View source for 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}}

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