Eli Aljadeff

Graded polynomial identities and exponential growth

Abstract Let A be a finite dimensional algebra over a field of characteristic zero graded by a finite abelian group G. Here we study a growth function related to the graded polynomial identities satisfied by A by computing the exponential rate of growth of the sequence of graded codimensions of A. We prove that the G-exponent of A exists and is an integer related in an explicit way to the dimension of a suitable semisimple subalgebra of A.

REPRESENTABILITY AND SPECHT PROBLEM FOR G-GRADED ALGEBRAS

Abstract Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let id G ( W ) denote the T-ideal of G-graded identities of W. We prove: 1. [G-graded PI-equivalence] There exists a field extension K of F and a finite-dimensional Z / 2 Z × G -graded algebra A over K such that id G ( W ) = id G ( A ∗ ) where A ∗ is the Grassmann envelope of A. 2. [G-graded Specht problem] The T-ideal id G ( W ) is finitely generated as a T-ideal. 3. [G-graded PI-equivalence for affine algebras] Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite-dimensional algebra A over K such that id G ( W ) = id G ( A ) .

Graded identities of matrix algebras and the universal graded algebra

In the last decade, group gradings and graded identities of finite dimensional central simple algebras have been an active area of research. We refer the reader to Bahturin, et al [6] and [7]. There are two basic kinds of group grading, elementary and fine. It was proved by Bahturin and Zaicev [7] that any group grading of Mn(C) is given by a certain composition of an elementary grading and a fine grading. In this paper we are concerned with fine gradings on Mn(C) and their corresponding graded identities. Let R be a simple algebra, finite dimensional over its center k and G a finite group. We say that R is fine graded by G if R ∼= ⊕g∈GRg is a grading and dimk(Rg) ≤ 1. Thus any component is either 0 or isomorphic to k as a k–vector space. It is easy to show that Supp(R), the subset of elements of G for which Rg is not 0, is a subgroup of G. Moreover

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.

POLYNOMIAL IDENTITIES AND NONCOMMUTATIVE VERSAL TORSORS

Abstract To any cleft Hopf Galois object, i.e., any algebra H α obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras” A H α and U H α . The algebra A H α is obtained by twisting the multiplication of H with the most general two-cocycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construction, A H α is a cleft H-Galois extension of a “big” commutative algebra B H α . Any “form” of H α can be obtained from A H α by a specialization of B H α and vice versa. If the algebra H α is simple, then A H α is an Azumaya algebra with center B H α . The algebra U H α is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of H α are satisfied. We construct an embedding of U H α into A H α ; this embedding maps the center Z H α of U H α into B H α when the algebra H α is simple. In this case, under an additional assumption, A H α ≅ B H α ⊗ Z H α U H α , thus turning A H α into a central localization of U H α . We completely work out these constructions in the case of the four-dimensional Sweedler algebra.

On twisting of finite-dimensional Hopf algebras

In this paper we study the properties of Drinfeld’s twisting for finite-dimensional Hopf algebras. We determine how the integral of the dual to a unimodular Hopf algebra H changes under twisting of H . We show that the classes of cosemisimple unimodular, cosemisimple involutive, cosemisimple quasitriangular finite-dimensional Hopf algebras are stable under twisting. We also prove the cosemisimplicity of a coalgebra obtained by twisting of a cosemisimple unimodular Hopf algebra by two different twists on two sides (such twists are closely related to bi-Galois extensions), and describe the representation theory of its dual. Next, we define the notion of a non-degenerate twist for a Hopf algebra H , and set up a bijection between such twists for H and H ∗ . This bijection is based on Miyashita–Ulbrich actions of Hopf algebras on simple algebras. It generalizes to the non-commutative case the procedure of inverting a non-degenerate skew-symmetric bilinear form on a vector space. Finally, we apply these results to classification of twists in group algebras and of cosemisimple triangular finite-dimensional Hopf algebras in positive characteristic, generalizing the previously known classification in characteristic zero.

Simple

Let G be any group and F an algebraically closed field of characteristic zero. We show that any G-graded finite dimensional associative G-simple algebra over F is determined up to a G-graded isomorphism by its G-graded polynomial identities. This result was proved by Koshlukov and Zaicev in case G is abelian.

G

LetK be a commutative ring with a unit element 1. Let Γ be a finite group acting onK via a mapt: Γ→Aut(K). For every subgroupH≤Γ define trH:K→KH by trh(x)=Σσ∈Hσ(x). We proveTheorem: trΓis surjective ontoKΓif and only if trPis surjective onto KPfor every (cyclic) prime order subgroup P of Γ.This is false for certain non-commutative ringsK.

-graded algebras and their polynomial identities

Let K be any field of characteristic p > 0 and let G be a finite group acting on K via a map T. The skew group algebra K,G may be non-semisimple (precisely when pIIH/, H = Kerr). We provide necessary conditions for the existence of a class c( E H’(G, K*) which “twists” the skew group algebra K, G into a semisimple crossed product Kr G. Further, we give a thorough analysis of the converse problem namely whether these conditions are also sufficient for the existence of a “semisimple 2-cocycle”. As a consequence we show this it is indeed so in many cases, in particular whenever G is a p-group.

On the surjectivity of some trace maps

Abstract We present a proof of Kemers representability theorem for affine PI algebras over a field of characteristic zero.