S. Mishchenko

POLYNOMIAL IDENTITIES ON SUPERALGEBRAS AND ALMOST POLYNOMIAL GROWTH

Let A be a superalgebra over a field of characteristic zero. In this paper we investigate the graded polynomial identities of A through the asymptotic behavior of a numerical sequence called the sequence of graded codimensions of A. Our main result says that such sequence is polynomially bounded if and only if the variety of superalgebras generated by A does not contain a list of five superalgebras consisting of a 2-dimensional algebra, the infinite dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with trivial and nontrivial gradings. Our main tool is the representation theory of the symmetric group.

POLYNOMIAL GROWTH OF THE *-CODIMENSIONS AND YOUNG DIAGRAMS

Let A be an algebra with involution * over a field F of characteristic zero and Id(A, *) the ideal of the free algebra with involution of *-identities of A. By means of the representation theory of the hyperoctahedral group Z 2wrS n we give a characterization of Id(A, *) in case the sequence of its *-codimensions is polynomially bounded. We also exhibit an algebra G 2 with the following distinguished property: the sequence of *-codimensions of Id(G 2, *) is not polynomially bounded but the *-codimensions of any T-ideal U properly containing Id(G 2, *) are polynomially bounded.

Exponents of varieties of lie algebras with a nilpotent commutator subalgebra

We prove that the growth exponent of any variety of Lie algebras with a nilpotent commutator subalgebra is integral.

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 .

Codimension growth of two-dimensional non-associative algebras

Let F be a field of characteristic zero and let A be a two-dimensional non-associative algebra over F. We prove that the sequence c n (A), n =1,2,..., of codimensions of A is either bounded by n + 1 or grows exponentially as 2 n . We also construct a family of two-dimensional algebras indexed by rational numbers with distinct T-ideals of polynomial identities and whose codimension sequence is n + 1, n > 2.

Polynomial growth of the codimensions: a characterization

Let A be a not necessarily associative algebra over a field of characteristic zero. Here we characterize the T-ideal of identities of A in case the corresponding sequence of codimensions is polynomially bounded.