PIRSA:18010097

Elliptic quantum groups and affine Grassmannians over an elliptic curve

Zhao, G. (2018). Elliptic quantum groups and affine Grassmannians over an elliptic curve . Perimeter Institute. https://pirsa.org/18010097

Zhao, Gufang. Elliptic quantum groups and affine Grassmannians over an elliptic curve . Perimeter Institute, Jan. 29, 2018, https://pirsa.org/18010097 

          @misc{ pirsa_PIRSA:18010097,
            doi = {10.48660/18010097},
            url = {https://pirsa.org/18010097},
            author = {Zhao, Gufang},
            keywords = {Mathematical physics},
            language = {en},
            title = {Elliptic quantum groups and affine Grassmannians over an elliptic curve },
            publisher = {Perimeter Institute},
            year = {2018},
            month = {jan},
            note = {PIRSA:18010097 see, \url{https://pirsa.org}}
          }

Gufang Zhao University of Melbourne

DOI 10.48660/18010097
Collection
Talk Type Scientific Series
Subject

Abstract

This is based on my joint work with Yaping Yang. In this talk, we use the equivariant elliptic cohomology theory to study the elliptic quantum groups.  We define a sheafified elliptic quantum group for any symmetric Kac-Moody Lie algebra. This definition is naturally obtained using the cohomological Hall algebra associated to the equivariant elliptic cohomology. After taking suitable rational sections, the sheafified elliptic quantum group becomes a quantum algebra consisting of the elliptic Drinfeld currents. The Drinfeld currents satisfy the relations of the elliptic quantum group studied by Felder and Gautam-Toledano Laredo. We show the elliptic quantum group acts on the equivariant elliptic cohomology of Nakajima quiver varieties.

In particular, the sheafified elliptic quantum group is an algebra object in a certain monoidal category of sheaves on the colored Hilbert scheme of an elliptic curve. This monoidal structure is related to Mirkovic’s refinement of the factorization structure on semi-infinite affine Grassmannian over an elliptic curve. If time permits, I will also talk about a work in progress, joint with Mirkovic and Yang, towards a construction of a double loop Grassmannian and vertex representations of the toroidal algebra, which in turn is related to representations of the elliptic quantum groups