Leibniz algebra
An algebra over a field generalising the properties of a Lie algebra. A Leibniz algebra L is a K-algebra with multiplication denoted by [\cdot,\cdot] satisfying [ x, [y,z]] = [[x,y],z] - [[x,z],y] \ . Every Lie algebra is a Leibniz algebra, and a Leibniz algebra is a Lie algebra if in addition [x,x] = 0.
The free Leibniz algebra on a generating set X may be defined as the quotient of the free non-associative algebra over K (cf. Free algebra over a ring) by the ideal generated by all elements of the form [ x, [y,z]] - [[x,y],z] + [[x,z],y] . The standard Leibniz algebra on X is obtained from the vector space V = KX and forming the tensor module T(X) = V \oplus V^{{}\otimes 2} \oplus \cdots \oplus V^{{}\otimes n} \oplus \cdots with the multiplication [x,v] = x \otimes v when v \in V and [x, y\otimes v] = [x,y] \otimes v - [x \otimes v, y] \ . The standard algebra is then a presentation of the free algebra on X.
See also: Leibniz–Hopf algebra, Non-associative rings and algebras.
References
- Mikhalev, Alexander A.; Shpilrain, Vladimir; Yu, Jie-Tai, Combinatorial methods. Free groups, polynomials, and free algebras, CMS Books in Mathematics 19 Springer (2004) ISBN 0-387-40562-3 Zbl 1039.16024
Leibniz algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leibniz_algebra&oldid=54550