Leibniz algebra

An algebra over a field $K$ 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