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Centers and Azumaya loci for finite W-algebras in positive characteristic

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  24 May 2021

BIN SHU  and
YANG ZENG
BIN SHU
Affiliation:
School of Mathematical Sciences, East China Normal University, No. 500, Dongchuan Road, Shanghai 200241, P. R. China e-mail: bshu@@math.ecnu.edu.cn
YANG ZENG
Affiliation:
School of Statistics and Data Science, Nanjing Audit University, No. 86 West Yushan Road, Jiangpu Street, Pukou District, Nanjing, Jiangsu Province 211815, P. R. China e-mail: zengyang@nau.edu.cn
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Abstract

In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

MSC classification

Primary: 17B50: Modular Lie (super)algebras
Secondary: 17B05: Structure theory 17B08: Coadjoint orbits; nilpotent varieties
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 1 , July 2022 , pp. 35 - 66
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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