Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
restricted Lie superalgebra; $\chi $-reduced representation; indecomposable module; simple module; $p$-character
restricted Lie superalgebra; $\chi $-reduced representation; indecomposable module; simple module; $p$-character
Summary:
Simple modules for restricted Lie superalgebras are studied. The indecomposability of baby Kac modules and baby Verma modules is proved in some situation. In particular, for the classical Lie superalgebra of type $A(n|0)$, the baby Verma modules $Z_{\chi }(\lambda )$ are proved to be simple for any regular nilpotent $p$-character $\chi $ and typical weight $\lambda $. Moreover, we obtain the dimension formulas for projective covers of simple modules with $p$-characters of standard Levi form.
References:
[1] Brundan, J.: Modular representations of the supergroup $Q(n)$. II. Pac. J. Math. 224 65-90 (2006). DOI 10.2140/pjm.2006.224.65 | MR 2231652 | Zbl 1122.20022
[2] Brundan, J., Kleshchev, A.: Modular representations of the supergroup $Q(n)$. I. J. Algebra 260 64-98 (2003). DOI 10.1016/S0021-8693(02)00620-8 | MR 1973576 | Zbl 1027.17004
[3] Brundan, J., Kleshchev, A.: Projective representations of symmetric groups via Sergeev duality. Math. Z. 239 27-68 (2002). DOI 10.1007/s002090100282 | MR 1879328 | Zbl 1029.20008
[4] Brundan, J., Kujawa, J.: A new proof of the Mullineux conjecture. J. Algebr. Comb. 18 13-39 (2003). DOI 10.1023/A:1025113308552 | MR 2002217 | Zbl 1043.20006
[5] Friedlander, E. M., Parshall, B. J.: Modular representation theory of Lie algebras. Am. J. Math. 110 1055-1093 (1988). DOI 10.2307/2374686 | MR 0970120 | Zbl 0673.17010
[6] Jacobson, N.: Lie Algebras. Interscience Tracts in Pure and Applied Mathematics 10 Interscience Publishers (a division of John Wiley & Sons), New York (1962). MR 0143793 | Zbl 0121.27504
[7] Jantzen, J. C.: Representations of Lie algebras in prime characteristic. (Notes by Ian Gordon). A. Broer et al. Representation Theories and Algebraic Geometry NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514 Kluwer Academic Publishers, Dordrecht 185-235 (1998). MR 1649627 | Zbl 0974.17022
[8] Kac, V. G.: Lie superalgebras. Adv. Math. 26 8-96 (1977). DOI 10.1016/0001-8708(77)90017-2 | MR 0486011 | Zbl 0367.17007
[9] Kujawa, J.: The Steinberg tensor product theorem for $ {GL}(m|n)$. G. Benkart et al. Representations of Algebraic Groups, Quantum Groups, and Lie Algebras. AMS-IMS-SIAM Joint Summer Research Conference, Snowbird, USA, 2004 Contemp. Math. 413 AMS, Providence 123-132 (2006). MR 2263092 | Zbl 1127.20032
[10] Petrogradski, V. M.: Identities in the enveloping algebras for modular Lie superalgebras. J. Algebra 145 1-21 (1992). DOI 10.1016/0021-8693(92)90173-J | MR 1144655
[11] Shu, B., Wang, W.: Modular representations of the ortho-symplectic supergroups. Proc. Lond. Math. Soc. (3) 96 251-271 (2008). DOI 10.1112/plms/pdm040 | MR 2392322 | Zbl 1219.20031
[12] Shu, B., Yao, Y.-F.: Character formulas for restricted simple modules of the special superalgebras. Math. Nachr. 285 1107-1116 (2012). DOI 10.1002/mana.201000064 | MR 2928401 | Zbl 1293.17022
[13] Shu, B., Zhang, C.: Representations of the restricted Cartan type Lie superalgebra $W(m,n,1)$. Algebr. Represent. Theory 14 463-481 (2011). DOI 10.1007/s10468-009-9198-6 | MR 2785918 | Zbl 1281.17020
[14] Shu, B., Zhang, C.: Restricted representations of the Witt superalgebras. J. Algebra 324 652-672 (2010). DOI 10.1016/j.jalgebra.2010.04.032 | MR 2651562 | Zbl 1217.17010
[15] Wang, W., Zhao, L.: Representations of Lie superalgebras in prime characteristic. I. Proc. Lond. Math. Soc. 99 145-167 (2009). DOI 10.1112/plms/pdn057 | MR 2520353 | Zbl 1176.17013
[16] Wang, W., Zhao, L.: Representations of Lie superalgebras in prime characteristic. II. The queer series. J. Pure Appl. Algebra 215 2515-2532 (2011). DOI 10.1016/j.jpaa.2011.02.011 | MR 2793955 | Zbl 1225.17021
[17] Yao, Y.-F.: Non-restricted representations of simple Lie superalgebras of special type and Hamiltonian type. Sci. China, Math. 56 239-252 (2013). DOI 10.1007/s11425-012-4486-8 | MR 3015372 | Zbl 1275.17034
[18] Yao, Y.-F.: On restricted representations of the extended special type Lie superalgebra ${\bar{S}(m, n, 1)}$. Monatsh. Math. 170 239-255 (2013). DOI 10.1007/s00605-012-0414-9 | MR 3041742
[19] Yao, Y.-F., Shu, B.: Restricted representations of Lie superalgebras of Hamiltonian type. Algebr. Represent. Theory 16 615-632 (2013). DOI 10.1007/s10468-011-9322-2 | MR 3049662
[20] Zhang, C.: On the simple modules for the restricted Lie superalgebra $\mathop sl(n|1)$. J. Pure Appl. Algebra 213 756-765 (2009). DOI 10.1016/j.jpaa.2008.09.005 | MR 2494368
[21] Zhang, Y.-Z., Liu, W.-D.: Modular Lie Superalgebras. Scientific Press Beijing, 2001 Chinese. Zbl 1163.17019
Search
Partner of
