Mikhail Zaicev

Codimension growth of special simple Jordan algebras

Let R be a special simple Jordan algebra over a field of characteristic zero. We exhibit a noncommutative Jordan polynomial f multialternating on disjoint sets of variables which is not a polynomial identity of R. We then study the growth of the polynomial identities of the Jordan algebra R through an analysis of its sequence of Jordan codimensions. By exploiting the basic properties of the polynomial f , we are able to compute the exponential rate of growth of the sequence of Jordan codimensions of R and prove that it equals the dimension of the Jordan algebra over its center. We also show that for any finite dimensional special Jordan algebra, such an exponential rate of growth cannot be strictly between 1 and 2.

Конечномерные простые градуированные алгебры@@@Finite-dimensional simple graded algebras

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Minimal varieties of algebras of exponential growth

The exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer. Through this scale we can now classify all minimal varieties of a given exponent and of finite basic rank. As a consequence we describe the corresponding T-ideals of the free algebra, and we compute the asymptotics of the related codimension sequences. We then verify in this setting some known conjectures.

Polynomial Identities of Finite Dimensional Simple Algebras

Let F be an algebraically closed field and let A and B be arbitrary finite dimensional simple algebras over F. We prove that A and B are isomorphic if and only if they satisfy the same identities.

Abelian Gradings on Upper Block Triangular Matrices

Let G be an arbitrary finite abelian group. We describe all possible G-gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero. Dipartimento di Metodi e Modelli Matematici, Universita di Palermo, Palermo, Italy e-mail: avalenti@unipa.it Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119992, Russia e-mail: zaicev@mech.math.msu.su Received by the editors December 22, 2008. Published electronically March 23, 2011. The first author was partially supported by MIUR of Italy. The second author was partially supported by RFBR grant No 06-01-00485 and SSC-5666.2006.1 AMS subject classification: 16W50.

ON THE COLENGTH OF A VARIETY OF LIE ALGEBRAS

We study the variety of Lie algebras defined by the identity over a field of characteristic zero. We prove that, as in the associative case, in the nth cocharacter χn of this variety, every irreducible Sn-character appears with polynomially bounded multiplicity (not greater than n2). Anyway, surprisingly enough, we also show that the colength of this variety, i.e. the total number of irreducibles appearing in χn is asymptotically equal to .

Polynomial Identities of Algebras of Small Dimension

It is well known that given an associative algebra or a Lie algebra A, its codimension sequence c n (A) is either polynomially bounded or grows at least as fast as 2 n . In [2] we proved that for a finite dimensional (in general nonassociative) algebra A, dim A = d, the sequence c n (A) is also polynomially bounded or c n (A) ≥ a n asymptotically, for some real number a > 1 which might be less than 2. Nevertheless, for d = 2, we may take a = 2. Here we prove that for d = 3 the same conclusion holds. We also construct a five-dimensional algebra A with c n (A) < 2 n .

GRADED IDENTITIES OF GROUP ALGEBRAS

ABSTRACT Let H be a normal subgroup of G. Then the group algebra can be naturally graded by where the homogeneous components are cosets. We prove that if A satisfies a -graded identity than it also satisfies an ordinary polynomial identity under the assumption that is finite.

Classifying the Minimal Varieties of Polynomial Growth

Let V be a variety of associative algebras generated by an algebra with 1 over a field of characteristic zero. This paper is devoted to the classification of the varieties V which are minimal of polynomial growth (i.e., their sequence of codimensions growth like n but any proper subvariety grows like n with t < k). These varieties are the building blocks of general varieties of polynomial growth. It turns out that for k ≤ 4 there are only a finite number of varieties of polynomial growth n, but for each k > 4, the number of minimal varieties is at least |F|, the cardinality of the base field and we give a recipe of how to construct them.

Group gradings on finite dimensional Lie algebras

We study gradings by noncommutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if