Canonical bases arising from quantum symmetric pairs of Kac-Moody type

Abstract

For quantum symmetric pairs $(\mathbf{U},{\text{U}}^{ı})$ of Kac-Moody type, we construct $ı$canonical bases for the highest weight integrable $\mathbf{U}$-modules and their tensor products regarded as ${\mathbf{U}}^{ı}$-modules, as well as an $ı$canonical basis for the modified form of the $ı$quantum group ${\mathbf{U}}^{ı}$. A key new ingredient is a family of explicit elements called $ı$divided powers, which are shown to generate the integral form of ${\dot{\mathbf{U}}}^{ı}$. We prove a conjecture of Balagovic-Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi-K-matrix and the constructions of $ı$canonical bases, by avoiding a case-by-case rank one analysis and removing the strong constraints on the parameters in a previous work.

v2, 31 pages, mild corrections, to appear in Compositio Math

References in corpus (5)

• Canonical bases for tensor products and super Kazhdan-Lusztig theory
• Categorification of quantum symmetric pairs I
• A Serre presentation for the $ı$quantum groups
• Formulae of $ı$-divided powers in ${\mathbf{U}}_q({\mathfrak{sl}}_2)$
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Cited by in corpus (7)

• Universal K-matrices for quantum Kac-Moody algebras
• A Serre presentation for the $ı$quantum groups
• Based modules over the $ı$quantum group of type AI
• Lectures on dualities ABC in representation theory
• Representations of weakly triangular categories
• A Serre presentation for the $ı$quantum covering groups
• Canonical bases arising from $ı$quantum covering groups