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Algebras and Representation Theory

, Volume 21, Issue 4, pp 787–806 | Cite as

Irreducible Modules Over Witt Algebras \(\mathcal {W}_{n}\) and Over \(\mathfrak {sl}_{n+1}(\mathbb {C})\)

  • Haijun Tan
  • Kaiming Zhao
Article
  • 34 Downloads

Abstract

In this paper, by using the “twisting technique” we obtain a class of new modules A b over the Witt algebras \(\mathcal {W}_{n}\) from modules A over the Weyl algebras \(\mathcal {K}_{n}\) (of Laurent polynomials) for any \(b\in \mathbb {C}\). We give necessary and sufficient conditions for A b to be irreducible, and determine necessary and sufficient conditions for two such irreducible \(\mathcal {W}_{n}\)-modules to be isomorphic. Since \(\mathfrak {sl}_{n+1}(\mathbb {C})\) is a subalgebra of \(\mathcal {W}_{n}\), all the above irreducible \(\mathcal {W}_{n}\)-modules A b can be considered as \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules. For a class of such \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules, denoted by Ω1−a (λ 1, λ 2, ⋯ ,λ n ) where \(a\in \mathbb {C}, \lambda _{1},\lambda _{2},\cdots ,\lambda _{n} \in \mathbb {C}^{*}\), we determine necessary and sufficient conditions for these \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules to be irreducible. If the \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-module Ω1−a (λ 1, λ 2,⋯ ,λ n ) is reducible, we prove that it has a unique nontrivial submodule W 1−a (λ 1, λ 2,...λ n ) and the quotient module is the finite dimensional \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-module with highest weight mΛ n for some non-negative integer \(m\in \mathbb {Z}_{+}\). We also determine necessary and sufficient conditions for two \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules of the form Ω1−a (λ 1, λ 2,⋯ ,λ n ) or of the form W 1−a (λ 1, λ 2,...λ n ) to be isomorphic.

Keywords

Witt algebra Weyl algebra \(\mathcal {K}_{n}\) \( \mathfrak {sl}_{n+1}(\mathbb {C})\) Non-weight module Irreducible module 

Mathematics Subject Classification (2010)

17B10 17B20 17B65 17B66 17B68 

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Notes

Acknowledgments

The authors would like to thank the referee for many nice suggestions. The first author is partially supported by CSC (Grant 201608220004), China Postdoctoral Science Foundation (Grants 111900302, 111900350), and NSF of Jilin Province (Grant 20160520111JH). The second author is partially supported by NSF of China (Grants 11271109, 11471233) and NSERC (Grant 311907-2015). The authors thank Prof. R. Lu for many helpful discussions during the preparation of the paper. We thank Prof. V. Mazorchuk to send us the paper [28] right after we have finished the present paper.

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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsNortheast Normal UniversityChangchunPeople’s Republic of China
  2. 2.Department of Applied MathematicsChangchun University of Science and TechnologyChangchunPeople’s Republic of China
  3. 3.College of Mathematics and Information ScienceHebei Normal (Teachers) UniversityShijiazhuangPeople’s Republic of China
  4. 4.Department of MathematicsWilfrid Laurier UniversityWaterlooCanada

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