M. Zaicev

Minimal varieties of algebras of exponential growth

Abstract 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 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, verifying in this setting some known conjectures. We also show that the number of these minimal varieties is finite for any given exponent. We finally point out some relations between the exponent of a variety and the Gelfand–Kirillov dimension of the corresponding relatively free algebras of finite rank.

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.

CODIMENSION GROWTH AND MINIMAL SUPERALGEBRAS

A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope G(A) of a finite dimensional superalgebra A. In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field. The importance of such algebras is readily proved: A is a minimal superalgebra if and only if the ideal of identities of G(A) is a product of verbally prime T-ideals. Also, such superalgebras allow us to classify all minimal varieties of a given exponent i.e., varieties V such that exp(V) = d > 2 and exp(U) < d for all proper subvarieties U of V. This proves in the positive a conjecture of Drensky (1988). As a corollary we obtain that there is only a finite number of minimal varieties for any given exponent. A classification of minimal varieties of finite basic rank was proved by the authors (2003). As an application we give an effective way for computing the exponent of a T-ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras.

Simple and semisimple Lie algebras and codimension growth

We study the exponential growth of the codimensions cn(B) of a finite dimensional Lie algebra B over a field of characteristic zero. In the case when B is semisimple we show that limn→∞ n √ cn(B) exists and, when F is algebraically closed, is equal to the dimension of the largest simple summand of B. As a result we characterize central-simplicity: B is central simple if and only if dimB = limn→∞ n √ cn(B).

Identities of graded algebras and codimension growth

Let A = ○+ g ∈ G A g be a G-graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component A e to that of the whole of A, in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where A is finite dimensional and A e has polynomial growth.

identities on associative algebras

Let R be an algebra over a field and G a finite group of automorphisms and anti-automorphisms of R. We prove that if R satisfies an essential G-polynomial identity of degree d, then the G-codimensions of R are exponentially bounded and R satisfies a polynomial identity whose degree is bounded by an explicit function of d. As a consequence we show that if R is an algebra with involution ∗ satisfying a ∗-polynomial identity of degree d, then the ∗-codimensions of R are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case R must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity. §

ASYMPTOTICS FOR THE STANDARD AND THE CAPELLI IDENTITIES

AbstractLet {cn(Stk)} and {cn(Ck)} be the sequences of codimensions of the T-ideals generated by the standard polynomial of degreek and by thek-th Capelli polynomial, respectively. We study the asymptotic behaviour of these two sequences over a fieldF of characteristic zero. For the standard polynomial, among other results, we show that the following asymptotic equalities hold:

The exponential growth of codimensions for Capelli identities

\begin{gathered} c_n \left( {St_{2k} } \right) \simeq c_n \left( {C_{k^2 + 1} } \right) \simeq c_n \left( {M_k \left( F \right)} \right), \hfill \\ c_n \left( {St_{2k + 1} } \right) \simeq c_n \left( {M_{k \times 2k} \left( F \right) \oplus M_{2k \times k} \left( F \right)} \right), \hfill \\ \end{gathered}