Equivalence of finite-dimensional representations of Yangians and quantum loop algebras
Abstract: Quantum groups were discovered in the mid-eighties as symmetries of 1 and 2-dimensional Statistical Mechanical models. After a period of intense development, they reemerged more recently as the symmetries of 4-dimensional supersymmetric quantum gauge theories, through the work of Nekrasov-Shatashvilii, and as the constraints governing the enumerative geometry of Nakajima quiver varieties, through the work of Maulik-Okounkov.
This talk will centre on two closely related infinite dimensional quantum groups associated to a complex, simple Lie algebra g: the Yangian Yg, and the quantum loop algebra U_q(Lg). These deform respectively the Lie algebras g[s] and g[z,z^{-1}] of Taylor or Laurent polynomials with values in g.
I will sketch the classification of their finite-dimensional irreducible representations, and then construct an equivalence of categories of fd representations of Yg and U_q(Lg).
The talk is based on joint work with S. Gautam.