Antonio Giambruno

Lie nilpotence of group rings

Let FG be the group algebra of a group G over a field F. Denote by ∗ the natural involution, (∑fi gi -1. Let S and K denote the set of symmetric and skew symmetric and skew symmetric elements respectively with respect to this involutin. It is proved that if the characteristic of F is zero p≠2 and G has no 2-elements, then the Lie nilpotence of S or K implies the Lie nilpotence of FG.

Group algebras whose units satisfy a group identity

Let FG be the group algebra of a torsion group over an infinite field F . Let U be the group of units of FG. We prove that if U satisfies a group identity, then FG satisfies a polynomial identity. This confirms a conjecture of Brian Hartley.

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.

Derivations with nilpotent values

IfR is a semiprme ring andd a derivation ofR such thatd(x)n=0 for allx∈R, wheren≥1 is a fixed integer, thend=0.

Group-graded algebras with polynomial identity

LetG be a finite group and letR=Σg∈GRg be any associative algebra over a field such that the subspacesRg satisfyRgRh⊆Rgh. We prove that ifR1 satisfies a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the order ofG. This result implies the following: ifH is a finite-dimensional semisimple commutative Hopfalgebra andR is anyH-module algebra withRH satisfying a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the dimension ofH.

On the identities of the Grassmann algebras in characteristicp>0

In this note we exhibit bases of the polynomial identities satisfied by the Grassmann algebras over a field of positive characteristic. This allows us to answer the following question of Kemer: Does the infinite dimensional Grassmann algebra with 1, over an infinite fieldK of characteristic 3, satisfy all identities of the algebraM2(K) of all 2×2 matrices overK? We give a negative answer to this question. Further, we show that certain finite dimensional Grassmann algebras do give a positive answer to Kemers question. All this allows us to obtain some information about the identities satisfied by the algebraM2(K) over an infinite fieldK of positive odd characteristic, and to conjecture bases of theidentities ofM2(K).

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 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. §

Multialternating graded polynomials and growth of polynomial identities

Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by mean of Regev central polynomials we construct multialternating graded polynomials of arbitrarily large degree non vanishing on A. As a consequence we compute the exponential rate of growth of the sequence of graded codimensions of an arbitrary G-graded algebra satisfying an ordinary polynomial identity. In particular we show it is an integer. The result was proviously known in case G is abelian.

A characterization of algebras with polynomial growth of the codimensions

Let A be an associative algebras over a field of characteristic zero. We prove that the codimensions of A are polynomially bounded if and only if any finite dimensional algebra B with Id(A) = Id(B) has an explicit decomposition into suitable subalgebras; we also give a decomposition of the n-th cocharacter of A into suitable Sn-characters. We give similar characterizations of finite dimensional algebras with involution whose ∗-codimension sequence is polynomially bounded. In this case we exploit the representation theory of the hyperoctahedral group. §