A new involution for quantum loop algebras

Abstract

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

12 pages