Diagram automorphisms and quantum groups
| Autor: | Zhiping Zhou, Toshiaki Shoji |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Diagram (category theory)
General Mathematics Fixed point 01 natural sciences canonical bases PBW-bases Combinatorics Mathematics::Quantum Algebra Mathematics - Quantum Algebra 0103 physical sciences Lie algebra FOS: Mathematics Quantum Algebra (math.QA) 20G42 0101 mathematics Mathematics::Representation Theory Mathematics quantum groups Quantum group 010102 general mathematics Subalgebra 17B37 81R50 Automorphism 17B37 81R50 Standard basis Bijection 010307 mathematical physics |
| Zdroj: | J. Math. Soc. Japan 72, no. 2 (2020), 639-671 |
| ISSN: | 0025-5645 |
| DOI: | 10.2969/jmsj/81488148 |
| Popis: | 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. Comment: 35 pages, final version, to appear in J. of Math. Soc. of Japan |
| Databáze: | OpenAIRE |
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