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April, 2020 Diagram automorphisms and quantum groups
Toshiaki SHOJI, Zhiping ZHOU
J. Math. Soc. Japan 72(2): 639-671 (April, 2020). DOI: 10.2969/jmsj/81488148

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.

Citation

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Toshiaki SHOJI. Zhiping ZHOU. "Diagram automorphisms and quantum groups." J. Math. Soc. Japan 72 (2) 639 - 671, April, 2020. https://doi.org/10.2969/jmsj/81488148

Information

Received: 17 October 2018; Published: April, 2020
First available in Project Euclid: 2 November 2019

zbMATH: 07196915
MathSciNet: MR4090349
Digital Object Identifier: 10.2969/jmsj/81488148

Subjects:
Primary: 17B37
Secondary: 20G42 , 81R50

Keywords: canonical bases , PBW-bases , quantum groups

Rights: Copyright © 2020 Mathematical Society of Japan

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Vol.72 • No. 2 • April, 2020
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