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

TL;DR

This paper derives explicit formulas for the multiplication and comultiplication structure constants of $ ext{ extit{i}}$-divided powers in the split rank 1 $ ext{ extit{i}}$-quantum group, establishing their integrality and positivity.

Contribution

It provides the first closed-form expressions for the structure constants of $ ext{ extit{i}}$-divided powers, advancing understanding of their algebraic properties.

Findings

Structure constants are integral and positive.

Closed formulas for multiplication and comultiplication are obtained.

Enhances the algebraic framework of $ ext{ extit{i}}$-quantum groups.

Abstract

The $\imath$-divided powers (depending on a parity) form the $\imath$canonical basis for the split rank 1 $\imath$quantum group and they are a basic ingredient for $\imath$quantum groups of higher rank. We obtain closed formulae for the structure constants for multiplication of the $\imath$-divided powers. Closed formulae for the comultiplication of the $\imath$-divided powers are also obtained. These structure constants are integral and positive.