Georgia Benkart is Professor of Mathematics at the University of Wisconsin, Madison. She received her Ph.D. from Yale University, the 29th of Nathan Jacobson's 33 students. Her main research interests are Lie theory (including the classification of simple Lie algebras in prime characteristic and infinite-dimensional Lie algebras), representation theory, and combinatorics. Georgia is currently writing a book, Combinatorial Representation Theory, with Arun Ram. More information can be found on her curriculum vitae.
 
 
 
 

The  down and up operators on a partially ordered set (poset) encode many of the enumerative and combinatorial properties of the poset. The algebra they generate often is some nice algebra such as a Weyl algebra or a universal enveloping algebra of a Lie algebra. This talk will focus on certain generalizations of these algebras called down-up algebras. The talk will explore connections with enveloping algebras of Lie algebras, with Hopf algebras, and with quantum groups. Down-up algebras exhibit many striking properties including a beautiful representation theory.
 
 

In the theory of infinite-dimensional Lie algebras, the affine Lie algebras play a pre-eminent role. Their applications can be found throughout mathematics: in combinatorics, number theory, the theory of quivers, singularity theory, and many branches of mathematical physics. This talk introduces some natural generalizations called extended affine Lie algebras.