EVENT DETAILS AND ABSTRACT

Title: A q-analogue of the algebra of symmetric functions in infinitely many variables
Speaker: Professor Edward Frenkel
Speaker Info: University of California, Berkeley
Brief Description:
Special Note:
Abstract:

It is well-known that there is a natural Hopf algebra structure on the ring of symmetric functions in infinitely many variables. This structure has a clear representation theoretic meaning if one thinks of the ring of symmetric functions as the direct limit of Rep GL(N), the Grothendieck groups of (polynomial) representations of the group GL(N). The symmetric functions appear if one considers characters of these representations. We will introduce a q-analogue of this Hopf algebra: the direct limit of Grothendieck groups of finite-dimensional representations of the affine quantum group of GL(N). It turns out that this Hopf algebra is isomorphic to a familiar Hopf algebra, namely, the Hall algebra of an infinite linear (resp., cyclic) quiver if q is generic (resp., root of unity). In addition, to each representation one can attach its "q-character", so one obtains an algebra of "q-symmetric functions". The q-characters have already found interesting applications in representation theory and mathematical physics.