Abstract
Given a basis of a vector space V over a field \(\mathbb{K}\) and a multiplication table which defines a bilinear map on V, we develop a computer program on Mathematica which checks if the bilinear map satisfies the Leibniz identity, that is, if the multiplication table endows V with a Leibniz algebra structure. In case of a positive answer, the program informs whether the structure corresponds to a Lie algebra or not, that is, if the bilinear map is skew-symmetric or not.
The algorithm is based on the computation of a Gröbner basis of an ideal, which is employed in the construction of the universal enveloping algebra of a Leibniz algebra. Finally, we describe a program in the NCAlgebra package which permits the construction of Gröbner bases in non commutative algebras.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Albeverio, S., Ayupov, S.A., Omirov, B.A.: On nilpotent and simple Leibniz algebras. Comm. Algebra 33(1), 159–172 (2005)
Albeverio, S., Omirov, B.A., Rakhimov, I.S.: Varieties of nilpotent complex Leibniz algebras of dimension less than five. Comm. Algebra 33(5), 1575–1585 (2005)
Albeverio, S., Omirov, B.A., Rakhimov, I.S.: Classification of 4-dimensional nilpotent complex Leibniz algebras. Extracta Math. 21(3), 197–210 (2006)
Ayupov, S.A., Omirov, B.A.: On Leibniz algebras. Algebra and Operator Theory (Tashkent, 1997), pp. 1–12. Kluwer Acad. Publ., Dordrecht (1997)
Ayupov, S.A., Omirov, B.A.: On a description of irreducible component in the set of nilpotent Leibniz algebras containing the algebra of maximal nilindex, and classification of graded filiform Leibniz algebras. In: Computer algebra in scientific computing (Samarkand, 2000), pp. 21–34. Springer, Berlin (2000)
Ayupov, S.A., Omirov, B.A.: On some classes of nilpotent Leibniz algebras. Siberian Math. J. 42(1), 15–24 (2001)
Bergman, G.M.: The diamond lemma for ring theory. Adv. in Math. 29, 178–218 (1978)
Cuvier, C.: Algèbres de Leibnitz: définitions, propriétés. Ann. Sci. École Norm. Sup. 27(1) (4), 1–45 (1994)
de Graaf, W.: Lie algebras: theory and algorithms. North-Holland Publishing Co., Amsterdam (2000)
de Graaf, W.: Classification of solvable Lie algebras. Experiment. Math. 14, 15–25 (2005)
Goze, M., Khakimdjanov, Y.: Nilpotent Lie algebras. Mathematics and its Applications, vol. 361. Kluwer Acad. Publ., Dordrecht (1996)
Helton, J.W., Miller, R.L., Stankus, M.: NCAlgebra: A Mathematica package for doing noncommuting algebra (1996), http://math.ucsd.edu/~ncalg
Insua, M.A.: Varias perspectivas sobre las bases de Gröbner: forma normal de Smith, algoritmo de Berlekamp y álgebras de Leibniz. Ph. D. Thesis (2005)
Insua, M.A., Ladra, M.: Gröbner bases in universal enveloping algebras of Leibniz algebras. J. Symbolic Comput. 44, 517–526 (2009)
Jacobson, N.: Lie algebras. Interscience Publ., New York (1962)
Kurdiani, R.: Cohomology of Lie algebras in the tensor category of linear maps. Comm. Algebra 27(10), 5033–5048 (1999)
Loday, J.-L.: Une version non commutative des algèbres de Lie: les algèbres de Leibniz. Enseign. Math. 39(2), 269–293 (1993)
Loday, J.-L.: Algèbres ayant deux opérations associatives (digèbres). C. R. Acad. Sci. Paris Sér. I Math. 321(2), 141–146 (1995)
Loday, J.-L., Pirashvili, T.: Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann. 296, 139–158 (1993)
Mora, T.: An introduction to commutative and noncommutative Gröbner bases. Theor. Comp. Sci. 134, 131–173 (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Casas, J.M., Insua, M.A., Ladra, M., Ladra, S. (2009). Algorithm for Testing the Leibniz Algebra Structure. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds) Computer Aided Systems Theory - EUROCAST 2009. EUROCAST 2009. Lecture Notes in Computer Science, vol 5717. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04772-5_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-04772-5_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04771-8
Online ISBN: 978-3-642-04772-5
eBook Packages: Computer ScienceComputer Science (R0)