Algorithm for testing the Leibniz algebra structure

Abstract

Given a basis of a vector space V over a field 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. © 2009 Springer-Verlag Berlin Heidelberg.