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\centerline {\bf Leibniz algebra}
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A {\it Leibniz algebra} over the field $K$ is a vector space $g$ equipped with a binary
operation $[-,-] : g\otimes g \to g$ satisfying the Leibniz identity
$$ [x,[y,z]]= [[x,y],z] - [[x,z],y]\ .$$
Equivalently,  the map $[-,z]$ is a derivation for any $z\in g$. If the bracket is
skew-symmetric, then the Leibniz identity is equivalent to the Jacobi identity, hence we
get a Lie algebra. So a Leibniz algebra is a noncommutative version of Lie algebra. For a Lie
algebra the universal enveloping object is an associative algebra, for a Leibniz algebra the 
 universal enveloping object is a {\it dialgebra}.
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\noindent {\bf Examples}

\noindent 0. Any Lie algebra is a Leibniz algebra.

\noindent 1. Let $(g, d)$ be a differential Lie algebra. Define a new bracket
 (called derived bracket)  on $g$ by
$[x,y]_d:=[x,dy]$. Then $[-,-]_d$ satisfies the Leibniz identity and we have defined a
Leibniz structure on $g$. Observe that it suffices to start with a differential Leibniz
algebra. Many brackets in differential geometry (Schouten-Nijenhuis bracket, Gerstenhaber
bracket, Vinogradov bracket) are constructed this way.
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\noindent {\bf Leibniz homology.} For any Leibniz algebra (e.g. Lie algebra) $g$ there is
defined a chain complex $(CL_*(g), d)$ as follows:
$CL_n(g) := g^{\otimes n}$ and the boundary map $d: g^{\otimes n}\to g^{\otimes n-1}$
is given by
$$
d(x_1,\ldots ,x_n)=\sum_{1\leq i<j\leq n}(-1)^j(x_1,\ldots ,x_{i-1},[x_i,x_j],x_{i+1},
\ldots ,\widehat x_j,\ldots ,x_n).
$$
 Its homology is denoted $HL_*(g)$ and called {\it Leibniz
homology}. Observe that the formula for $d$ has almost the same form as the boundary map
in the classical Chevalley-Eilenberg complex, except that the bracket is put at the
$i$th place instead of the first. As a consequence for a Lie algebra, there is a natural
map $HL_*(g)\to H_*(g)$.

Here are a few computation. Let $A$ be an associative algebra over a characteristic zero
field, and let $gl(A)$ denotes its Lie algebra of finite dimensional matrices. Then there
is an isomorphism (Cuvier-Loday):
$$
HL_*(gl(A)) \cong T(HH_{*-1}(A))\, ,
$$
where $T$ is the tensor module functor and $HH$ the Hochschild homology. This
isomorphism has been generalized to matrices over dialgebras by A. Frabetti.

If  $g$ be a semi-simple Lie algebra, then $HL_n(g)=0$ for $n>0$ (Ntolo, Pirashvili).
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\noindent {\bf Ariane's thread:}

\noindent J.-L. Loday,  {\it Une version non commutative des alg\`ebres de
Lie: les alg\`ebres de Leibniz}. Enseign. Math. (2) 39 (1993), no. 3-4, 269--293.
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\noindent References on Leibniz algebras : http://irmasrv1.u-strasbg.fr/~loday/refLeibniz.html
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