples of FB-spaces and NFB-spaces.
Proposition 1.1 [1]. Let F be an infinite field of arbitrary characteristics. The L-spaces Al = (ell,el2)F and A2 = {ell,e2l)F over the field F are FB-spaces with bases of identities {[x, y]z} and {x[y, z]} respectively.
Theorem 1.1 [1]. Let F be an infinite field of arbitrary characteristics. The vector space A = Al © A2 over the field F is an NFB-space with a basis of identities:
{St3(x,y,z),x[y,u]v, [x,y][u,v], [x,y]zlZ2 ... zm[u,v]\m = 1, 2,... }.
Note that the space A = Al © A2 is an NFB-space over an arbitrary infinite field. However, any linear associative algebra over a field of characteristic zero has a finite basis for its identities [5].
Proposition 1.2 [11]. Let F = GF(q) be a finite field of q elements. The L-spaces Al = (ell,el2)p and A2 = {ell,e2l)F over the field F are FB-spaces with bases of identities {[x, y]z, (xq — x)y} and {x[y, z],x(yq — y)} respectively.
Theorem 1.2 [11]. Let F = GF(q) be a finite field of q elements. The vector space A = Al © A2 over the field F is an NFB-space with a basis of identities:
{St3(x,y,z),x[y,u}v, [x,y][u,v],x(y — yq)z, (x — xq)(y — yq), [x,y](z — zq),
(x — xq )[y, z], [x, y]zlZ2 ...zm [u,v]\m = 1, 2,... }.
Theorem 1.3 [1]. Let F be an infinite field of arbitrary characteristics. The multiplicative vector space T2(F) of upper triangular matrices over the field F is an NFB-space with a basis of identities:
{[x,y][u,v], [x,y]zlZ2 ... zm[u,v]\m = 1, 2,... }.
Theorem 1.4 [12]. Let F = GF(q) be a finite field of q elements. The multiplicative vector space T2(F) of upper triangular matrices over the field F is an NFB-space with a basis of identities:
G = {[x,y}[u,v}, (x — xq)(y — yq), [X,y](Z — Z9),
(x — xq )[y, z], [x, y]zlZ2 ...zm [u,v]\m = 1, 2,... }.
Theorem 1.5 [13]. Let F be a field of characteristic zero. The multiplicative vector space E0 = {ell + el2, e22)F is an NFB-space with a basis of identities:
{St3(x,y,z), [x,y][u,v], [x,y]zlZ2 . ..Zm[u,v]\m =1, 2,... }.
Theorem 1.6 [14]. Let F = GF(q). The multiplicative vector space E0 = {ell + zl2, e22)f over the field F is an NFB-space with a basis of identities:
{xq2-q+l — x, St3(x,y,z), [x,y][u,v], (x — xq)(y — yq),
[x,y](z — zq), (x — xq )[y,z], [x,y]ZlZ2 ...Zk [u,v]\k = 1, 2,... }.
Note that the space E0 is not an F-algebra.
3. Inherently Nonfinitely Based Spaces and Strongly Nonfinitely Based Spaces
A finite NFB-algebra (of arbitrary signature) is called inherently nonfinitely based (INFB-algebra) if any locally finite variety containing this algebra does not have a finite basis of identities. Sometimes inherently nonfinitely based algebras are called essentially nonfinitely based.
Any finite algebra containing an INFB-algebra as a subalgebra has no finite basis of identities. Thus, if we construct an example of an INFB-algebra, then a series of examples of NFB-algebras is automatically constructed.
There are known examples of INFB-algebras in various classes of algebraic systems:
• in the class of groupoids, an example was constructed by R. C. Lyndon [7];
• in the class of semigroups — M. V. Sapir [15];
• in the class of rings and linear algebras — I. M. Isaev [12].
In 1987, M. V. Sapir gave a complete description of INFB-semigroups.
Let us give examples of INFB-spaces.
Theorem 2.1 [11]. Let F = GF(g) be a finite field of q elements. The multiplicative vector space T2(F) of upper triangular matrices over the field F is an INFB-space.
It follows from this theorem that the multiplicative vector space of matrices of any order over a finite field has no finite basis of identities. However, inherently nonfinitely based algebras and multiplicative vector spaces can be considered only for algebras and L-spaces over a finite field.
Consider a monomial w = w(x1, x2,..., xn; y1, y2,..., yk) G F{X) that is linear in each of the variables x1, x2, . . . , xn. Let C,lw) = 0 be the Capelli identity and Cap(n) = Var{ C,(r) = 0) the variety of linear algebras satisfying all possible Capelli identities for a fixed n.
A variety M of linear algebras over a field F is called strongly nonfinitely based (SNFB-variety) if M C Cap( fc) for some k and any variety of F-algebras containing M and contained in Cap(n) for some n is NFB-variety.
The algebra generating the SNFB-variety of algebras will be called the SNFB-algebra. Any finite dimensional algebra containing an SNFB-algebra as a subalgebra has no finite basis of identities. Strongly nonfinitely based multiplicative vector spaces can be defined similarly.
Let us give examples of SNFB-spaces.
Theorem 2.2 [11]. Let F be an arbitrary field. The multiplicative vector space T2(F) of upper triangular matrices over the field F is an SNFB-space.
Theorem 2.3 [13]. Let F be an field of characteristic zero. The multiplicative vector space E0 = {e 11 + e12, e22)^ over the field F is an SNFB-space.
Theorem 2.4 [11]. Let F be an arbitrary field. The multiplicative vector space A = A1 © A2 over the field F is an NFB space, but it is not an SNFB-space.
4. L-varieties
Let G C F{X). The class of all multiplicative vector pairs of the form (A, E) satisfying all the identities of the set G is called an L-variety defined by the set of identities G and is denoted by VarL(g = 0 | g G G).
If G is a basis of identities in the space E, then the L-variety VarL{g = 0 | g G G) is denoted by Var^E and called the L-varieties generated by the space E. If we need information about the enveloping algebra A of the space E, then VarL(g = 0 | g G G) is denoted as VarL(A, E) and called the L-varieties generated by the pair (A, E).
Using the concept of an L-variety, it was possible to partially describe the FB-space and SNFB-spaces.
Theorem 3.1 [13]. Let F be a field of characteristic zero, and A be a finite dimensional L-space over the field F, which is also an F-algebra with a unity element. An L-space A has a finite basis of identities iff T2(F) G Var^A.
Theorem 3.2 [13]. Let F be a field of characteristic zero, and A be a finite dimensional L-space over the field F, which is also an F-algebra with a unity element. An L-space A is strongly infinitely based iff T2 (F) G Var^A.
It can be shown that any class of multiplicative vector pairs is an L-variety iff it is closed with respect to taking subpairs, homomorphic images of pairs, and direct products of pairs; that is, for L-varieties, an analogue of Birkhof's theorem holds. In the study of L-varieties, we can pose the same problems as for varieties of linear algebras.
An L-variety M is called Specht if any pair (A, E) G M has a finite basis of identities. The union of the L-varieties M1 and M2 is the smallest L-variety containing M1 and M2.
Let F be an arbitrary field; A1 = {e 11, e21 )F, A2 = {e 11, e12)^ is the vector space over field F, M1 = VarLA1, M2 = VarLA2, M = VarL(A1 © A2). It's obvious that M = M1 U M2. Note that M1 = VarL{x[y, z] = 0), M2 = VarL{[x, y]z = 0).
Theorem 3.3 [16]. Let F be an arbitrary field. An infinitely based L-variety M is the union of the Specht L-varieties M1 and M2.
Corollary 3.1. Let F be an infinite field. An arbitrary L-space over the field F satisfying either the identity [x, y]z = 0 or the identity x[y, z] = 0 has a finite basis of identities.
Corollary 3.2. Let F = GF(g). An arbitrary L-space over the field F satisfying either the identities [x, y]z = 0 and (x9 — x)y = 0 or the identities x[y, z] = 0 and x(yq — y) = 0 has a finite basis of identities.
The following result of V. S. Drensky strengthens corollary 3.1 and corollary 3.2.
Theorem 3.4 [17]. Over an arbitrary field, every L-ideal that contains one of the weak polynomial identities [x, y]z or x[y, z] is finitely generated.
V. S. Drensky also received a description of all L-ideals containing a polynomial [x, y]z (or x[y, z]).
Theorem 3.5 [17]. Over a field of characteristic zero, the following L-ideals are all L-ideals that contain the weak identity [x, y] z:
• The L-ideal generated by [x, y] z;
• The L-ideal generated by [x, y]z and the weak identity xra[x, y], n > 0;
• The L-ideal generated by [x, y]z and the weak identity xm, m > 1;
• The L-ideal generated by [x, y]z and the weak identities xm, m > 2 and xn[x, y],
0 < n < m — 2.
There is also a dual theorem for the weak identity x[y, z].
Let further F be a field of characteristic zero and Ml = V&rL{xy[z,t] = 0), M2 =
VQXl([x, y]zt = 0).
Theorem 3.6 [18]. Let F be a field of characteristic zero. L-varieties M\_ and M2 are Specht.
Corollary 3.3. Let F be a field of characteristic zero. An arbitrary L-space over the field F satisfying either the identity [x, y]zt = 0 or the identity xy[z, t] = 0 has a finite basis of identities.
It can be shown that M = Ml U A2 is not a Specht L-variety.
Recall that a T-ideal generated by the set G (in the notation T(G)) is an ideal of the algebra F {X) such that f (wl,w2,... ,wn) E F{X) for all f(xl,x2,...,xn) E T(G), wl,w2,... ,wn E F{X) and T(G) is the smallest ideal containing G. The set of all identities in linear algebra forms a T-ideal.
An ideal of the algebra F {X) that is closed under linear permutations of variables will be called an L-ideal. The smallest L-ideal containing the set G will be called the L-ideal generated by the set G. It is clear that the set of identities of a multiplicative vector space forms an L-ideal.
Proposition 3.1 [16,18]. Let be G C F{X). If [x,y]z E G (x[y,z] E G) then L(G) = T(G). However, if [x, y]zt E G (xy[z, t] E G) then L(G) C T(G)
An L-variety M is called locally finite if for any pair (A, E) e M, where E is a finite dimensional vector space, the algebra A is finite. A pair (A, E) will be called critical if the algebra A is finite and the pair (A, E) does not belong to the L-variety generated by its proper factors. Any locally finite L-variety is generated by its critical pairs.
Critical rings and algebras are well studied. They are an effective means for the study of varieties of rings and linear algebras. Theorems 3.7-3.9 were proven by I. M. Isaev and the author.
Theorem 3.7. Let M2(F) be the algebra of second-order matrices over a finite field F = GF(q). Pair (M2(F), M2(F)) is critical.
Theorem 3.8. Let T2(F) be the algebra of upper triangular matrices of second-order over a finite field F = GF(q). Pair (T2(F),T2(F)) is critical.
Theorem 3.9. Let F = GF(q), Al = {ell,el2)F, and A2 = {ell,e2l)F. Pairs (.Al, Ai) and (A2, A2) are critical.
A variety M of linear algebras is called a almost 9-variety if M does not satisfy the 9 property, but any proper subvariety of the variety M satisfies this property.
Almost ^-varieties of algebras play an important role in the study of varieties of linear algebras. In the theory of associative rings and linear algebras, almost ^-varieties are related to the indicator characterization of varieties.
If the property 9 is a concrete identity, then in the class of associative rings and linear algebras there are descriptions of almost ^-varieties:
• almost commutative varieties of rings [19];
• almost commutative varieties of ^-algebras, where $ is a Noetherian commutative Jacobson ring with unity element [20];
• almost Engel varieties of the linear algebras [21];
• almost permutative varieties of algebras over an infinite field [22].
Description problems can be formulated for almost ^-varieties of L-spaces. Theorem 3.10 [23]. Let F be a finite field and M a nonnilpotent L-variety generated by an F-algebra considered as a vector space. Then an L-variety M is almost commutative if and only if it is generated by one of the following spaces:
where a G AutP, a = 1 and the field of invariants Pa is the only maximal subfield of P containing F. The L-variety M3 = VarLA3 contains a noncommutative proper L-subvariety.
Theorem 3.11 [24]. Let F be an arbitrary field and M be a L-variety generated by an F-algebra considered as a vector space. Then an L-variety M is almost Engel if and only if it is generated by one of the following spaces:
where a G AutP, a =1 and the field of invariants Pis the only maximal subfield of P containing F. The L-variety M3 = VarLA3 contains a non-Engel proper L-subvariety.
An L-variety M is called a minimal nonzero L-variety (with respect to the inclusion) or an atom if for any L-variety N it follows from inclusion N C M that either M = N or M is the zero L-variety.
In 1956, A. Tarski showed that atoms in the class of rings are generated either by a simple field GF(p) or by a ring with zero multiplication [25]. The problem of describing atoms in the class of L-spaces is of interest for study.
Theorem 3.12 [26]. An L-variety of multiplicative vector spaces over a field GF(2) is an atom iff it coincides with either M0, or M1, or Mp(x), where
A1 = {e 11, 612)^, A2 = {e 11, 621)^
Corollary 3.4. Let
A1 = {e 11, 612)^, A2 = {e 11, 621)^
Corollary 3.5. Let
Mo = VarL{xy = 0), M1 = VarL{[x, y] = 0,x + x2 = 0), Mp(x) = VarL{[x,y] = 0,x2y = xy2,x ■ p(x) = 0),
p(x) is an irreducible polynomial over the field GF(2).
Note that the identities of the L-variety Mp(x) do not define an atom in the class of linear algebras over the field GF(g).
5. Identities of Nonassociative Linear Algebras
Let P = Var{x(y z) = 0) be the variety of left-nilpotent algebras of index 3. This variety of linear algebras was first considered in 1976 by S. V. Polin [9].
Let V be a vector space, and E is the (sub)space of linear transformations of the space V. Consider the algebra V = V © E. The nonzero products of basis elements of this algebra are given by the rule: ^e ij = Vj for Vi G V and e^ G E. It is easy to see that V G p. As we said earlier, the identities of the algebra V = V © E and L-space E are very closely related.
Theorem 4.1 [2]. The nonassociative polynomial z/ ( RX1, RX2,..., RXn) is an identity of the algebra V = V © E iff the associative polynomial /(x1, x2,..., xn) is an identity of the L-space E.
Corollary 4.1. Let G = {/,( xil , xt2 , . . . , xik ) N G
C F{X),
z G = {z /,( RXn ,...,Rifc )|*G/}. _
The set zG is the basis of an identities for the algebra V = V © E iff the set G is a basis of an identities for the L-space E.
Using Corollary 4.1 and the results obtained earlier for multiplicative vector spaces and their identities, we can obtain a number of consequences for nonassociative algebras of the form V = V © E.
We formulate some corollaries of theorems proved for L-spaces. We assume that modulo x(yz) = 0 the brackets in the arbitrary word x1x2... are placed according to the rule: x1x2... = ((... ((x1x2)x3)... )x^-1)xfc and the writing z/(x1, x2,..., xn) is a short form of the writing of z/( RX1, RX2,..., RXn).
Theorem 4.2 [11]. Let F be an infinite field of arbitrary characteristics. The nonassociative algebra A = {w 1, i>2, w3, v4)f © {e 11, e12, e33, e43)^ is an NFB-algebra with a basis of identities:
{x(y z), zSt3(x, y, i), zx[y, , z[x, y][w, 1>],
z[x, y]Z1Z2 ... zm[m, w]|m =1, 2,... }.
Theorem 4.3 [11]. Let F = GF(g) be a finite field. The nonassociative algebra
A = {w 1, i>2, w3, v4)f © {e 11, e12, e33, e43)^ is an NFB-algebra with a basis of identities:
{x(y z), zSt3(x, y, i), zx[y, , z[x, y][w, w], zx(y — yq)i, z(x — xq)(y — yq),
z[x, y](i — tq), z(x — xq)[y, i], z[x, y]Z1Z2 ... zm[m, w]|m = 1, 2,... }.
Theorem 4.4 [13]. Let F be an infinite field of characteristic zero. The nonassociative algebra A = {w 1, w2, e11 + e12, e22)F is an NFB-algebra with a basis of identities:
{x(yz), zSt3(x,y, i), z[x, y][w, w], z[x, y]Z1Z2 ... zm[w, w]|m = 1, 2,... }.
Theorem 4.5 [14]. Let F = GF(g) be a finite field. The nonassociative algebra A = {w 1, i>2, e11 + e12, e22)F is an NFB-algebra with a basis of identities:
{x(y z), z(xq2-q+l — x), zSt3(x, y, i), z[x, y][w, w], z(x — xq)(y — y9),
z[x, y](i — i9), z(x — x9)[y, i], z[x, y]Z1Z2 ... Zfc[m, = 1, 2,... }.
The nonassociative algebras constructed in Theorems 4.4 and 4.5 are examples of four-dimensional NFB-algebras. Examples of five-dimensional NFB-algebras were previously known [2,27]. If we put F = GF(2) in Theorem 4.5, then we obtain an example of an NFB-ring containing 16 elements.
Also, by having examples of INFB-spaces and SNFB-spaces, we can construct examples of non-associative INFB-algebras and SNFB-algebras, respectively.
Theorem 4.6 [11]. Let F be an arbitrary field. The algebra A = V © T2(F) is an SNFB-algebra.
Theorem 4.7 [13]. Let F be a field of characteristic zero; E0 = (ell + el2, e22)f. The algebra A = V © E0 over the field F is an SNFB-algebra.
Corollary 4.2. Any finite dimensional F-algebra (over the corresponding field F) containing the algebra A as a subalgebra has no finite basis of identities.
Above, we gave an example of an L-variety that does not have a finite basis of identities and is the union of two Specht varieties. A similar construction can be constructed for algebras from the variety p = (x(yz) = 0). Let F be a field of characteristic zero. Let Bl = (vi,v2,en,ei2)F, B2 = (vl,v2 + ell,e2l )p be F-algebras from the variety p.
Theorem 4.8 [28]. Let F be a field of characteristic zero. Varieties Ml = Var Bl and M2 = Var B2 are Specht.
Theorem 4.9 [28]. Let F be a field of characteristic zero. The variety M = Ml U M2 = Var Bl © B2 is an NFB-variety with a basis of identities
[x(yz), zSt3(x, y, t),zx[y, u]v, x[y, u]v — v[y, u]x,
z[x,y]ziz2 ... zm [u,v]lm = 1, 2,... }.
In 1993, I.P. Shestakov formulated in Dniester notebook a question [29]: do there exist finite dimensional central simple algebras over a field of characteristic zero that do not have a finite basis of identities? Using the found SNFB-algebras and INFB-algebras, the required example is constructed for an arbitrary field.
Note that any finite dimensional simple algebra over an algebraically closed field is uniquely determined by their identities up to isomorphism [30].
Theorem 4.10 [31]. Let A = {1,vl,v2,ell ,el2,e22,p)F be an algebra over an arbitrary field F, where 1 is a unity element of A, and nonzero products of basis elements that are not equal to unity elements are defined by the rules: ^e^ = Vj, v2p = 1. Then the algebra A is a central simple F-algebra and A has no finite basis of identities.
After giving this example, the question was posed about the existence of a finite dimensional simple commutative or anticommutative NFB-algebra. In the case of characteristic zero, a positive answer to this question was obtained.
Theorem 4.11 [32]. Let A = (1,vl,v2,ell,el2,e22,p)F be an algebra over field F of characteristic zero, where 1 is a unity element of A, and nonzero products of basis elements that are not equal to unity elements are defined by rules: ^e^ = e^Vi = Vj, v2p = pv2 = 1. Then the algebra A is a central simple commutative F-algebra and A has no finite basis of identities.
Theorem 4.12 [33]. Let A = (e, vl,v2, ell,el2, e22,p)p be an algebra over field F of characteristic zero, where nonzero products of basis elements are defined by rules: ^e^ = —eijVi = Vj, V2P = —PV2 = e, Vie = —evi = ei:je = —eei:j = ei:j, pe = —ep = p.
Then the algebra A is a simple anticommutative F-algebra and A has no finite basis of identities.
Theorems 4.10-4.12 give examples of nonfinitely based simple algebras of dimension seven. In the case of a field of characteristic zero, the dimension of the algebra from Theorem 4.10 can be decreased to six.
Theorem 4.13 [34]. Let A = {1, w1, w2, e11 + e12, e22,p)F be an algebra over the field F of characteristic zero, where 1 is a unity element of A, and nonzero products of basis elements that are not equal to unity elements (taking into account the law of distributive) are defined by the rules: ^e ij = Vj, = 1. Then the algebra A is a central simple F-algebra and A has no finite basis of identities.
6. Some Unsolved Problems
• Description of NFB-spaces (SNFB-spaces, INFB-spaces).
• Description of minimal nonzero L-varieties of vector spaces.
• Study the structure of the lattice of L-subvarieties.
• Full description of almost commutative (almost nilpotent, almost Engel, etc.) L-varieties of vector spaces.
• When is T(G) = L(G) for the set G of associative polynomials?
• Construct an example of a not finitely based L-variety, any proper L-subvariety of which is given by a finite number of identities.
• Is there a two- and three-dimensional nonassociative NFB-algebra? Acknowledgments
The author expresses deep appreciation to Prof. V. A. Romankov for his attention to the paper. Supported by Russian Science Foundation, project No. 22-21-00745.
References
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ТОЖДЕСТВА ПРОСТРАНСТВ ЛИНЕЙНЫХ ПРЕОБРАЗОВАНИЙ И НЕАССОЦИАТИВНЫХ ЛИНЕЙНЫХ АЛГЕБР
А. В. Кислицын
аспирант, e-mail: avkislitsin@gmail.com
Омский государственный университет им. Ф. М. Достоевского, г. Омск, Россия Алтайский государственный педагогический университет, г. Барнаул, Россия
Аннотация. В работе представлены основные результаты о мультипликативных векторных пространствах, тождествах мультипликативных векторных пространств и l-многообразиях. Также приведены следствия полученных результатов для неассоциативных линейных алгебр
Ключевые слова: мультипликативное векторное пространство, тождество мультипликативного векторного пространства, базис тождеств, конечно базируемое пространство, не конечно базируемое пространство, L-многообразие, существенно бесконечно базируемое многообразие, сильно бесконечно базируемое многообразие..
Дата поступления в редакцию: 14.10.2023
