Home Simple modules over the Lie algebras of divergence zero vector fields on a torus
Article
Licensed
Unlicensed Requires Authentication

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

  • Brendan Frisk Dubsky , Xiangqian Guo EMAIL logo , Yufeng Yao and Kaiming Zhao
Published/Copyright: March 7, 2019
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 3

Abstract

Let n≥2 be an integer, 𝕊n the Lie algebra of divergence zero vector fields on an n-dimensional torus, and 𝒦n the Weyl algebra over the Laurent polynomial algebra An=ℂ⁢[x1±1,x2±1,…,xn±1]. For any 𝔰⁢𝔩n-module V and any module P over 𝒦n, we define an 𝕊n-module structure on the tensor product P⊗V. In this paper, necessary and sufficient conditions for the 𝕊n-modules P⊗V to be simple are given, and an isomorphism criterion for nonminuscule 𝕊n-modules is provided. More precisely, all nonminuscule 𝕊n-modules are simple, and pairwise nonisomorphic. For minuscule 𝕊n-modules, minimal and maximal submodules are concretely determined.

Keywords: Lie algebra of divergence zero vector fields; Weyl algebra; irreducible module; (non)minuscule module; density theorem
MSC 2010: 17B10; 17B65; 17B66

Communicated by Karl-Hermann Neeb

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

Funding statement: 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).

Acknowledgements

The authors are grateful to the referees for pointing out inaccuracies and providing valuable suggestions to make the paper easier to follow.

References

[1] V. V. Bavula and T. Lu, Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules, J. Math. Phys. 58 (2017), no. 1, Article ID 011701. 10.1063/1.4973378Search in Google Scholar

[2] V. V. Bavula and T. Lu, Classification of simple weight modules over the Schrödinger algebra, Canad. Math. Bull. 61 (2018), no. 1, 16–39. 10.4153/CMB-2017-017-7Search in Google Scholar

[3] Y. Billig, Jet modules, Canad. J. Math. 59 (2007), no. 4, 712–729. 10.4153/CJM-2007-031-2Search in Google Scholar

[4] Y. Billig and V. Futorny, Classification of irreducible representations of Lie algebra of vector fields on a torus, J. Reine Angew. Math. 720 (2016), 199–216. 10.1515/crelle-2014-0059Search in Google Scholar

[5] Y. Billig, A. Molev and R. Zhang, Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus, Adv. Math. 218 (2008), no. 6, 1972–2004. 10.1016/j.aim.2008.03.026Search in Google Scholar

[6] Y. Billig and J. Talboom, Classification of category 𝒥 modules for divergence zero vector fields on a torus, J. Algebra 500 (2018), 498–516. 10.1016/j.jalgebra.2017.05.022Search in Google Scholar

[7] R. E. Block, The irreducible representations of the Lie algebra 𝔰⁢𝔩⁢(2) and of the Weyl algebra, Adv. in Math. 39 (1981), no. 1, 69–110. 10.1016/0001-8708(81)90058-XSearch in Google Scholar

[8] E. Cartan, Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull. Soc. Math. France 41 (1913), 53–96. 10.24033/bsmf.916Search in Google Scholar

[9] J. Dixmier, Enveloping Algebras, North-Holland Math. Libr. 14, North-Holland, Amsterdam, 1977. Search in Google Scholar

[10] S. Eswara Rao, Irreducible representations of the Lie-algebra of the diffeomorphisms of a d-dimensional torus, J. Algebra 182 (1996), no. 2, 401–421. 10.1006/jabr.1996.0177Search in Google Scholar

[11] S. Eswara Rao, Partial classification of modules for Lie algebra of diffeomorphisms of d-dimensional torus, J. Math. Phys. 45 (2004), no. 8, 3322–3333. 10.1063/1.1769104Search in Google Scholar

[12] X. Guo, G. Liu, R. Lu and K. Zhao, Simple Witt modules that are finitely generated over the Cartan subalgebra, preprint (2017), https://arxiv.org/abs/1705.03393; to appear in Mosc. Math. J. 10.17323/1609-4514-2020-20-1-43-65Search in Google Scholar

[13] X. Guo, G. Liu and K. Zhao, Irreducible Harish-Chandra modules over extended Witt algebras, Ark. Mat. 52 (2014), no. 1, 99–112. 10.1007/s11512-012-0173-9Search in Google Scholar

[14] X. Guo and K. Zhao, Irreducible weight modules over Witt algebras, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2367–2373. 10.1090/S0002-9939-2010-10679-2Search in Google Scholar

[15] N. Jacobson, Basic Algebra. II, W. H. Freeman, San Francisco, 1980. Search in Google Scholar

[16] T. A. Larsson, Conformal fields: A class of representations of Vect⁢(N), Internat. J. Modern Phys. A 7 (1992), no. 26, 6493–6508. 10.1142/S0217751X92002970Search in Google Scholar

[17] T. A. Larsson, Lowest-energy representations of non-centrally extended diffeomorphism algebras, Comm. Math. Phys. 201 (1999), no. 2, 461–470. 10.1007/s002200050563Search in Google Scholar

[18] T. A. Larsson, Extended diffeomorphism algebras and trajectories in jet space, Comm. Math. Phys. 214 (2000), no. 2, 469–491. 10.1007/s002200000280Search in Google Scholar

[19] G. Liu, R. Lu and K. Zhao, Irreducible Witt modules from Weyl modules and 𝔤⁢𝔩n-modules, J. Algebra 511 (2018), 164–181. 10.1016/j.jalgebra.2018.06.021Search in Google Scholar

[20] G. Liu and K. Zhao, New irreducible weight modules over Witt algebras with infinite-dimensional weight spaces, Bull. Lond. Math. Soc. 47 (2015), no. 5, 789–795. 10.1112/blms/bdv048Search in Google Scholar

[21] X. Liu, X. Guo and Z. Wei, Irreducible modules over the divergence zero algebras and their q-analogues, preprint (2017), https://arxiv.org/abs/1709.02972. 10.1016/j.geomphys.2020.104050Search in Google Scholar

[22] R. Lü, X. Guo and K. Zhao, Irreducible modules over the Virasoro algebra, Doc. Math. 16 (2011), 709–721. 10.4171/dm/349Search in Google Scholar

[23] R. Lü, V. Mazorchuk and K. Zhao, Classification of simple weight modules over the 1-spatial ageing algebra, Algebr. Represent. Theory 18 (2015), no. 2, 381–395. 10.1007/s10468-014-9499-2Search in Google Scholar

[24] R. Lu and K. Zhao, Irreducible Virasoro modules from irreducible Weyl modules, J. Algebra 414 (2014), 271–287. 10.1016/j.jalgebra.2014.04.029Search in Google Scholar

[25] O. Mathieu, Classification of Harish-Chandra modules over the Virasoro Lie algebra, Invent. Math. 107 (1992), no. 2, 225–234. 10.1007/BF01231888Search in Google Scholar

[26] O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 537–592. 10.5802/aif.1765Search in Google Scholar

[27] V. Mazorchuk and K. Zhao, Supports of weight modules over Witt algebras, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 1, 155–170. 10.1017/S0308210509000912Search in Google Scholar

[28] V. Mazorchuk and K. Zhao, Characterization of simple highest weight modules, Canad. Math. Bull. 56 (2013), no. 3, 606–614. 10.4153/CMB-2011-199-5Search in Google Scholar

[29] D. K. Nakano, Projective modules over Lie algebras of Cartan type, Mem. Amer. Math. Soc. 98 (1992), no. 470. 10.1090/memo/0470Search in Google Scholar

[30] G. Y. Shen, Graded modules of graded Lie algebras of Cartan type. I. Mixed products of modules, Sci. Sinica Ser. A 29 (1986), no. 6, 570–581. Search in Google Scholar

[31] J. Talboom, Irreducible modules for the Lie algebra of divergence zero vector fields on a torus, Comm. Algebra 44 (2016), no. 4, 1795–1808. 10.1080/00927872.2015.1027396Search in Google Scholar

[32] H. Tan and K. Zhao, 𝒲n+- and 𝒲n-module structures on U⁢(𝔥n), J. Algebra 424 (2015), 357–375. 10.1016/j.jalgebra.2014.09.031Search in Google Scholar

[33] H. Tan and K. Zhao, Irreducible modules over Witt algebras 𝒲n and over 𝔰⁢𝔩n+1⁢(ℂ), Algebr. Represent. Theory 21 (2018), no. 4, 787–806. 10.1007/s10468-017-9738-4Search in Google Scholar

[34] J. Zhang, Non-weight representations of Cartan type S Lie algebras, Comm. Algebra 46 (2018), no. 10, 4243–4264. 10.1080/00927872.2018.1424885Search in Google Scholar

[35] K. Zhao, Weight modules over generalized Witt algebras with 1-dimensional weight spaces, Forum Math. 16 (2004), no. 5, 725–748. 10.1515/form.2004.034Search in Google Scholar

Received: 2018-04-28
Revised: 2018-10-29
Published Online: 2019-03-07
Published in Print: 2019-05-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics
  3. Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds
  4. Variable weak Hardy spaces WHLp(·)(ℝn) associated with operators satisfying Davies–Gaffney estimates
  5. Sublinear operators on weighted Hardy spaces with variable exponents
  6. Hereditarily minimal topological groups
  7. Rational torsion of generalized Jacobians of modular and Drinfeld modular curves
  8. Morita homotopy theory for (∞,1)-categories and ∞-operads
  9. Homomorphisms into totally disconnected, locally compact groups with dense image
  10. On the Ramanujan–Petersson conjecture for modular forms of half-integral weight
  11. Uniform continuity of nonautonomous superposition operators in ΛBV-spaces
  12. Simple modules over the Lie algebras of divergence zero vector fields on a torus
  13. Effective bounds for the Andrews spt-function
  14. Free subgroups in k(x1,... ,xn)(X;σ) and k(x,y)(k;σ)
  15. A geometric proof of an equivariant Pieri rule for flag manifolds
  16. On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces
  17. Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝN
https://doi.org/10.1515/forum-2018-0096
Keywords for this article
Lie algebra of divergence zero vector fields; Weyl algebra; irreducible module; (non)minuscule module; density theorem

Articles in the same Issue

  1. Frontmatter
  2. Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics
  3. Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds
  4. Variable weak Hardy spaces WHLp(·)(ℝn) associated with operators satisfying Davies–Gaffney estimates
  5. Sublinear operators on weighted Hardy spaces with variable exponents
  6. Hereditarily minimal topological groups
  7. Rational torsion of generalized Jacobians of modular and Drinfeld modular curves
  8. Morita homotopy theory for (∞,1)-categories and ∞-operads
  9. Homomorphisms into totally disconnected, locally compact groups with dense image
  10. On the Ramanujan–Petersson conjecture for modular forms of half-integral weight
  11. Uniform continuity of nonautonomous superposition operators in ΛBV-spaces
  12. Simple modules over the Lie algebras of divergence zero vector fields on a torus
  13. Effective bounds for the Andrews spt-function
  14. Free subgroups in k(x1,... ,xn)(X;σ) and k(x,y)(k;σ)
  15. A geometric proof of an equivariant Pieri rule for flag manifolds
  16. On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces
  17. Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝN
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 20.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2018-0096/html
Scroll to top button