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

TL;DR

This paper proves conjectured explicit formulae for the $ extit{i}$-divided powers in the quantum $ extit{sl}_2$ algebra, establishing their integrality and positivity, and providing a foundation for understanding the $ extit{i}$-canonical basis.

Contribution

It provides the first proof of the conjectured explicit formulae for $ extit{i}$-divided powers in quantum $ extit{sl}_2$, including their basis expressions and positivity properties.

Findings

Confirmed explicit formulae for $ extit{i}$-divided powers.

Established integrality and positivity of the formulae.

Connected $ extit{i}$-canonical basis with Lusztig's divided powers.

Abstract

The existence of the $\imath$-canonical basis (also known as the $\imath$-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 $\imath$-divided powers. This is achieved by first establishing closed formulae of the $\imath$-divided powers in basis for quantum $\mathfrak{sl}_2$ and then formulae for the $\imath$-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.