Antonino Giambruno

Wreath products and P.I. algebras

Abstract The representation theory of wreath products G ∼ S n is applied to study algebras satisfying polynomial identities that involve a group G of (anti)automorphisms, in the same way the representation theory of S n was applied earlier to study ordinary P.I. algebras. Some of the basic results of the ordinary case are generalized to the G -case.

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.

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.

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.

A LIE PROPERTY IN GROUP RINGS

Let A be an additive subgroup of a group ring R over a field K. DeIlOte by (A. R) the additive subgroup generated by the Lie products (a. r) = ar - ra, a E A, r E R. Inductively, let (A. Rn) = ((A.Rn-l). R). We prove that (A. Rn) = 0 for some n =:> (A. R)R is a nilpotent ideal.

Some generalizations of the center of a ring

RiassuntoSi generalizza la nozione di ipercentro introdotta da Herstein in [3] e si trova una forma equivalente alla congettura di Köethe.

Anomalies on codimension growth of algebras

Abstract This paper deals with the asymptotic behavior of the sequence of codimensions c n u2062 ( A )

Polynomial growth of the codimensions: a characterization

{c_{n}(A)}

Periodicn-th commutators of traces in rings with involution

, n = 1 , 2 , … ,