A new involution for quantum loop algebras

TL;DR

This paper introduces a completion of the positive part of quantum affine algebras, defines a new involution and Kashiwara's operators, and conjectures the existence of a crystal basis and canonical basis.

Contribution

It constructs a new bar-involution and Kashiwara's operators on the completed quantum affine algebra, proposing a conjecture on crystal and canonical bases.

Findings

Defined a new bar-involution on the completed algebra

Constructed Kashiwara's operators in this setting

Conjectured the existence of crystal and canonical bases

Abstract

In this article, we introduce a completion $\widehat{U}^+_v(\mathcal{L}\mathfrak{g})$ of the positive half of the quantum affinization $U^+_v(\mathcal{L}\mathfrak{g})$ of a symmetrizable Kac-Moody algebra $\mathfrak{g}$. On $\widehat{U}^+_v(\mathcal{L}(\mathfrak{g}))$, we define a new "bar-involution" and construct the analogue Kashiwara's operators. We conjecture that the resulting pair $(\widehat{\mathcal{L}},\widehat{\mathcal{B}})$ is a crystal basis which provides the existence of the "canonical basis" on the (completion of the) of the positive half of the quamtum affinization.