Crystal bases of modified $\imath$quantum groups of certain quasi-split types

TL;DR

This paper introduces the concept of $ ext{ extit{i}}$-crystals to analyze the behavior of $ ext{ extit{i}}$-canonical bases of certain quasi-split $ ext{ extit{i}}$-quantum groups at $q = ext{ extit{infinity}}$, revealing their structural properties.

Contribution

It defines $ ext{ extit{i}}$-crystals for quasi-split $ ext{ extit{i}}$-quantum groups and constructs a projective system capturing the $ ext{ extit{i}}$-canonical basis at $q = ext{ extit{infinity}}$.

Findings

$ ext{ extit{i}}$-crystals explain why $ ext{ extit{i}}$-canonical bases are not always preserved.

A projective system of $ ext{ extit{i}}$-crystals is constructed.

The projective limit models the $ ext{ extit{i}}$-canonical basis at $q = ext{ extit{infinity}}$.

Abstract

In order to see the behavior of $\imath$canonical bases at $q = \infty$, we introduce the notion of $\imath$crystals associated to an $\imath$quantum group of certain quasi-split type. The theory of $\imath$crystals clarifies why $\imath$canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of $\imath$crystals whose projective limit can be thought of as the $\imath$canonical basis of the modified $\imath$quantum group at $q = \infty$.