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A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas

  • Yan-an Cai , Hongjia Chen , Xiangqian Guo EMAIL logo , Yao Ma and Mianmian Zhu
Published/Copyright: March 21, 2021
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 3

Abstract

In this paper, we construct a class of new modules for the quantum group Uq⁢(s⁢l2) which are free of rank 1 when restricted to C⁢[K±1]. The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C⁢[K±1]-free Uq⁢(s⁢l2)-module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C⁢[K±1]-free modules with finite-dimensional simple modules over Uq⁢(s⁢l2), and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over Uq⁢(s⁢l2).

Keywords: Quantum group; irreducible module; non-weight module; tensor product; Clebsch–Gordan formula
MSC 2010: 17B37; 20G42; 81R50

Funding source: China Postdoctoral Science Foundation

Award Identifier / Grant number: 2016M600140

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11801390

Award Identifier / Grant number: 11771410

Award Identifier / Grant number: 11931009

Award Identifier / Grant number: 11971440

Award Identifier / Grant number: 11801066

Funding statement: Y. Cai is partially supported by the China Postdoctoral Science Foundation (Grant 2016M600140) the NSF of China (Grant 11801390) and High-level Innovation and Entrepreneurship Talents Introduction Program of Jiangsu Province of China. H. Chen is partially supported by the NSF of China (Grants 11771410 and 11931009) and Anhui Initiative in Quantum Information Technologies (Grant AHY150000). X. Guo is partially supported by the NSF of China (Grant 11971440). Y. Ma is partially supported by the NSF of China (Grant 11801066).

Acknowledgements

The authors would like to thank the referee for valuable suggestions to improve the paper.

  1. Communicated by: Jan Frahm

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Received: 2020-12-04
Revised: 2021-03-03
Published Online: 2021-03-21
Published in Print: 2021-05-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Weighted value distributions of the Riemann zeta function on the critical line
  3. Equivariant prequantization and the moment map
  4. Covering classes and 1-tilting cotorsion pairs over commutative rings
  5. Higher pullbacks of modular forms on orthogonal groups
  6. Distribution of root numbers of Hecke characters attached to some elliptic curves
  7. Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions
  8. On the number of zeros of diagonal cubic forms over finite fields
  9. Generic cuspidal representations of 𝑈(2, 1)
  10. A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas
  11. Set-theoretic solutions to the Yang–Baxter equation and generalized semi-braces
  12. Integral foliated simplicial volume and S1-actions
  13. Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature
  14. Vector bundles 𝐸 on ℙ3 with homological dimension 2 and 𝜒(End 𝐸) = 1
  15. On counting cuspidal automorphic representations for GSp(4)
  16. Algebraic cycles and intersections of a quadric and a cubic
https://doi.org/10.1515/forum-2020-0345
Keywords for this article
Quantum group; irreducible module; non-weight module; tensor product; Clebsch–Gordan formula

Articles in the same Issue

  1. Frontmatter
  2. Weighted value distributions of the Riemann zeta function on the critical line
  3. Equivariant prequantization and the moment map
  4. Covering classes and 1-tilting cotorsion pairs over commutative rings
  5. Higher pullbacks of modular forms on orthogonal groups
  6. Distribution of root numbers of Hecke characters attached to some elliptic curves
  7. Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions
  8. On the number of zeros of diagonal cubic forms over finite fields
  9. Generic cuspidal representations of 𝑈(2, 1)
  10. A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas
  11. Set-theoretic solutions to the Yang–Baxter equation and generalized semi-braces
  12. Integral foliated simplicial volume and S1-actions
  13. Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature
  14. Vector bundles 𝐸 on ℙ3 with homological dimension 2 and 𝜒(End 𝐸) = 1
  15. On counting cuspidal automorphic representations for GSp(4)
  16. Algebraic cycles and intersections of a quadric and a cubic
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