Enveloping Lie Algebras of Low Dimensional Leibniz Algebras
ABSTRACT
We calculate the enveloping Lie algebras of Leibniz algebras of dimensions two and three. We show how these Lie algebras could be used to distinguish non-isomorphic (nilpotent) Leibniz algebras of low dimension in some cases. These results could be used to associate geometric objects (loop spaces) to low dimensional Leibniz algebras.
We calculate the enveloping Lie algebras of Leibniz algebras of dimensions two and three. We show how these Lie algebras could be used to distinguish non-isomorphic (nilpotent) Leibniz algebras of low dimension in some cases. These results could be used to associate geometric objects (loop spaces) to low dimensional Leibniz algebras.
Cite this paper
nullM. Amini, I. Rakhimov and S. Langari, "Enveloping Lie Algebras of Low Dimensional Leibniz Algebras," Applied Mathematics, Vol. 2 No. 8, 2011, pp. 1027-1030. doi: 10.4236/am.2011.28142.
nullM. Amini, I. Rakhimov and S. Langari, "Enveloping Lie Algebras of Low Dimensional Leibniz Algebras," Applied Mathematics, Vol. 2 No. 8, 2011, pp. 1027-1030. doi: 10.4236/am.2011.28142.
References
[1] J. L. Loday, “Une version non-commutative des algebras de Lie, Les algebres de Leibniz,” Mathematics at Ecole Normale Supérieure, Vol. 39, 1993, pp. 269-293. [2] R. E. Beck and B. Kolman, “Constructions of Nilpotent Lie Algebras over Arbitrary Fields,” In: P. S. Wang, Ed., Proceedings of 1981 ACM Symposium on Symbolic and Algebraic Computation, New York, 1981, pp. 169-174. [3] M. K. Kinyon and A. Weinestein, “Leibniz Algebras, Courant Algebroids, and Multiplications on Homogeneous Spaces,” American Journal of Mathematics, Vol. 123, No. 3, 2001, pp. 525-550. doi:10.1353/ajm.2001.0017 [4] S. Albeverio, B. A. Omirov and I. S. Rakhimov, “Varieties of Nilpotent Complex Leibniz Algebras of Dimension Less Than Five,” Communications in Algebra, Vol. 33, No. 5, 2005, pp. 1575-1585. doi:10.1081/AGB-200061038
[1] J. L. Loday, “Une version non-commutative des algebras de Lie, Les algebres de Leibniz,” Mathematics at Ecole Normale Supérieure, Vol. 39, 1993, pp. 269-293. [2] R. E. Beck and B. Kolman, “Constructions of Nilpotent Lie Algebras over Arbitrary Fields,” In: P. S. Wang, Ed., Proceedings of 1981 ACM Symposium on Symbolic and Algebraic Computation, New York, 1981, pp. 169-174. [3] M. K. Kinyon and A. Weinestein, “Leibniz Algebras, Courant Algebroids, and Multiplications on Homogeneous Spaces,” American Journal of Mathematics, Vol. 123, No. 3, 2001, pp. 525-550. doi:10.1353/ajm.2001.0017 [4] S. Albeverio, B. A. Omirov and I. S. Rakhimov, “Varieties of Nilpotent Complex Leibniz Algebras of Dimension Less Than Five,” Communications in Algebra, Vol. 33, No. 5, 2005, pp. 1575-1585. doi:10.1081/AGB-200061038