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.