On representations of restricted Lie superalgebras

Yu-Feng Yao

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 3, page 845-856
  • ISSN: 0011-4642

Abstract

top
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 χ ( λ ) are proved to be simple for any regular nilpotent p -character χ and typical weight λ . Moreover, we obtain the dimension formulas for projective covers of simple modules with p -characters of standard Levi form.

How to cite

top

Yao, Yu-Feng. "On representations of restricted Lie superalgebras." Czechoslovak Mathematical Journal 64.3 (2014): 845-856. <http://eudml.org/doc/262191>.

@article{Yao2014,
abstract = {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.},
author = {Yao, Yu-Feng},
journal = {Czechoslovak Mathematical Journal},
keywords = {restricted Lie superalgebra; $\chi $-reduced representation; indecomposable module; simple module; $p$-character; restricted Lie superalgebra; $\chi $-reduced representation; indecomposable module; simple module; -character},
language = {eng},
number = {3},
pages = {845-856},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On representations of restricted Lie superalgebras},
url = {http://eudml.org/doc/262191},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Yao, Yu-Feng
TI - On representations of restricted Lie superalgebras
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 3
SP - 845
EP - 856
AB - 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.
LA - eng
KW - restricted Lie superalgebra; $\chi $-reduced representation; indecomposable module; simple module; $p$-character; restricted Lie superalgebra; $\chi $-reduced representation; indecomposable module; simple module; -character
UR - http://eudml.org/doc/262191
ER -

References

top
  1. Brundan, J., 10.2140/pjm.2006.224.65, Pac. J. Math. 224 65-90 (2006). (2006) Zbl1122.20022MR2231652DOI10.2140/pjm.2006.224.65
  2. Brundan, J., Kleshchev, A., 10.1016/S0021-8693(02)00620-8, J. Algebra 260 64-98 (2003). (2003) Zbl1027.17004MR1973576DOI10.1016/S0021-8693(02)00620-8
  3. Brundan, J., Kleshchev, A., 10.1007/s002090100282, Math. Z. 239 27-68 (2002). (2002) Zbl1029.20008MR1879328DOI10.1007/s002090100282
  4. Brundan, J., Kujawa, J., 10.1023/A:1025113308552, J. Algebr. Comb. 18 13-39 (2003). (2003) Zbl1043.20006MR2002217DOI10.1023/A:1025113308552
  5. Friedlander, E. M., Parshall, B. J., 10.2307/2374686, Am. J. Math. 110 1055-1093 (1988). (1988) Zbl0673.17010MR0970120DOI10.2307/2374686
  6. Jacobson, N., Lie Algebras, Interscience Tracts in Pure and Applied Mathematics 10 Interscience Publishers (a division of John Wiley & Sons), New York (1962). (1962) Zbl0121.27504MR0143793
  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). (1998) Zbl0974.17022MR1649627
  8. Kac, V. G., 10.1016/0001-8708(77)90017-2, Adv. Math. 26 8-96 (1977). (1977) Zbl0367.17007MR0486011DOI10.1016/0001-8708(77)90017-2
  9. Kujawa, J., The Steinberg tensor product theorem for G L ( 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). (2006) Zbl1127.20032MR2263092
  10. Petrogradski, V. M., 10.1016/0021-8693(92)90173-J, J. Algebra 145 1-21 (1992). (1992) MR1144655DOI10.1016/0021-8693(92)90173-J
  11. Shu, B., Wang, W., 10.1112/plms/pdm040, Proc. Lond. Math. Soc. (3) 96 251-271 (2008). (2008) Zbl1219.20031MR2392322DOI10.1112/plms/pdm040
  12. Shu, B., Yao, Y.-F., 10.1002/mana.201000064, Math. Nachr. 285 1107-1116 (2012). (2012) Zbl1293.17022MR2928401DOI10.1002/mana.201000064
  13. Shu, B., Zhang, C., 10.1007/s10468-009-9198-6, Algebr. Represent. Theory 14 463-481 (2011). (2011) Zbl1281.17020MR2785918DOI10.1007/s10468-009-9198-6
  14. Shu, B., Zhang, C., 10.1016/j.jalgebra.2010.04.032, J. Algebra 324 652-672 (2010). (2010) Zbl1217.17010MR2651562DOI10.1016/j.jalgebra.2010.04.032
  15. Wang, W., Zhao, L., 10.1112/plms/pdn057, Proc. Lond. Math. Soc. 99 145-167 (2009). (2009) Zbl1176.17013MR2520353DOI10.1112/plms/pdn057
  16. Wang, W., Zhao, L., 10.1016/j.jpaa.2011.02.011, The queer series. J. Pure Appl. Algebra 215 2515-2532 (2011). (2011) Zbl1225.17021MR2793955DOI10.1016/j.jpaa.2011.02.011
  17. Yao, Y.-F., 10.1007/s11425-012-4486-8, Sci. China, Math. 56 239-252 (2013). (2013) Zbl1275.17034MR3015372DOI10.1007/s11425-012-4486-8
  18. Yao, Y.-F., 10.1007/s00605-012-0414-9, Monatsh. Math. 170 239-255 (2013). (2013) MR3041742DOI10.1007/s00605-012-0414-9
  19. Yao, Y.-F., Shu, B., 10.1007/s10468-011-9322-2, Algebr. Represent. Theory 16 615-632 (2013). (2013) MR3049662DOI10.1007/s10468-011-9322-2
  20. Zhang, C., 10.1016/j.jpaa.2008.09.005, J. Pure Appl. Algebra 213 756-765 (2009). (2009) MR2494368DOI10.1016/j.jpaa.2008.09.005
  21. Zhang, Y.-Z., Liu, W.-D., Modular Lie Superalgebras, Scientific Press Beijing, 2001 Chinese. Zbl1163.17019

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.