Taiwanese Journal of Mathematics

Character Formulas for Simple Modules of Hamiltonian Lie Superalgebras of Odd Type

Wende Liu, Jixia Yuan, and Shujuan Wang

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Abstract

In this paper, character formulas are explicitly characterized for all simple restricted modules of Hamiltonian Lie superalgebras of odd type over an algebraically closed field of characteristic $p \gt 3$. In the process we use the lengths and highest weights of simple quotients of restricted Kac modules of atypical weights with respect to a series of Borel subalgebras to determine the composition factors, composition series and the character formulas for the restricted Kac modules of atypical weights for the Lie superalgebras under consideration.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 23 pages.

Dates
First available in Project Euclid: 30 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1543546838

Digital Object Identifier
doi:10.11650/tjm/181111

Subjects
Primary: 17B10: Representations, algebraic theory (weights) 17B50: Modular Lie (super)algebras 17B66: Lie algebras of vector fields and related (super) algebras

Keywords
Hamiltonian Lie superalgebras of odd type restricted Kac modules atypical weights character formulas

Citation

Liu, Wende; Yuan, Jixia; Wang, Shujuan. Character Formulas for Simple Modules of Hamiltonian Lie Superalgebras of Odd Type. Taiwanese J. Math., advance publication, 30 November 2018. doi:10.11650/tjm/181111. https://projecteuclid.org/euclid.twjm/1543546838


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References

  • W. Bai and W. Liu, Superderivations for modular graded Lie superalgebras of Cartan-type, Algebr. Represent. Theory 17 (2014), no. 1, 69–86.
  • J. E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category $\mathscr{O}$, Graduate Studies in Mathematics 94, American Mathematical Society, Providence, RI, 2008.
  • J. C. Jantzen, Representations of Lie algebras in prime characteristic, in: Representation Theories and Algebraic Geometry (Montreal, PQ, 1997), 185–235, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, Kluwer Acad., Dordrecht, 1998.
  • V. G. Kac, Classification of infinite-dimensional simple linearly compact Lie superalgebras, Adv. Math. 139 (1998), no. 1, 1–55.
  • A. Lebedev, Analogs of the orthogonal, Hamiltonian, Possion, and contact Lie superalgebras in characteristic 2, J. Nonlinear Math. Phys. 17 (2010), suppl. 1, 217–251.
  • D. Leites, J. Bernstein, S. Bouarroudj, B. Clarke, P. Grozman, A. Lebedev, D. Leites and I. Shchepochkina, Representation Theory, (Vol. 2. Nonholonomic Distributions in Quest for Simple Modular Lie Superalgebras), A. Salam School of Mathematical Sciences, Lahore, (2009)
  • W.-Liu and Y. He, Finite-dimensional special odd Hamiltonian superalgebras in prime characteristic, Commun. Contemp. Math. 11 (2009), no. 4, 523–546.
  • W. Liu and Y. Zhang, Finite-dimensional odd Hamiltonian superalgebras over a field of prime characteristic, J. Aust. Math. Soc. 79 (2005), no. 1, 113–130.
  • V. Serganova, On representations of Cartan type Lie superalgebras, in: Lie Groups and Invariant Theory, 223–239, Amer. Math. Soc. Transl. Ser. 2 213, Adv. Math. Sci. 56, Amer. Math. Soc., Providence, RI, 2005.
  • B. Shu and Y.-F. Yao, Character formulas for restricted simple modules of the special superalgebras, Math. Nachr. 285 (2012), no. 8-9, 1107–1116.
  • B. Shu and C. Zhang, Restricted representations of the Witt superalgebras, J. Algebra 324 (2010), no. 4, 652–672.
  • ––––, Representations of the restricted Cartan type Lie superalgebra $W(m,n,1)$, Algebr. Represent. Theory 14 (2011), no. 3, 463–481.
  • Y.-F. Yao, On restricted representations of the extended special type Lie superalgebra $\overline{S}(m,n,1)$, Monatsh. Math. 170 (2013), no. 2, 239–255.
  • ––––, Non-restricted representations of simple Lie superalgebras of special type and Hamiltonian type, Sci. China Math. 56 (2013), no. 2, 239–252.
  • Y.-F. Yao and B. Shu, Restricted representations of Lie superalgebras of Hamiltonian type, Algebr. Represent. Theory, 16 (2013), no. 3, 615–632.
  • J. Yuan and W. Liu, Restricted Kac modules of Hamiltonian Lie superalgebras of odd type, Monatsh. Math. 178 (2015), no. 3, 473–488.
  • Y. Zhang, Finite-dimensional Lie superalgebras of Cartan type over fields of prime characteristic, Chinese Sci. Bull. 42 (1997), no. 9, 720–724.