BIPARTITE GRAPHS AND THE STRUCTURE OF FINITE-DIMENSIONAL SEMISIMPLE LEIBNIZ ALGEBRAS

Year 2019, Volume: 26 Issue: 26, 122 - 130, 11.07.2019

Rustam Turdibaev

https://doi.org/10.24330/ieja.587009

Cited By: 1

Abstract

Given a nite connected bipartite graph, finite-dimensional indecomposable

semisimple Leibniz algebras are constructed. Furthermore, any

finite-dimensional indecomposable semisimple Leibniz algebra admits a similar

construction.

Keywords

Semisimple Leibniz algebra, connected bipartite graph

References

• Sh. Ayupov, K. Kudaybergenov, B. Omirov and K. Zhao, Semisimple Leibniz algebras, their derivations and automorphisms, Linear Multilinear Algebra, (2019), accepted.
• D. W. Barnes, On Levi's theorem for Leibniz algebras, Bull. Aust. Math. Soc., 86(2) (2012), 184-185.
• A. Bloh, On a generalization of the concept of Lie algebra, Dokl. Akad. Nauk SSSR, 165 (1965), 471-473.
• A. Ja. Bloh, A certain generalization of the concept of Lie algebra, Moskov. Gos. Ped. Inst. Ucen. Zap., 375 (1971), 9-20 (in Russian).
• A. S. Dzhumadil'daev and S. A. Abdykassymova, Leibniz algebras in characteristic p, C. R. Acad. Sci. Paris Ser. I Math., 332(12) (2001), 1047-1052.
• N. Jacobson, Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers, New York-London, 1962.
• M. K. Kinyon and A.Weinstein, Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces, Amer. J. Math., 123(3) (2001), 525-550.
• K. Kudaybergenov, M. Ladra and B. Omirov, On Levi-Malcev theorem for Leibniz algebras, Linear Multilinear Algebra, 67(7) (2019), 1471-1482.
• J.-L. Loday, Cyclic Homology, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 301, Springer- Verlag, Berlin, 1992.
• J.-L. Loday, Une version non commutative des algebres de Lie: les algebres de Leibniz [A noncommutative version of Lie algebras: the Leibniz algebras], Enseign. Math., 39(2) (1993), 269-293.
• G. Mason and G. Yamskulna, Leibniz algebras and Lie algebras, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), 063 (10 pp).
• T. Pirashvili, On Leibniz homology, Ann. Inst. Fourier (Grenoble), 44(2) (1994), 401-411.
• Z. X. Wan, Lie Algebras, International Series of Monographs in Pure and Applied Mathematics, Vol. 104, Pergamon Press, Oxford-New York-Toronto, Ont., 1975.