Formulae of $ı$-divided powers in ${\mathbf{U}}_q({\mathfrak{sl}}_2)$

Abstract

The existence of the $ı$-canonical basis (also known as the $ı$-divided powers) for the coideal subalgebra of the quantum ${\mathfrak{sl}}_2$ were established by Bao and Wang, with conjectural explicit formulae. In this paper we prove the conjectured formulae of these $ı$-divided powers. This is achieved by first establishing closed formulae of the $ı$-divided powers in basis for quantum ${\mathfrak{sl}}_2$ and then formulae for the $ı$-canonical basis in terms of Lusztig's divided powers in each finite-dimensional simple module of quantum ${\mathfrak{sl}}_2$. These formulae exhibit integrality and positivity properties.

v3, 38 pages, a typo in (3.7) in the published version corrected

References in corpus (1)

• Canonical bases for tensor products and super Kazhdan-Lusztig theory

Cited by in corpus (5)

• Canonical bases arising from quantum symmetric pairs of Kac-Moody type
• A Serre presentation for the $ı$quantum groups
• Formulae of $ı$-divided powers in ${\mathbf{U}}_q({\mathfrak{sl}}_2)$, II
• A Serre presentation for the $ı$quantum covering groups
• Canonical bases arising from $ı$quantum covering groups