Evgeny Mukhin
Purdue, USA
The quantum toroidal gl(1).
Abstract: The quantum toroidal gl(1) algebra is a Hopf algebra quantization of the Lie algebra of difference operators on a line which recently appeared in several geometrical studies. We study this algebra by algebraic means. In particular, we present a large family of tame irreducible representations with combinatorial bases given by sets of partitions with explicit conditions. In appropriate limits, these representations reduce to well-known representations of different classical objects -- W_n algebras, gl_{\infty}, etc. As a result we obtain a new information about such classical representations.

This is a joint project with B. Feigin, M. Jimbo and T. Miwa.