## Journal of the Mathematical Society of Japan

### Diagram automorphisms and quantum groups

#### Abstract

Let $\mathbf{U}^-_q = \mathbf{U}^-_q(\mathfrak{g})$ 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 $\underline{\mathbf{U}}^-_q = \mathbf{U}^-_q(\mathfrak{g}^{\sigma})$. Let $\mathbf{B}$ be the canonical basis of $\mathbf{U}_q^-$ and $\underline{\mathbf{B}}$ the canonical basis of $\underline{\mathbf{U}}_q^-$. $\sigma$ induces a natural action on $\mathbf{B}$, and we denote by $\mathbf{B}^{\sigma}$ the set of $\sigma$-fixed elements in $\mathbf{B}$. Lusztig proved that there exists a canonical bijection $\mathbf{B}^{\sigma} \simeq \underline{\mathbf{B}}$ 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.

#### Article information

Source
J. Math. Soc. Japan, Advance publication (2019), 33 pages.

Dates
First available in Project Euclid: 2 November 2019

https://projecteuclid.org/euclid.jmsj/1572660116

Digital Object Identifier
doi:10.2969/jmsj/81488148

#### Citation

SHOJI, Toshiaki; ZHOU, Zhiping. Diagram automorphisms and quantum groups. J. Math. Soc. Japan, advance publication, 2 November 2019. doi:10.2969/jmsj/81488148. https://projecteuclid.org/euclid.jmsj/1572660116

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