Abstract
Let W(n) be the Witt Lie superalgebra over an algebraically closed field k of characteristic p>2. We study the representations of W(n),n≥3 with p-character of height one. We prove that the Kac module always has a unique simple quotient and all simple modules are given in this way. In fact, most Kac modules are simple. We will determine the sufficient and necessary conditions such that the Kac module is simple. Moreover, composition factors of each Kac module can be determined. It is proved that the number of composition factors in each Kac module is at most 2.
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References
Cheng, S.-J., Wang, W.: Dualities and representations of Lie superalgebras, volume 144 of Graduate Studies in Mathematics American Mathematical Society. RI, Providence (2012)
Duan, F., Shu, B.: Representations of the Witt superalgebra W(2). J. Algebra 396, 272–286 (2013)
Gordon, R., Green, E.L.: Graded artin algebras, vol. 76 (1982)
Jantzen, J.C.: Representations of Lie algebras in prime characteristic. In: Representation theories and algebraic geometry (Montreal, PQ, 1997), volume 514 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 185–235. Kluwer Acad. Publ., Dordrecht. Notes by Iain Gordon (1998)
Jantzen, J.C.: Representations of Lie algebras in positive characteristic. In: Representation theory of algebraic groups and quantum groups, volume 40 of Adv. Stud. Pure Math., pages 175–218. Math. Soc. Japan, Tokyo (2004)
Kac, V.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)
Musson, I.M.: Lie superalgebras and enveloping algebras Volume 131 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2012)
Premet, A.: Irreducible representations of lie algebras of reductive groups and the Kac-Weisfeiler conjecture. Invent. Math. 121, 79–117 (1995)
Serganova, V.: On representations of Cartan type Lie superalgebras. In: Lie groups and invariant theory, volume 213 of Amer. Math. Soc. Transl. Ser. 2, pages 223–239. Amer. Math. Soc., Providence, RI (2005)
Shu, B., Zhang, C.: Restricted representations of the Witt superalgebras. J. Algebra 324(4), 652–672 (2010)
Wang, W., Zhao, L.: Representations of Lie superalgebras in prime characteristic I. P. Lond Math Soc 99, 145–167 (2009)
Wang, W., Zhao, L.: Representations of lie superalgebras in prime characteristic II The queer series. J. Pure Appl. Algebra 215(10), 2515–2532 (2011)
Yao, Y.F.: Non-restricted representations of simple Lie superalgebras of special type and Hamiltonian type. Sci. China Math. 56(2), 239–252 (2013)
Zhao, L.: Representations of Lie superalgebras in prime characteristic, III. Pacific J. Math 248(2), 493–510 (2010)
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Presented by Peter Littelmann.
The work is supported by the National Natural Science Foundation of China (No.11271318, No.11171296 and No. 11401522) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No.20110101110010) and the Zhejiang Provincial Natural Science Foundation of China (No.LZ13A010001)
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Duan, F. Representations of the Witt Lie Superalgebra with p-character of Height One. Algebr Represent Theor 19, 873–893 (2016). https://doi.org/10.1007/s10468-016-9602-y
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DOI: https://doi.org/10.1007/s10468-016-9602-y