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# Forum Mathematicum

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Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 31, Issue 3

# Simple modules over the Lie algebras of divergence zero vector fields on a torus

Brendan Frisk Dubsky
/ Xiangqian Guo
/ Yufeng Yao
/ Kaiming Zhao
• College of Mathematics and Information Science, Hebei Normal (Teachers) University, Shijiazhuang, Hebei, 050016, P. R. China; and Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada N2L 3C5
• Email
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Published Online: 2019-03-07 | DOI: https://doi.org/10.1515/forum-2018-0096

## Abstract

Let $n\ge 2$ be an integer, ${𝕊}_{n}$ the Lie algebra of divergence zero vector fields on an n-dimensional torus, and ${\mathcal{𝒦}}_{n}$ the Weyl algebra over the Laurent polynomial algebra ${A}_{n}=ℂ\left[{x}_{1}^{±1},{x}_{2}^{±1},\mathrm{\dots },{x}_{n}^{±1}\right]$. For any $𝔰{𝔩}_{n}$-module V and any module P over ${\mathcal{𝒦}}_{n}$, we define an ${𝕊}_{n}$-module structure on the tensor product $P\otimes V$. In this paper, necessary and sufficient conditions for the ${𝕊}_{n}$-modules $P\otimes V$ to be simple are given, and an isomorphism criterion for nonminuscule ${𝕊}_{n}$-modules is provided. More precisely, all nonminuscule ${𝕊}_{n}$-modules are simple, and pairwise nonisomorphic. For minuscule ${𝕊}_{n}$-modules, minimal and maximal submodules are concretely determined.

MSC 2010: 17B10; 17B65; 17B66

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Revised: 2018-10-29

Published Online: 2019-03-07

Published in Print: 2019-05-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11271109

Award identifier / Grant number: 11471294

Award identifier / Grant number: 11571008

Award identifier / Grant number: 11671138

Award identifier / Grant number: 11771279

Funding Source: Natural Science Foundation of Shanghai

Award identifier / Grant number: 16ZR1415000

Funding Source: Natural Sciences and Engineering Research Council of Canada

Award identifier / Grant number: 311907-2015

The research was carried out during the visit of the first two authors to Wilfrid Laurier University in the summer of 2017. The hospitality of Wilfrid Laurier University is gratefully acknowledged. Brendan Frisk Dubsky is partially supported by the Lundström-Åman foundation. Xianqian Guo is partially supported by NSF of China (Grant No. 11471294) and the Outstanding Young Talent Research Fund. Yufeng Yao is partially supported by NSF of China (Grant Nos. 11771279, 11571008 and 11671138) and NSF of Shanghai (Grant No. 16ZR1415000). Kaiming Zhao is partially supported by NSF of China (Grant No. 11871190) and NSERC (Grant 311907-2015).

Citation Information: Forum Mathematicum, Volume 31, Issue 3, Pages 727–741, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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