Sequence Of Codimensions Research Articles

On H-simple not necessarily associative algebras

An algebra [Formula: see text] with a generalized [Formula: see text]-action is a generalization of an [Formula: see text]-module algebra where [Formula: see text] is just an associative algebra with [Formula: see text] and a relaxed compatibility condition between the multiplication in [Formula: see text] and the [Formula: see text]-action on [Formula: see text] holds. At first glance, this notion may appear too general, however, it enables to work with algebras endowed with various kinds of additional structures (e.g. comodule algebras over Hopf algebras, graded algebras, algebras with an action of a semigroup by anti-endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if [Formula: see text] is a finite dimensional (not necessarily associative) algebra over a field of characteristic [Formula: see text] and [Formula: see text] is simple with respect to a generalized [Formula: see text]-action, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of polynomial [Formula: see text]-identities of [Formula: see text]. In particular, if [Formula: see text] is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of graded polynomial identities of [Formula: see text]. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized [Formula: see text]-actions.

Varieties of special Jordan algebras of almost polynomial growth

Let J be a special Jordan algebra and let cn(J) be its corresponding codimension sequence. The aim of this paper is to prove that in case J is finite dimensional, such a sequence is polynomially bounded if and only if the variety generated by J does not contain UJ2, the special Jordan algebra of 2×2 upper triangular matrices. As an immediate consequence, we prove that UJ2 is the only finite dimensional special Jordan algebra that generates a variety of almost polynomial growth.

Combinatorics on Binary Words and Codimensions of Identities in Left Nilpotent Algebras

Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with specified properties of the codimension sequence. However, the class of the infinite words used was confined to periodic words and Sturm words. Here the previously proposed approach is generalized to a considerably more general case. It is proved that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exists a PI-exponent, and its precise value is computed.

Lie algebras simple with respect to a Taft algebra action

We classify finite dimensional Hm2(ζ)-simple Hm2(ζ)-module Lie algebras L over an algebraically closed field of characteristic 0 where Hm2(ζ) is the mth Taft algebra. As an application, we show that despite the fact that L can be non-semisimple in ordinary sense, limn→∞⁡cnHm2(ζ)(L)n=dim⁡L where cnHm2(ζ)(L) is the codimension sequence of polynomial Hm2(ζ)-identities of L. In particular, the analog of Amitsur's conjecture holds for cnHm2(ζ)(L).

Growth of central polynomials of verbally prime algebras

We compute the exact asymptotics of the codimension sequence for the central polynomials of k × k matrices and show that it is asymptotic to $$\frac{1}{k^2}$$ times the ordinary cocharacter. For the other verbally prime algebras we show that these sequences are bounded above and below by constants times the ordinary codimensions.

Some characterizations of algebras with involution with polynomial growth of their codimensions

Let A be an associative algebra endowed with an involution of the first kind and let denote the sequence of -codimensions of A. In this paper, we are interested in algebras with involution such that the -codimension sequence is polynomially bounded. We shall prove that A is of this kind if and only if it satisfies the same identities of a finite direct sum of finite dimensional algebras with involution , each of which with Jacobson radical of codimension less than or equal to one in . We shall also relate the condition of having polynomial codimension growth with the sequence of cocharacters and with the sequence of colengths. Along the way, we shall show that the multiplicities of the irreducible characters in the decomposition of the cocharacters are eventually constants. Finally, we shall give a classification of the algebras with involution whose -codimensions are at most of linear growth.

Sturmian words and overexponential codimension growth

Let A be a non necessarily associative algebra over a field of characteristic zero satisfying a non-trivial polynomial identity. If A is a finite dimensional algebra or an associative algebra, it is known that the sequence cn(A), n=1,2,…, of codimensions of A is exponentially bounded.If A is an infinite dimensional non associative algebra such sequence can have overexponential growth. Such phenomenon is present also in the case of Lie or Jordan algebras. In all known examples the smallest overexponential growth of cn(A) is (n!)12.Here we construct a family of algebras whose codimension sequence grows like (n!)α, for any real number α with 0<α<1.

Growth of codimensions of metabelian algebras

Numerical invariants of identities of nonassociative algebras are considered. It is proved that the codimension sequence of any finitely generated metabelian algebra has an exponentially bounded codimension growth. It is shown that the upper PI-exponent increases at most by 1 after adjoining an external unit. It is proved that for two-step left-nilpotent algebras the lower PI-exponent increases at least by 1.

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.

ЧИСЛОВЫЕ ХАРАКТЕРИСТИКИ АЛГЕБР ЛЕЙБНИЦА-ПУАССОНА

The paper is survey of recent results of investigations on varieties of Leibniz-Poisson algebras. We show that a variety of Leibniz-Poisson algebras has either polynomial growth or growth with exponential not less than 2, the eld being arbitrary. We show that every variety of Leibniz-Poisson algebras of polynomial growth over a eld of characteristic zero has a nite basis for its polynomial identities.We construct a variety of Leibniz-Poisson algebras with almost polynomial growth. We give equivalent conditions of the polynomial codimension growth of a variety of Leibniz-Poisson algebras over a eld of characteristic zero. We show all varieties of Leibniz-Poisson algebras with almost polynomial growth in one class of varieties. We study varieties of Leibniz-Poisson algebras, whose ideals of identities contain the identity fx; yg fz; tg = 0, we study an interrelation between such varieties and varieties of Leibniz algebras. We show that from any Leibniz algebra L one can construct the Leibniz-Poisson algebra A and the properties of L are close to the properties of A. We show that if the ideal of identities of a Leibniz-Poisson variety V does not contain any Leibniz polynomial identity then V has overexponential growth of the codimensions. We construct a variety of Leibniz-Poisson algebras with almost exponential growth. Let f n(V)gn1 be the sequence of proper codimension growth of a variety of Leibniz-Poisson algebras V. We give one class of minimal varieties of Leibniz-Poisson algebras of polynomial growth of the sequence f n(V)gn1, i.e. the sequence of proper codimensions of any such variety grows as a polynomial of some degree k, but the sequence of proper codimensions of any proper subvariety grows as a polynomial of degree strictly less than k.

Identities of metabelian alternative algebras

We study metabelian alternative (in particular, associative) algebras over a field of characteristic 0. We construct additive bases of the free algebras of mentioned varieties, describe some centers of these algebras, compute the values of the sequence of codimensions of corresponding T-ideals, and find unitarily irreducible components of the decomposition of mentioned varieties into a union and their bases of identities. In particular, we find a basis of identities for the metabelian alternative Grassmann algebra. We prove that the free algebra of a variety that is generated by the metabelian alternative Grassmann algebra possesses the zero associative center.

Comparing the ℤ 2-Graded Identities of Two Minimal Superalgebras with the Same Superexponent

Let F be a field of characteristic zero. We study two minimal superalgebras A and B having the same superexponent but such that T 2 (A) ⫋ T 2 (B), thus providing the first example of a minimal superalgebra generating a non minimal supervariety. We compare the structures and codimension sequences of A and B.

The polynomial part of the codimension growth of affine PI algebras

Let F be a field of characteristic zero and W an associative affine F-algebra satisfying a polynomial identity (PI). The codimension sequence {cn(W)} associated to W is known to be of the form Θ(ntdn), where d is the well known PI-exponent of W. In this paper we establish an algebraic interpretation of the polynomial part (the constant t) by means of Kemer's theory. In particular, we show that in case W is a basic algebra (hence finite dimensional), t=q−d2+s, where q is the number of simple component in W/J(W) and s+1 is the nilpotency degree of J(W) (the Jacobson radical of W). Thus proving a conjecture of Giambruno.

Proper T-ideals of poisson algebras with extreme properties

Let {γn(V)}n≥1 be the sequence of proper codimensions of some variety V of Poisson algebras over a field of characteristic zero. A class of minimal varieties of Poisson algebras of polynomial growth of the sequence {γn(V)}n≥1 is presented, i.e., the sequence {γn(V)}n≥1 of any such variety V grows as a polynomial of some degree k, but the sequence {γn(W)}n≥1 of any proper subvariety W in V grows as a polynomial of degree strictly less than k.

Polynomial codimension growth and the Specht problem

We construct a continuous family of algebras over a field of characteristic zero with slow codimension growth bounded by a polynomial of degree 4. This is achieved by building, for any real number α∈(0,1) a commutative nonassociative algebra Aα whose codimension sequence cn(Aα), n=1,2,… , is polynomially bounded and lim⁡logn⁡cn(Aα)=3+α.As an application we are able to construct a new example of a variety with an infinite basis of identities.

О МНОГООБРАЗИЯХ С ТОЖДЕСТВАМИ ОДНОПОРОЖДЕННОЙ СВОБОДНОЙ МЕТАБЕЛЕВОЙ АЛГЕБРЫ

A set of linear algebras where a fixed set of identities takes place, following A.I. Maltsev, is called a variety. In the case of zero characteristic of the main field all the information about the variety is contained in multilinear parts of relatively free algebra of the variety. We can study the identities of variety by means of investigations of multilinear part of degree n as module of the symmetric group Sn. Using the language of Lie algebras we say that an algebra is metabelian if it satisfies the identity (xy)(zt) ≡ 0. In this paper we study the identities of non-associative one-generated free metabelian algebra and its factors. In particular, the infinite set of the varieties with different fractional exponents between one and two was constructed. Note that the sequence of codimensions of these varieties asymptotically formed by using colength, and not by using the dimension of some irreducible module of the symmetric group what was for all known before examples.

On algebras of polynomial codimension growth

Let A be an associative algebra over a field F of characteristic zero and let \(c_n(A), n=1, 2, \ldots \), be the sequence of codimensions of A. It is well-known that \(c_n(A), n=1, 2, \ldots \), cannot have intermediate growth, i.e., either is polynomially bounded or grows exponentially. Here we present some results on algebras whose sequence of codimensions is polynomially bounded.

Standard polynomials and matrices with superinvolutions

Let Mn(F) be the algebra of n×n matrices over a field F of characteristic zero. The superinvolutions ⁎ on Mn(F) were classified by Racine in [12]. They are of two types, the transpose and the orthosymplectic superinvolution. This paper is devoted to the study of ⁎-polynomial identities satisfied by Mn(F). The goal is twofold. On one hand, we determine the minimal degree of a standard polynomial vanishing on suitable subsets of symmetric or skew-symmetric matrices for both types of superinvolutions. On the other, in case of M2(F), we find generators of the ideal of ⁎-identities and we compute the corresponding sequences of cocharacters and codimensions.

On the codimension sequence of G-simple algebras

Let F be an algebraically closed field of characteristic zero and let G be a finite group. In this paper we will show that the asymptotics of cnG(A), the G-graded codimension sequence of a finite dimensional G-simple F-algebra A, has the form αn1−dimF⁡Ae2(dimF⁡A)n (as conjectured by E. Aljadeff, D. Haile and M. Natapov), where α is some positive real number and Ae denotes the identity component of A. In the special case where A is the algebra of m×m matrices with an arbitrary elementary G-grading we succeeded in calculating the constant α explicitly.

Codimension and colength sequences of algebras and growth phenomena

We consider non necessarily associative algebras over a field of characteristic zero and their polynomial identities. Here we describe some of the results obtained in recent years on the sequence of codimensions and the sequence of colengths of an algebra.