»ã±¨±êÌâ (Title)£º·ÂÉäÀî´úÊý¡¢ÃÀÂú¾§ÌåÓ뼸ºÎ¾§Ì壨Affine Lie algebras, Perfect Crystals and Geometric Crystals£©

»ã±¨ÈË (Speaker)£ºKailash Misra (NC State University, USA)

»ã±¨¹¦·ò (Time)£º2025Äê5ÔÂ20ÈÕ (Öܶþ) 16:00

»ã±¨µØÖ· (Place)£ºÐ£±¾²¿GJ303

Ô¼ÇëÈË(Inviter)£ºÕźìÁ« ½ÌÊÚ

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»ã±¨ÌáÒª£ºCrystal 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.