Crystal bases of modified $ı$quantum groups of certain quasi-split types

Abstract

In order to see the behavior of $ı$canonical bases at $q=\infty$, we introduce the notion of $ı$crystals associated to an $ı$quantum group of certain quasi-split type. The theory of $ı$crystals clarifies why $ı$canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of $ı$crystals whose projective limit can be thought of as the $ı$canonical basis of the modified $ı$quantum group at $q=\infty$.

arXiv comment74 pages. Presentation is modified in order to make it clearer

References in corpus (7)

• Quantum symmetric Kac-Moody pairs
• A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs
• Canonical bases for tensor products and super Kazhdan-Lusztig theory
• A Serre presentation for the $ı$quantum groups
• Formulae of $ı$-divided powers in ${\mathbf{U}}_q({\mathfrak{sl}}_2)$, II
• Based modules over the $ı$quantum group of type AI
• A new tableau model for irreducible polynomial representations of the orthogonal group