Canonical bases arising from $\imath$quantum covering groups

TL;DR

This paper constructs canonical bases for $ ext{U}^ ext{i}$-modules and the modified form of $ ext{i}$quantum covering groups of super Kac-Moody type, advancing the understanding of their algebraic structures.

Contribution

It introduces $ ext{i}$-canonical bases for modules and the algebra itself in the super Kac-Moody setting, using $ ext{i}^ extpi$-divided powers and rank one bases.

Findings

Constructed $ ext{i}$-canonical bases for highest weight modules.

Established bases for tensor products as $ ext{U}^ ext{i}$-modules.

Provided a canonical basis for the modified form of $ ext{U}^ ext{i}$.

Abstract

For $\imath$quantum covering groups $(\mathbf{U}, \mathbf{U}^\imath)$ of super Kac-Moody type, we construct $\imath$-canonical bases for the highest weight integrable $\mathbf{U}$-modules and their tensor products regarded as $\mathbf{U}^\imath$-modules, as well as a canonical basis for the modified form of the $\imath$quantum group $\mathbf{U}^\imath$, using the $\imath^\pi$-divided powers, rank one canonical basis for $\mathbf{U}^\imath$.