A new involution for quantum loop algebras

In this article, we introduce a completion Uˆv+(Lg) of the positive half of the quantum affinization Uv+(Lg) of a symmetrizable Kac-Moody algebra g. On Uˆv+(L(g)), we define a new “bar-involution” and construct the analogue Kashiwara's operators. We conjecture that the resulting pair (Lˆ,Bˆ) is a crystal basis which provides the existence of the “canonical basis” on the (completion of the) of the positive half of the quantum affinization.