报告题目 (Title)：仿射李代数、完美晶体与几何晶体(Affine Lie algebras, Perfect Crystals and Geometric Crystals)
报告人 (Speaker)：Kailash Misra (NC State University, USA)
报告时间 (Time)：2025年5月20日 (周二) 16:00
报告地点 (Place)：校本部GJ303
邀请人(Inviter)：张红莲教授
报告摘要：Crystal bases for integrable representations of affine Lie algebras was introduced around 1990. In 1991, we introduced the concept of a perfect representation whose associated crystal is called a perfect crystal. In 1994, we introduced the notion of a coherent family of perfect crystals which admits a projective limit. In 2000, Berenstein and Kazhdan introduced the notion of a geometric crystal for reductive algebraic groups which was generalized by Nakashima to symmetrizable Kac-Moody groups in 2005. A remarkable relation between geometric crystals and algebraic crystals is the Ultra-discretization functor. In 2008, Kashiwara, Nakashima and Okado conjectured that there exists an affine geometric crystal at each nonzero Dynkin node of an affine Lie algebra whose ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for its Langland dual. So far this conjecture has been proved for some specific affine Lie algebras at some specific Dynkin nodes. In this talk I will review these concepts and go over the results for the affine Lie algebra $A_n^{(1)}$ at any Dynkin node $k$ which is a joint work with T. Nakashima.