Ph.D Thesis
Ph.D Student Igor Karasik On Graded and H-module Algebras Satisfying a PI Department of Mathematics Full Professor Aljadeff Eli

Abstract

The purpose of this thesis is to extend classical results from ordinary PI theory to a more general framework. We start with a semisimple Hopf algebra H which is finite dimensional over an algebraically closed field F of characteristic zero and consider an H-module F-algebra W which satisfies an ordinary polynomial identity. We show that W is H-PI representable. That is, there is a finite dimensional H-module algebra A having the same H-polynomial identities as W, assuming W is affine. More generally, if W is not assumed affine, then there is a super H-module algebra A of finite dimension, such that W has the same H-identities as the Grassmann envelope of A. Thus, we obtain a generalization of Kemer's PI representability theorem. This generalization includes two impotent cases: (finite) group graded algebras and (finite) group acted algebras - the first is already known whereas the second is a new result of this thesis. As a result we are able to conclude that for an H-module W which satisfies an ordinary PI, there are a finite set of H-polynomials which generates the T-ideal of H-identities of W. In other words, we solve the Specht problem for H-module algebras satisfying an ordinary PI. Furthermore, using the representability theorem, we prove that the exponent of the codimension series of W, that is the sequence of dimensions of the space of n-multilinear H-polynomial non-identities of W, is an integer.

In the rest of the thesis we consider group (G) graded algebras W which satisfy an ordinary PI. We show that the analog theorems to Kaplansky PI theorem and Posner's theorem also hold in this framework. More precisely, we show that for any group G (including infinite groups) every G-primitive ideal of W is, in fact, G-simple. Thus, we are able to determine exactly the structure of G-primitive, ordinary PI algebras. Furthermore, for groups G which are residually finite we are able to prove that any G -prime W, having a field as its e-component of its center, is G-simple.

The last parts of the thesis are devoted to the study of primeness properties of graded T-ideals. We consider three kinds of primness: prime, verbally prime and absolutely verbally prime G-graded T-ideals (all the T-ideals are considered to contain a non-zero ordinary polynomial). The first two properties are well known in the ordinary case (trivial grading) due to the work of Kemer. Moreover, the third property that we introduce is equivalent to the second one, in the ordinary case. However, in the case of non-trivial (finite) group gradings the third property becomes the most interesting (and complicated) of the three. All in all, we manage to classify completely all prime, verbally prime T -ideals and absolutely verbally prime T-ideals which contain a Cappeli polynomials. It is remarkable to mention that the third family corresponds to T-ideals of finite dimensional G-division algebras. That is, G-graded algebras whose non-zero homogenous elements are inevitable.