Diagram automorphisms and quantum groups

Abstract

Let $U_q^{-}=U_q^{-}(\mathfrak{g})$ be the negative part of the quantum group associated to a finite dimensional simple Lie algebra $\mathfrak{g}$, and $\sigma :\mathfrak{g}\to \mathfrak{g}$ be the automorphism obtained from the diagram automorphism. Let ${\mathfrak{g}}^{\sigma }$ be the fixed point subalgebra of $\mathfrak{g}$, and put ${\underline{U}}_q^{-}=U_q^{-}({\mathfrak{g}}^{\sigma })$. Let $B$ be the canonical basis of $U_q^{-}$ and $\underline{B}$ the canonical basis of ${\underline{U}}_q^{-}$. $\sigma$ induces a natural action on $B$, and we denote by $B^{\sigma }$ the set of $\sigma$-fixed elements in $B$. Lusztig proved that there exists a canonical bijection $B^{\sigma }\simeq \underline{B}$ by using geometric considerations. In this paper, we construct such a bijection in an elementary way. We also consider such a bijection in the case of certain affine quantum groups, by making use of PBW-bases constructed by Beck and Nakajima.

35 pages, final version, to appear in J. of Math. Soc. of Japan