Midwest Algebra and Representation Theory Conference

DePaul University, December 1-2, 2001

ABSTRACTS

John Beachy, Northern Illinois University

M-injective modules and prime M-ideals"

Abstract: I look at the category sigma[M] of modules subgenerated by a fixed module M, and look at a correspondence between indecomposable M-injective modules and a notion of "prime" sumodules in M.  There's a copy of the paper on my website:  www.math.niu.edu/~beachy/papers

Esther Beneish, Northwestern University

Stable rationality of the center of the generic division ring"

Steve Doty, Loyola University

An algebra based on derangements"

Abstract: This talk is based on joint work with Georgia Benkart. The group algebra of the symmetric group is an algebra with basis elements indexed by permutations. I will describe an algebra with basis elements indexed by derangements (permutations having no fixed points). This algebra is the centralizer algebra for tensor powers of the adjoint action of the special linear Lie algebra on itself, and it can be realized as a subalgebra of the walled Brauer algebra. One can classify the representations of the derangement algebra. (All of this is over the complex field.)

Kent Fuller, University of Iowa

Generalizations of Morita Duality Related to Cotilting Modules''

Abstract: Given an S-R-bimodule W, the contravariant functors D = Hom(_,W) induce a duality between the categories of reflexive right R-modules and  left S-modules. If these categories are closed under submodules, factor modules and (necessarily) extensions, then D is a Morita duality. This notion dates back to 1958. Over the ensuing years various generalizations have been investigated.  For example, if W is the vector space dual of a tilting module over a finite dimensional algebra, then D gives a duality between categories of finitely generated R and S-modules that are closed under submodules and extensions. Various related dualities will be discussed.

Esther Garcia Gonzalez, University of Oviedo, Spain

Simple, Primitive and Strongly Prime Jordan 3-graded Lie algebras".

Abstract: A special class of 3-graded Lie algebras is that of Jordan 3-graded Lie algebras. They are closely related to Jordan pairs and can be seen as central extensions of Tits-Kantor-Koecher algebras. Indeed, under conditions of regularity (namely, simplicity, strong primeness and primitivity) they are Tits-Kantor-Koecher algebras of their associated Jordan pairs and, moreover, those Jordan pairs satisfy the same regularity condition. This will allow us to give full (if and only if) descriptions of regular Jordan 3-graded Lie algebras.

Fred Goodman, University of Iowa

A path algorithm for affine Kazhdan Lusztig polynomials"

Darrell Haile, Indiana University

On weakly Azumaya algebras and Frobenius algebras"

Abstract: If $K/F$ is a finite Galois extension of degree $n$ with Galois group $G$ and $f$ is a weak two-cocycle from $G\times G$ to $K$ (that is, a cocycle in which the value zero is allowed), the resulting crossed-product algebra $A_f$ is associative of dimension $n^2$ over its center $F$. The algebra $A_f$ is not necessarily simple but has a
Wedderburn decomposition that can be understood in terms of the cocycle. Associated to each algebra is a graph on a set of cosets of $G$. The graphs that can arise in this way are called lower subtractive. The collection of algebras with the same graph forms a group under a suitable product. In this talk we will discuss these algebras and graphs.  In particular we will prove  that the algebras in this group are Frobenius if and only if the graph has a unique maximal element. An interesting example is the so-called weak ordering (related to the Bruhat ordering) on the Weyl group $W$ of a semisimple Lie algebra. We show that the resulting graph on $W$ is lower subtractive. Since this orering has a unique maximal element the resulting algebra (for a Galois extension with group $W$) is Frobenius and we examine its properties.

David Hemmer, University of Georgia

Support Varieties for Symmetric Groups'' ( joint work with Dan Nakano)

Abstract: We introduce some techniques to compute complexity and support varieties for symmetric group modules. We determine the complexity and support varieties for Young modules and permutation modules, using a result which
reduces the computation of support varieties to that of relative support varieties over Young subgroups. Finally we will apply these techniques to determine the complexity and support variety for a class of simple modules known as completely splittable modules."

The reduced Clifford algebra of a binary form and its Brauer class''

Martin Lorenz, Temple University

Euler classes of crystallographic groups and Hopf algebras''

Abstract: We show that the Euler class of a finitely generated abelian-by-finite group $G$ over a field of characteristic $p\ge 0$ has finite order if and only if every $p$-regular element of $G$ has infinite centralizer in
$G$. We also give a lower bound for the order of the Euler class in terms of suitable finite subgroups of $G$. This lower bound is derived from a more general result on finite-dimensional representations of smash products of Hopf algebras.

Izuru Mori, Purdue University

B\'ezout's Theorem for Projectively Cohen-Macaulay Schemes''

Abstract: Let $k$ be an algebraically closed field.  The Fundamental Theorem of Algebra says that if $f\in k[x]$ is a polynomial, then the equation $f=0$ has $\deg f$ many solutions in $k$ counting multiplicity.  A tempting but false expectation is that if $f, g\in k[x,y]$ are polynomials, then the system of equations $f=0, g=0$ has $\deg f\cdot \deg g$ many solutions in $k$ counting multiplicity. In geometric terms, this statement is equivalent to saying that if $C, D$ are curves in the affine plane, then $C, D$ intersect $\deg C\cdot \deg D$ times counting multiplicity. Fortunately, this geometric statement is true if we replace the affine plane by the projective plane, which is known as B\'ezout's Theorem. In this talk, we will extend B\'ezout's Theorem to a projectively Cohen-Macaulay scheme, which is a projective scheme associated to a (not necessarily commutative) Cohen-Macaulay graded algebra.  The theory typically applies to an enveloping algebra of a finite dimensional Lie algebra.

Marc Renault, Temple University

Multiplicative Invariants of Reflection Groups’’

Abstract: Let G be a finite group acting by automorphism on a lattice A. Then G acts by automorphisms on the group algebra R = k[A] as well. It is known that if G is a reflection group then the algebra of G-invariants in R, denoted R^G, is a semigroup algebra. We will discuss an algorithm for computing the algebra generators of R^G and explore the structure of R^G when G is a subgroup of a reflection group.

Yi Ming Zou, University of Wisconsin-Milwaukee

Constructing Crystal Bases for Lie Superalgebras’’

Abstract: The crystal bases introduced by Kashiwara for the quantized enveloping algebras of Kac-Moody algebras with symmetrizable Cartan matrices carry interesting combinatorial properties, in particular, they can be used to describe the decompositions of tensor products of representations. Recently, Kashiwara's construction has been generalized to the quantized enveloping algebras of certain Lie superalgebras. This talk will survey these recent developments.