Yuri Bahturin, Memorial University of Newfoundland and Moscow State University

Abstract: Given a group G, a G-graded algebra A is of Lie type if for any three homogeneous elements a, b, c of degrees g, h, k, respectively, we have a(bc)=p(ab)c+q(ac)b, where p,q are the scalars depending on g, h, k, and p is nonzero. In particular, associative algebras, Lie algebras and (color) Lie superalgebras fall into this pattern. A number of recently developed combinatorial approaches to polynomial identities work in the general situation of Lie type algebras. In this talk I am going to describe some of recent results and techniques in the theory of graded algebras. I will mention our solution (with M.Zaicev) of a problem posed by A.Zalesskii of the existence of a non-trivial polynomial identity in a Lie algebra graded by a finite group in such a way that the identity component of the grading has this property. Then I will go on to one of our basic tools: the complete description of gradings of matrix algebras by finite abelian groups (with S. Sehgal and M.Zaicev). Among non-trivial consequences is the connection between the exponent of the growth of the identity component of an associative algebra graded by a finite group and that of the whole algebra (with M.Zaicev). Another consequence is the construction of the bases of graded identities of matrix rings in the whole number of various gradings (with V. Drensky).

Georgia Benkart, University of Wisconsin-Madison

Abstract: Down up algebras generalize the algebra generated by the down and up operators on a partially ordered set. They have connections with universal enveloping algebras, Hopf algebras, Drinfel'd (quantum) doubles, and quantum groups. This talk will survey these topics.

Roman Bezrukavnikov, University of Chicago

Abstract: This is a report on a joint work with Mirkovic and Rumynin (in progress). We apply (an appropriate version) of D-modules methods to study representations of semi-simple Lie algebras in prime characteristic. The results conform with conjectures by Lusztig on numerical invariants of the categories of non-restricted representations.

Anthony Giaquinto, Loyola University

Abstract: We obtain a presentation of Schur algebras (and q-Schur algebras) by generators and relations, one which is compatible with the usual presentation of the enveloping algebra (quantized enveloping algebra) corresponding to the Lie algebra gl_n of n x n matrices.  We also find new bases of Schur algebras and Hecke algebras.

Alexander Lichtman, University of Wisconsin-Parkside

Abstract: We consider skew fields generated by enveloping algebras or by some  classes of group rings. We construct valuations in these skew fields and use these valuations to study these skew fields. We prove that if H is a finitely generated residually finite p-group which is an extension of a torsion free nilpotent group U by a poly-{infinite cyclic} group then H has a p-series such that the associated restricted Lie algebra is a free abelian p-algebra. Hence the group ring KH over a field of characteristic p has a filtration whose associated graded ring gr(KH) is isomorphic to a polynomial algebra over K.

Ian Musson, University of Wisconsin-Milwaukee

Don Passman, University of Wisconsin-Madison

Abstract: A linear identity in a ring $R$ is an equation of the form

$r_1 xs_1+\cdots+ r_n x s_n=0$ that holds for all $x\in R$ or perhaps for all $x$ in some ``large'' subset of $R$. Such equations arise, for example, when determining the center of $R$, the extended centroid of the ring, or whether $R$ is prime, semiprime or satisfies a polynomial identity. In this talk, we discuss linear identities in group rings and various enveloping algebras. We indicate how they can be reduced to finite or finite-dimensional situations, and we mention a number of consequences of these reductions. The enveloping algebra results are, for the most part, joint work with Jeff Bergen.

Arun Ram, University of Wisconsin-Madison

Abstract: In joint work with Cathy Kriloff we have been explicitly computing the Springer correspondence for dihedral and icosahedral groups. Of course, this makes no sense, as we do not know what the appropriate varieties are to take the cohomology of for these cases. However, we are able to use tricks with graded Hecke algebras, to compute what these cohomologies have to be.
 
 

Michael Roitman, University of Michigan

Abstract: Strictly speaking, vertex algebras do not form a variety of algebras, because the so-called locality axiom is not an identity. In particular there is no free vertex algebras in general. However, a certain subcategory of vertex algebras, obtained by restricting the order of locality of generators, has a universal object, which would be the natural candidate for a free vertex algebra. We will present an explicit construction of free vertex algebras that allows to calculate their linear bases.

It turns out that free vertex algebras are closely related to the vertex algebras corresponding to integer lattices. The latter algebras play a very important role in different areas of mathematics and physics. This relation complies with the use of the word ``free'' in physical literature refering to some elements of lattice vertex algebras, like ``free field'', ``free bozon'' or ``free fermion''.

As a result, we find a nice presentation of lattice vertex algebras in terms of generators and relations, thus giving an alternative construction of these algebras, without using vertex operators. We remark that our construction works in a very general setting; we do not assume the lattice to be positive definite, neither non-degenerate, nor of a finite rank.