Diagram automorphisms and quantum groups

Abstract

Let 𝐔⁻_𝑞 = 𝐔⁻_𝑞(𝔤) be the negative part of the quantum group associated to a finite dimensional simple Lie algebra 𝔤, and 𝜎 : 𝔤 → 𝔤 be the automorphism obtained from the diagram automorphism. Let 𝔤^𝜎 be the fixed point subalgebra of 𝔤, and put \underline{𝐔}⁻_𝑞 = 𝐔⁻_𝑞(𝔤^𝜎). Let 𝐁 be the canonical basis of 𝐔⁻_𝑞 and \underline{𝐁} the canonical basis of \underline{𝐔}⁻_𝑞. 𝜎 induces a natural action on 𝐁, and we denote by 𝐁^𝜎 the set of 𝜎-fixed elements in 𝐁. Lusztig proved that there exists a canonical bijection 𝐁^𝜎 ≃ \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.