Varieties Of Associative Algebras Research Articles

Asymptotic Behavior in Lie Nilpotent Relatively Free Algebras and Extended Grassmann Algebras

The present paper is devoted to estimating the codimensions cn for the variety of associative algebras given by the identity [x1, …, xl] = 0 for odd l. The characteristic of the ground field is different from 2 and 3. The construction of the so-called extended Grassmann algebra is applied very effectively.

Minimal star-varieties of polynomial growth and bounded colength

Let V be a variety of associative algebras with involution ⁎ over a field F of characteristic zero. Giambruno and Mishchenko proved in [6] that the ⁎-codimension sequence of V is polynomially bounded if and only if V does not contain the commutative algebra D=F⊕F, endowed with the exchange involution, and M, a suitable 4-dimensional subalgebra of the algebra of 4×4 upper triangular matrices, endowed with the reflection involution. As a consequence the algebras D and M generate the only varieties of almost polynomial growth. In [20] the authors completely classify all subvarieties and all minimal subvarieties of the varieties var⁎(D) and var⁎(M). In this paper we exhibit the decompositions of the ⁎-cocharacters of all minimal subvarieties of var⁎(D) and var⁎(M) and compute their ⁎-colengths. Finally we relate the polynomial growth of a variety to the ⁎-colengths and classify the varieties such that their sequence of ⁎-colengths is bounded by three.

Varieties satisfying semigroup identities: Algebras over a finite field and rings

We study varieties of associative algebras over a finite field and varieties of associative rings satisfying semigroup or adjoint semigroup identities. We characterize these varieties in terms of “forbidden algebras” and discuss some corollaries of the characterizations.

On the primality property of central polynomials of prime varieties of associative algebras

In the paper, it is proved that, if f(x1,..., xn)g(y1,..., ym) is a multilinear central polynomial for a verbally prime T-ideal Γ over a field of arbitrary characteristic, then both polynomials f(x1,..., xn) and g(y1,..., ym) are central for Γ.

Minimal Varieties and Identities of Relatively Free Algebras

Let K be a field of characteristic zero and let 𝔐5 be the variety of associative algebras over K, defined by the identity [x 1, x 2][x 3, x 4, x 5]. It is well-known that such variety is a minimal variety and that it is generated by the algebra where E = E 0 ⊕ E 1 is the Grassmann algebra. In this article, for any positive integer k, we describe the polynomial identities of the relatively free algebras of rank k of 𝔐5, It turns out that such algebras satisfy the same polynomial identities of some algebras used in the description of the subvarieties of 𝔐5, given by Di Vincenzo, Drensky, and Nardozza.

Polynomial growth and star-varieties

Let V be a variety of associative algebras with involution over a field F of characteristic zero and let cn⁎(V), n=1,2,…, be its ⁎-codimension sequence. Such a sequence is polynomially bounded if and only if V does not contain the commutative algebra F⊕F, endowed with the exchange involution, and M, a suitable 4-dimensional subalgebra of the algebra of 4×4 upper triangular matrices. Such algebras generate the only varieties of ⁎-algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety is polynomially bounded. In this paper we completely classify all subvarieties of the ⁎-varieties of almost polynomial growth by giving a complete list of finite dimensional ⁎-algebras generating them.

On matrix types of prime varieties of associative algebras

Some answer is obtained to the problem by A. R. Kemer: We show that for a given number k and all sufficiently large p every prime variety of associative algebras of matrix type k over a field of characteristic p > 0 is regular.

Многообразия линейных алгебр полиномиального роста

The paper is survey of results of investigations on varieties of linear algebras of polynomial growth. We give equivalent conditions of the polynomial codimension growth of a variety of associative algebras, Lie algebras, Leibniz algebras, Poisson algebras, Leibniz-Poisson algebras. It is shown that in the study of varieties of linear algebras of polynomial growth varieties of almost polynomial growth play an important role.

VARIETIES OF REPRESENTATIONS OF GROUPS AND VARIETIES OF ASSOCIATIVE ALGEBRAS

This paper is tightly connected with the book [Varieties of Group Representations. General Theory, Connections and Applications (Zinatne, Riga, 1983) (in Russian)]. In the paper we prove new results in the spirit of the above-mentioned book. They are related to dimension subgroups, varieties of representations of groups and varieties of associative algebras. The main emphasis is put on the varieties of representations of groups induced by the varieties of associative algebras. We provide the reader with the list of open problems. For many reasons we consciously included in the text a brief review of the basic definitions and results from the theory of varieties of representations described in the book [Varieties of Group Representations. General Theory, Connections and Applications].

Codimension Growth of Strong Lie Nilpotent Associative Algebras

We study Lie nilpotent varieties of associative algebras. We explicitly compute the codimension growth for the variety of strong Lie nilpotent associative algebras. The codimension growth is polynomial and found in terms of Stirling numbers of the first kind. To achieve the result we take the free Lie algebra of countable rank L(X), consider its filtration by the lower central series and shift it. Next we apply generating functions of special type to the induced filtration of the universal enveloping algebra U(L(X)) = A(X).

Characterization of non-matrix varieties of associative algebras

A variety of associative algebras is called a non-matrix variety if it does not contain the algebra of 2 × 2 matrices over the base field K. There are some known characterizations of non-matrix varieties. We give some new characterizations in terms of properties of nilelements. Let V be a variety of associative algebras over an infinite field. Then the following conditions are equivalent: (1) V is a non-matrix variety, (2) any finitely generated algebra A ∈ V satisfies an identity of the form [x1, x2] … [x2s−1, x2s] ≡ 0, (3) let A ∈ V; then for any nilelements a, b ∈ A, the element a + b is again a nilelement. Let E be the Grassmann algebra in countable many generators. We also give similar characterizations for non-matrix varieties over fields of characteristic zero that do not contain E or E ⊗ E.

Identities in the varieties generated by the algebras of upper triangular matrices

Consider the algebra UTs of upper triangular matrices of size s over an arbitrary field. Petrogradsky proved that the exponent of an arbitrary subvariety in var(UTs) exists and is an integer. We strengthen the estimates for the growth of these varieties and provide equivalent conditions for finding these exponents. Kemer showed that in the case of a ground field of characteristic zero there exists no varieties of associative algebras with growth intermediate between polynomial and exponential. We prove that this property extends to the case of the fields of arbitrary characteristic distinct from 2.

The unitary closure property of the prime varieties of associative algebras

We prove that every prime variety of associative algebras over an infinite field of characteristic p>0 is generated by either a unital algebra or a nilalgebra of bounded index. We show that the Engel verbally prime T-ideals remain verbally prime as we impose the identity \( x^{p^N } = 0 \) for sufficiently large N. We then describe all prime varieties in an interesting class of varieties of associative algebras.

Limit T-spaces

Let F be a field of prime characteristic p and let V p be the variety of associative algebras over F without unity defined by the identities [[x, y], z] = 0 and x 4 = 0 if p = 2 and by the identities [[x, y], z] = 0 and x p = 0 if p > 2 (here [x, y] = xy − yx). Let A/V p be the free algebra of countable rank of the variety V p and let S be the T-space in A/V p generated by x 21 x 22 ⋯ x 2k + V 2, where k ∈ ℕ if p = 2, and by {ie4170-01}, where k ∈ ℕ and α 1, …, α 2k ∈ {0, p − 1} if p > 2. As is known, S is not finitely generated as a T-space. In the present paper, we prove that S is a limit T-space, i.e., a maximal nonfinitely generated T-space. As a corollary, we have constructed a limit T-space in the free associative F-algebra without unity of countable rank.

AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE ASSOCIATIVE ALGEBRA

Let [Formula: see text] be the variety of associative algebras over a field K and A = K 〈x1,…, xn〉 be a free associative algebra in the variety [Formula: see text] freely generated by a set X = {x1,…, xn}, End A the semigroup of endomorphisms of A, and Aut End A the group of automorphisms of the semigroup End A. We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End A. A similar result is obtained for the automorphism group Aut [Formula: see text], where [Formula: see text] is the subcategory of finitely generated free algebras of the variety [Formula: see text]. The later result solves Problem 3.9 formulated in [17].

An upper bound for the index of multiplicities in the cocharacters of PI-algebras

We give an upper bound for the index of the polynomial degree limiting the multiplicities in the cocharacter of a variety of associative algebras over a field of characteristic zero.

A Comparison of Deformations and Geometric Study of Varieties of Associative Algebras

The aim of this paper is to give an overview and to compare the different deformation theories of algebraic structures. In each case we describe the corresponding notions of degeneration and rigidity. We illustrate these notions by examples and give some general properties. The last part of this work shows how these notions help in the study of varieties of associative algebras. The first and popular deformation approach was introduced by M. Gerstenhaber for rings and algebras using formal power series. A noncommutative version was given by Pinczon and generalized by F. Nadaud. A more general approach called global deformation follows from a general theory by Schlessinger and was developed by A. Fialowski in order to deform infinite-dimensional nilpotent Lie algebras. In a nonstandard framework, M. Goze introduced the notion of perturbation for studying the rigidity of finite-dimensional complex Lie algebras. All these approaches share the common fact that we make an “extension” of the field. These theories may be applied to any multilinear structure. In this paper, we will be dealing with the category of associative algebras.

Algebras with involution with linear codimension growth

We study the ∗-varieties of associative algebras with involution over a field of characteristic zero which are generated by a finite-dimensional algebra. In this setting we give a list of algebras classifying all such ∗-varieties whose sequence of ∗-codimensions is linearly bounded. Moreover, we exhibit a finite list of algebras to be excluded from the ∗-varieties with such property. As a consequence, we find all possible linearly bounded ∗-codimension sequences.

On Identities and Varieties of Associative Algebras of Finite Type

Varieties in which each finitely generated relatively free algebra is finitely defined are considered. A description of such varieties with identities is obtained, and a number of their structural properties are found. Bibliography: 7 titles.

On Some Extremal Varieties of Associative Algebras

Suppose that F is a field of prime characteristic p and V p is the variety of associative algebras over F defined by the identities [[x, y], z] = 0 and x p = 0 if p > 2 and by the identities [[x, y], z] = 0 and x 4 = 0 if p = 2 (here [x, y] = xy − yx). As is known, the free algebras of countable rank of the varieties V p contain non-finitely generated T-spaces. We prove that the varieties V p are minimal with respect to this property.