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Simple 𝔰𝔩 d+1-modules from Witt algebra modules

  • Xiangqian Guo EMAIL logo , Xuewen Liu and Fenghua Zhang
Published/Copyright: October 27, 2023
Forum Mathematicum
From the journal Forum Mathematicum Volume 36 Issue 2

Abstract

Let d ≥ 1 be an integer and let 𝒲 d be the Witt algebra. For any admissible 𝒲 d -module P and any 𝔤 ⁢ 𝔩 d -module V, one can form a 𝒲 d -module ℱ ⁢ ( P , V ) , which as a vector space is P ⊗ V . Since 𝒲 d has a natural subalgebra isomorphic to 𝔰 ⁢ 𝔩 d + 1 , we can view ℱ ⁢ ( P , V ) as an 𝔰 ⁢ 𝔩 d + 1 -module. Taking P = Ω ⁢ ( 𝝀 ) , the rank-1 U ⁢ ( 𝔥 ) -free 𝒲 d -module, and V = V ⁢ ( 𝐚 , b ) , the simple cuspidal module over 𝔤 ⁢ 𝔩 d , we get the special 𝔰 ⁢ 𝔩 d + 1 -modules

ℱ ⁢ ( 𝝀 ; 𝐚 , b ) = ℱ ⁢ ( Ω ⁢ ( 𝝀 ) , V ⁢ ( 𝐚 , b ) )

which are U ⁢ ( 𝔥 ) -free modules of infinite rank. We determine the necessary and sufficient condition for the 𝔰 ⁢ 𝔩 d + 1 -module ℱ ⁢ ( 𝝀 ; 𝐚 , b ) to be simple, and for the non-simple case we construct their proper submodules explicitly. At last, using the above results, we deduce an explicit simplicity criterion for the generalized Verma modules induced from V ⁢ ( 𝐚 , b ) and obtain a family of simple affine modules from ℱ ⁢ ( 𝝀 ; 𝐚 , b ) , which can be viewed as the non-weight version of loop modules.

Keywords: Special linear algebra; Witt algebra; weight module; simple module
MSC 2020: 17B10; 17B20; 17B65; 17B66

Communicated by Jan Frahm

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11971440

Award Identifier / Grant number: 12001493

Funding statement: X. Guo is partially supported by NSF of China (Grant 11971440). X. Liu is partially supported by NSF of China (Grant 12001493).

Acknowledgements

The authors are grateful to the referees for pointing out inaccuracies and providing good suggestions to make the paper more readable.

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Received: 2023-04-29
Revised: 2023-08-03
Published Online: 2023-10-27
Published in Print: 2024-03-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Stability of solutions to obstacle problems with generalized Orlicz growth
  3. The Eisenstein cycles and Manin–Drinfeld properties
  4. Topological group actions by group automorphisms and Banach representations
  5. On abelian-by-cyclic Moufang loops
  6. Gromov ellipticity and subellipticity
  7. On the spectra of a class of Moran measures
  8. Finite-dimensional algebras not arising as blocks of group algebras
  9. A uniform Besov boundedness and the well-posedness of the generalized dissipative quasi-geostrophic equation in the critical Besov space
  10. Fourier integral operators with forbidden symbols on the Besov spaces
  11. Spectrality of Cantor–Moran measures with three-element digit sets
  12. Simple 𝔰𝔩 d+1-modules from Witt algebra modules
  13. Archimedean toroidal maps and their edge covers
  14. The smallest Mealy automaton generating an indicable regular branch group
  15. Global classical solution of the Cauchy problem to the 3D Benjamin–Bona–Mahony–Burgers-type equation with nonlocal control constraints
  16. Finite approximation properties of C*-modules III
https://doi.org/10.1515/forum-2023-0159
Keywords for this article
Special linear algebra; Witt algebra; weight module; simple module

Articles in the same Issue

  1. Frontmatter
  2. Stability of solutions to obstacle problems with generalized Orlicz growth
  3. The Eisenstein cycles and Manin–Drinfeld properties
  4. Topological group actions by group automorphisms and Banach representations
  5. On abelian-by-cyclic Moufang loops
  6. Gromov ellipticity and subellipticity
  7. On the spectra of a class of Moran measures
  8. Finite-dimensional algebras not arising as blocks of group algebras
  9. A uniform Besov boundedness and the well-posedness of the generalized dissipative quasi-geostrophic equation in the critical Besov space
  10. Fourier integral operators with forbidden symbols on the Besov spaces
  11. Spectrality of Cantor–Moran measures with three-element digit sets
  12. Simple 𝔰𝔩 d+1-modules from Witt algebra modules
  13. Archimedean toroidal maps and their edge covers
  14. The smallest Mealy automaton generating an indicable regular branch group
  15. Global classical solution of the Cauchy problem to the 3D Benjamin–Bona–Mahony–Burgers-type equation with nonlocal control constraints
  16. Finite approximation properties of C*-modules III
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