Journal of Hebei University(Natural Science Edition) >

2022 , Vol. 42 >Issue 6: 561 - 564

DOI: https://doi.org/10.3969/j.issn.1000-1565.2022.06.001

First cohomology of Lie superalgebra P^~(2)with coefficients to Kac modules

  • ZHANG Zilu ,
  • SUN Liping ,
  • WANG Shujuan
Expand
  • 1. College of Science, Harbin University of Science and Technology, Harbin 150080, China; 2. College of Arts and Sciences, Shanghai Maritime University, Shanghai 201306, China

Received date: 2022-03-24

  Online published: 2023-02-22

Fold

Abstract

Over a field of characteristic p≥3,firstly,according to the Z-degree structure of P^~(2)and the standard Cartan decomposition of the 0-degree P^~(2)0,for a given weight λ and the maximal vector vλ, finite-dimensional irreducible modules M(λ)of P^~(2)0 are constructed, and then its Kac structure is given. Secondly, using the definition of the first cohomology group, the first cohomology group from P^~(2)to its Kac module is transformed into the weight-derivation of the non-inner derivation of P^~(2)to Kac module. Finally, the first cohomology group of P^~(2)with coefficients to a class of Kac modules is determined.

Key words： Kac module; derivation; cohomology; weight

Cite this article

ZHANG Zilu , SUN Liping , WANG Shujuan . First cohomology of Lie superalgebra P^~(2)with coefficients to Kac modules[J]. Journal of Hebei University(Natural Science Edition), 2022 , 42(6) : 561 -564 . DOI: 10.3969/j.issn.1000-1565.2022.06.001

References

[1] KAC V G. Representations of classical Lie superalgebras[M]. Berlin: Lecture Notes in Mathematics, 1978.
[2] KAC V G. Characters of typical representations of classical Lie superalgebras[J]. Comm Algebra, 1977, 5:889-897.DOI: 10.1080/00927877708822201.
[3] SHU B, ZHANG C W. Representations of the restricted Cartan type Lie superalgebra W(m, n, 1)[J]. Algebras and Representation Theory, 2011, 14(3):463-481. DOI: 10.1007/S10468-009-9198-6.
[4] YAO Y F. Non-restricted representations of simple Lie superalgebras of special type and Hamiltonian Type[J]. Science in China, Series A, 2013, 56(2):239-252.DOI: 10.1007/S1425-012-4456-8.
[5] WANG S J, LIU W D. The first cohomology of sl(2,1)with coefficients in χ-reduced Kac modules and simple modules[J]. Journal of Pure and Applied Algebra, 2020, 224(11):106403.DOI: 10.1016/j.jpaa.2020.106403.
[6] 邱忠旬,王淑娟,刘文德.李超代数sl_(2|1)到Kac模的一次上同调[J].数学的实践与认识,2015,45(22):253-259.
[7] SU Y U, ZHANG R B. Cohomology of Lie superalgebras sl(m|n)and osp(2|2n)[J]. Proceedings of the London Mathematical Society, 2007, 94(1):91-136. DOI: 10.1112/plms/pdL005.
[8] FARNSTEINER R. Dual space derivations and H2(L, F)of modular Lie algebras[J]. Can J Math, 1987, 39(5):1078-1106.DOI: 10.4153/CJM-1987-055-0. (
Options
Outlines

/

〈 〉