[1] Wang Shengxiang, Zhou Jianhua,. Crossed modules of Lie color algebras [J]. Journal of Southeast University (English Edition), 2012, 28 (4): 502-504. [doi:10.3969/j.issn.1003-7985.2012.04.023]

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

28

Issue:

2012 4

Page:

502-504

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2012-12-30

Info

Title:

Crossed modules of Lie color algebras

Author(s):

Wang Shengxiang¹;  2;  Zhou Jianhua¹

¹ Department of Mathematics, Southeast University, Nanjing 211189, China
²School of Mathematics Sciences, Chuzhou University, Chuzhou 239000, China

Keywords:

crossed modules of Lie color algebras;  Witt type Lie color algebra;  third cohomology;  isomorphism

PACS:

O153.5

DOI:

10.3969/j.issn.1003-7985.2012.04.023

Abstract:

The linear operations of the equivalent classes of crossed modules of Lie color algebras are studied. The set of the equivalent classes of crossed modules is proved to be a vector space, which is isomorphic with the homogeneous components of degree zero of the third cohomology group of Lie color algebras. As an application of this theory, the crossed modules of Witt type Lie color algebras is described, and the result is proved that there is only one equivalent class of the crossed modules of Witt type Lie color algebras when the abelian group Γ is equal to Γ⁺. Finally, for a Witt type Lie color algebra, the classification of its crossed modules is obtained by the isomorphism between the third cohomology group and the crossed modules.

References:

[1] Scheunert M. Generalized Lie algebras[J]. J Math Phys, 1979, 20:712-720.
[2] Majid S. Algebras and Hopf algebras in braided categories[J]. Marcel Dekker Lecture Notes in Mathematics, 1994, 158(1):55-105.
[3] Scheunert M, Zhang N B. Cohomology of Lie superalgebras and their generalizations[J]. J Math Phys, 1998, 39(9):5024-5061.
[4] Pei F, Zhou J H. Crossed modules for Lie color algebras [J]. Journal of Southeast University: Natural Science Edition, 2007, 37(6):1137-1140.(in Chinese)
[5] Wang S X, Zhou J H. Crossed modules of Lie algebras [J]. Journal of Southeast University: Natural Science Edition, 2009, 39(1):185-190.(in Chinese)
[6] Kassel C, Loday J L. Extensions centrales d’algèbres de Lie[J]. Ann Inst Fourier, 1982, 32(4):119-142.
[7] Zhou J H. Lie color algebras of Witt type[J]. Nanjing University Mathematical Biquarterly, 2004, 21(2):219-225.

Memo

Memo:

Biographies: Wang Shengxiang(1979—), male, graduate; Zhou Jianhua(corresponding author), male, doctor, professor, jhzhou@seu.edu.cn.
Foundation items: The Natural Science Foundation of Jiangsu Province(No.BK2012736), the Natural Science Foundation of Chuzhou University(No.2010kj006Z).
Citation: Wang Shengxiang, Zhou Jianhua. Crossed modules of Lie color algebras[J].Journal of Southeast University(English Edition), 2012, 28(4):502-504.[doi:10.3969/j.issn.1003-7985.2012.04.023]

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