Some properties of generalized reduced Verma modules over ℤ -graded modular Lie superalgebras

Keli Zheng; Yongzheng Zhang

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 3, page 699-713
  • ISSN: 0011-4642

Abstract

top
We study some properties of generalized reduced Verma modules over ℤ -graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and coinduced modules are obtained. Moreover, invariant forms on the generalized reduced Verma modules are considered. In particular, for ℤ -graded modular Lie superalgebras of Cartan type we prove that generalized reduced Verma modules are isomorphic to mixed products of modules.

How to cite

top

Zheng, Keli, and Zhang, Yongzheng. "Some properties of generalized reduced Verma modules over $\mathbb {Z}$-graded modular Lie superalgebras." Czechoslovak Mathematical Journal 67.3 (2017): 699-713. <http://eudml.org/doc/294118>.

@article{Zheng2017,
abstract = {We study some properties of generalized reduced Verma modules over $\mathbb \{Z\}$-graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and coinduced modules are obtained. Moreover, invariant forms on the generalized reduced Verma modules are considered. In particular, for $\mathbb \{Z\}$-graded modular Lie superalgebras of Cartan type we prove that generalized reduced Verma modules are isomorphic to mixed products of modules.},
author = {Zheng, Keli, Zhang, Yongzheng},
journal = {Czechoslovak Mathematical Journal},
keywords = {modular Lie superalgebra; generalized reduced Verma module; coinduced module; invariant form; mixed product},
language = {eng},
number = {3},
pages = {699-713},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some properties of generalized reduced Verma modules over $\mathbb \{Z\}$-graded modular Lie superalgebras},
url = {http://eudml.org/doc/294118},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Zheng, Keli
AU - Zhang, Yongzheng
TI - Some properties of generalized reduced Verma modules over $\mathbb {Z}$-graded modular Lie superalgebras
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 3
SP - 699
EP - 713
AB - We study some properties of generalized reduced Verma modules over $\mathbb {Z}$-graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and coinduced modules are obtained. Moreover, invariant forms on the generalized reduced Verma modules are considered. In particular, for $\mathbb {Z}$-graded modular Lie superalgebras of Cartan type we prove that generalized reduced Verma modules are isomorphic to mixed products of modules.
LA - eng
KW - modular Lie superalgebra; generalized reduced Verma module; coinduced module; invariant form; mixed product
UR - http://eudml.org/doc/294118
ER -

References

top
  1. Bernshtein, I. N., Gel'fand, I. M., Gel'fand, S. I., 10.1007/BF01075841, Funct. Anal. Appl. 5 1-8. (1971), English. Russian original translation from Funkts. Anal. Prilozh. 5 1971 1-9. (1971) Zbl0246.17008MR0291204DOI10.1007/BF01075841
  2. Bernshtein, I. N., Gel'fand, I. M., Gel'fand, S. I., Differential operators on the base affine space and a study of 𝔤 -modules, Lie Groups and Their Representations Proc. Summer Sch. Bolyai Janos Math. Soc., Budapest, 1971; Halsted, New York (1975), 21-64. (1975) Zbl0338.58019MR0578996
  3. Cheng, Y., Su, Y., 10.1007/s11464-008-0008-y, Front. Math. China 3 (2008), 37-47. (2008) MR2373093DOI10.1007/s11464-008-0008-y
  4. Chiu, S., The invariant forms on the graded modules of the graded Cartan type Lie algebras, Chin. Ann. Math., Ser. B 13 (1992), 16-24. (1992) Zbl0754.17017MR1166870
  5. Coleman, A. J., Futorny, V. M., 10.1006/jabr.1994.1014, J. Algebra 163 (1994), 219-234. (1994) Zbl0817.17009MR1257315DOI10.1006/jabr.1994.1014
  6. Dixmier, J., Algèbres Enveloppantes, Gauthier-Villars, Paris (1974), French. (1974) Zbl0308.17007MR0498737
  7. Farnsteiner, R., 10.1007/BF01444560, Math. Ann. 288 (1990), 713-730. (1990) Zbl0731.17007MR1081272DOI10.1007/BF01444560
  8. Farnsteiner, R., Strade, H., 10.1007/BF02571333, Math. Z. 206 (1991), 153-168. (1991) Zbl0727.17010MR1086821DOI10.1007/BF02571333
  9. Futorny, V., Mazorchuk, V., 10.1006/jabr.1996.0229, J. Algebra 183 (1996), 456-482. (1996) Zbl0869.17006MR1399036DOI10.1006/jabr.1996.0229
  10. Kac, V. G., 10.1016/0001-8708(77)90017-2, Adv. Math. 26 (1977), 8-96. (1977) Zbl0366.17012MR0486011DOI10.1016/0001-8708(77)90017-2
  11. Khomenko, A., Mazorchuk, V., 10.1006/jabr.1998.7731, J. Algebra 215 (1999), 318-329. (1999) Zbl0931.17004MR1684154DOI10.1006/jabr.1998.7731
  12. Khomenko, O., Mazorchuk, V., 10.1006/jabr.2001.8815, J. Algebra 242 (2001), 561-576. (2001) Zbl0982.17003MR1848960DOI10.1006/jabr.2001.8815
  13. Mazorchuk, V., 10.1007/BF02567805, Manuscr. Math. 88 (1995), 59-72. (1995) Zbl0855.17003MR1348790DOI10.1007/BF02567805
  14. Mazorchuk, V. S., Ovsienko, S. A., 10.1023/A:1009929615175, Algebr. Represent. Theory 1 (1998), 3-26. (1998) Zbl0908.17004MR1654598DOI10.1023/A:1009929615175
  15. Ross, L. E., 10.2307/1994163, Trans. Am. Math. Soc. 120 (1965), 17-23. (1965) Zbl0145.25903MR0185043DOI10.2307/1994163
  16. Scheunert, M., 10.1007/bfb0070929, Lecture Notes in Mathematics 716, Springer, Berlin (1979). (1979) Zbl0407.17001MR0537441DOI10.1007/bfb0070929
  17. Shen, G., Graded modules of graded Lie algebras of Cartan type. I: Mixed products of modules, Sci. Sin., Ser. A 29 (1986), 570-581. (1986) Zbl0601.17013MR0862418
  18. Shen, G., Graded modules of graded Lie algebras of Cartan type. II: Positive and negative graded modules, Sci. Sin., Ser. A 29 (1986), 1009-1019. (1986) Zbl0615.17009MR0877285
  19. Shen, G., Graded modules of graded Lie algebras of Cartan type. III: Irreducible modules, Chin. Ann. Math., Ser. B 9 (1988), 404-417. (1988) Zbl0689.17011MR0998646
  20. Verma, D.-N., 10.1090/S0002-9904-1968-11921-4, Bull. Am. Math. Soc. 74 (1968), 160-166 errata ibid. 74 1968 628. (1968) Zbl0157.07604MR0218417DOI10.1090/S0002-9904-1968-11921-4
  21. Wang, Y., Zhang, Y., 10.1081/AGB-200029706, Commun. Algebra 32 (2004), 4117-4131. (2004) Zbl1084.17009MR2097451DOI10.1081/AGB-200029706
  22. Zhang, Y., Graded modules of the Cartan-type 𝐙 -graded Lie superalgebras W ( n ) and S ( n ) , Kexue Tongbao 40 (1995), 1829-1832 Chinese. (1995) MR1396205
  23. Zhang, Y., ℤ -graded module of Lie superalgebra H ( n ) of Cartan type, Chin. Sci. Bull. 41 (1996), 813-817. (1996) Zbl0859.17018MR1403021
  24. Zhang, Y., 10.1007/BF03186962, Chin. Sci. Bull. 42 (1997), 720-724. (1997) Zbl0886.17022MR1460613DOI10.1007/BF03186962
  25. Zhang, Y., Mixed products of modules of infinite-dimensional Lie superalgebras of Cartan type, Chin. Ann. Math., Ser. A 18 (1997), 725-732. (1997) Zbl0919.17004MR1620790
  26. Zhang, Y., Fu, H., 10.1081/AGB-120003981, Commun. Algebra 30 (2002), 2651-2673. (2002) Zbl1021.17017MR1908231DOI10.1081/AGB-120003981

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.