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\section*{Leibniz algebra}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{relation_to_lie_algebras_in_lodaypirashvili_category}{Relation to Lie algebras in Loday-Pirashvili category}\dotfill \pageref*{relation_to_lie_algebras_in_lodaypirashvili_category} \linebreak
\noindent\hyperlink{terminology}{Terminology}\dotfill \pageref*{terminology} \linebreak
\noindent\hyperlink{corepresentation_representation_crossed_module}{Corepresentation, representation, crossed module}\dotfill \pageref*{corepresentation_representation_crossed_module} \linebreak
\noindent\hyperlink{abelian_extensions}{Abelian extensions}\dotfill \pageref*{abelian_extensions} \linebreak
\noindent\hyperlink{homology_and_cohomology}{Homology and cohomology}\dotfill \pageref*{homology_and_cohomology} \linebreak
\noindent\hyperlink{the_related_kinds_of_algebras}{The related kinds of algebras}\dotfill \pageref*{the_related_kinds_of_algebras} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


\hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation}

Sometimes in the place where we expect [[Lie algebra]]s, some noncommutative phenomena occur and we need to drop out the requirement of antisymmetry of the brackets.

[[Jean-Louis Loday]] introduced Leibniz algebras, because of considerations in [[algebraic K-theory]]. Roughly speaking the [[Lie algebra homology]] is related to the appearance of cyclic homology (as it is manifest in the original work of Tsygan and then of Loday-Quillen).

\begin{itemize}%
\item [[Jean-Louis Loday]], [[Daniel Quillen]], \emph{Cyclic homology and the Lie algebra homology of matrices}, Comm. Math. Helv. \textbf{59}, n. 1, 565-591 (1984), \href{http://dx.doi.org/10.1007/BF02566367}{doi}

\end{itemize}
Lie algebra homology involves the [[Chevalley-Eilenberg chain complex]], which in turns involves the exterior powers of the Lie algebra. Loday found that there is a noncommutative generalization where roughly speaking one has the tensor and not the exterior powers of the Lie algebra in the complex; this new complex defines the \textbf{Leibniz homology} of Lie algebras. The Leibniz homology is related to the [[Hochschild homology]] the same way the Lie algebra homology is related to the cyclic homology.

\begin{itemize}%
\item C. Cuvier, \emph{Homologie de Leibniz et homologie de Hochschild}, C.R. Acad. Sci. Paris, Ser. A-B313, 569-572 (1991)

\end{itemize}
In fact this new complex for Leibniz homology further generalizes to the case of Leibniz algebras, where it computes certain [[Tor]] groups for corepresentations of Leibniz algebras.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Given a [[commutative unital ring]] $k$ (usually a [[field]]), a Lebniz $k$-algebra $A$ is a particular kind of [[nonassociative algebra]] over $k$ which is somewhat more general than a [[Lie algebra]] over $k$.

A \textbf{left Leibniz $k$-algebra} is $k$-[[module]] $L$ equipped with a bracket, which is a $k$-linear map $[,]:A\otimes A \to A$ satisfying the \textbf{left Leibniz identity}

\begin{displaymath}
[a, [b,c]] = [[a,b],c]+[b,[a,c]]
\end{displaymath}
In other words, the left $ad$-map, $a \mapsto (ad_l a = [a,-]:L\to L)$ is a [[derivation]] of $L$ as a nonassociative algebra. Similarly, there are right Leibniz algebras, for which the right $ad$-map $ad_r :a\mapsto [-,a]:L\to L$ is a derivation. In the presence of antisymmetry, the left Leibniz identity is equivalent to the [[Jacobi identity]], though this is not true in general; thus a Lie algebra is precisely an antisymmetric (or alternating) Leibniz algebra.

\hypertarget{relation_to_lie_algebras_in_lodaypirashvili_category}{}\subsection*{{Relation to Lie algebras in Loday-Pirashvili category}}\label{relation_to_lie_algebras_in_lodaypirashvili_category}

There is a remarkable observation of Loday and Pirashvili that in the [[Loday–Pirashvili tensor category]] of linear maps with (exotic) ``infinitesimal tensor product'', the category of internal Lie algebras has the category of, say left, Leibniz $k$-algebras as a full subcategory.

\hypertarget{terminology}{}\subsection*{{Terminology}}\label{terminology}

Some people dislike the term (left/right) Leibniz algebra (which is allegedly due to Loday), and prefer other names, including `Loday algebras' and many longer descriptive names.

\hypertarget{corepresentation_representation_crossed_module}{}\subsection*{{Corepresentation, representation, crossed module}}\label{corepresentation_representation_crossed_module}

Both a representation and a corepresentation of a right Leibniz $k$-algebra $\mathfrak{g}$ involve a $k$-module $M$ and two $k$-linear maps ``actions'' $M\otimes\mathfrak{g}\to M$ and $\mathfrak{g}\otimes M\to M$ with 3 axioms.

For representations:

\begin{displaymath}
[m, [x, y]] = [[m, x], y] - [[m, y], x]
\end{displaymath}
\begin{displaymath}
[x, [a, y]] = [[x, m], y] - [[x, y], m]
\end{displaymath}
\begin{displaymath}
[x, [y, m]] = [[x, y], m] - [[x, m], y]
\end{displaymath}
for $x,y\in\mathfrak{g}$ and for $m\in M$.

For corepresentatons:

\begin{displaymath}
[[x, y], m] = [x, [y, m]] - [y, [x, m]]
\end{displaymath}
\begin{displaymath}
[y, [a, x]] = [[y, m], x] - [m, [x, y]]
\end{displaymath}
\begin{displaymath}
[[m, x], y] = [m, [x, y]] - [[y, m], x].
\end{displaymath}
If the two ``actions'' are symmetric, i.e. $[x,m] + [m,x] = 0$ for all $m\in M$, $x\in\mathfrak{g}$ then all the 6 axioms of representation or corepresentation are equivalent. If $M$ is underlying a Leibniz algebra then an action of $\mathfrak{g}$ on $M$ is by definition symmetric, hence all the 6 equivalent conditions hold.

A map $t : \mathfrak{g}\to\mathfrak{b}$ together with an action of $\mathfrak{b}$ on $\mathfrak{g}$ is a Leibniz crossed module if

\begin{displaymath}
t([b,g])= [b,t(g)],\,\,\,t([g,b])=[t(g),b],\,\,\,\, for all\,\,\, b\in\mathfrak{b}, g' \in\mathfrak{g}
\end{displaymath}
\begin{displaymath}
[g, t(g')] = [g, g'] = [t(g), g'],\,\,\,\, for all\,\,\, g, g' \in\mathfrak{g}
\end{displaymath}
\hypertarget{abelian_extensions}{}\subsection*{{Abelian extensions}}\label{abelian_extensions}

Abelian extension of right Leibniz algebras is a split short exact sequence of $k$-modules

\begin{displaymath}
0\to M \to \mathfrak{h}\to \mathfrak{g}\to 0
\end{displaymath}
where the mapping $\mathfrak{h}\to\mathfrak{g}$ is a morphism of Leibniz algebras, and $M$ is equipped with induced action of $\mathfrak{g}$. The isomorphisms of extensions of $\mathfrak{g}$ by $M$ with fixed action are defined as usual. This way we obtain a set of equivalence classes $Ext(\mathfrak{g},M)$. To classify the extensions one looks for compatible Leibniz brackets on $M\oplus \mathfrak{g}$. The general form of a bracket is

\begin{displaymath}
[(m_1,x_1),(m_2,x_2)] = ([m_1, x_2] + [x_1, m_2] + f(x_1, x_2), [x_1, x_2]),
\end{displaymath}
where $f(x_1,x_2)$ satisfy the following 2-cocycle identity:

\begin{displaymath}
[x, f(y, z)] + [f(x, z), y] - [f(x, y), z] =
f([x, y], z) - f([x, z], y) - f(x, [y, z])
\end{displaymath}
The extension is \textbf{split} in the category of Leibniz algebras if $f$ is a \emph{boundary} i.e. there exists a $k$-module map $g:\mathfrak{g}\to M$ such that

\begin{displaymath}
f(x, y) = [x, g(y)] + [g(x), y] - g([x, y]), \,\,\,x,y,\in\mathfrak{g}
\end{displaymath}
As for the Lie algebras, the group of abelian extensions agrees with the 2-cohomology $HL^2(\mathfrak{g},M)$.

A $k$-linear \textbf{derivation} of a right Leibniz algebra $\mathfrak{g}$ with values in its representation $M$ is a $k$-linear map satisfying the Leibniz property with respect to the bracket:

\begin{displaymath}
\delta([x,y]) = [\delta(x),y]+[x,\delta(y)]
\end{displaymath}
Such derivations form a $k$-module $Der(\mathfrak{g},M)$.

\hypertarget{homology_and_cohomology}{}\subsection*{{Homology and cohomology}}\label{homology_and_cohomology}

The homology and cohomology of Leibniz algebra $\mathfrak{g}$ with abelian $k$-module of coefficients, which is a corepresentation $A$ in the case of homology and a representation $M$ in the case of cohomology:

\begin{displaymath}
HL_*(\mathfrak{g},A) = Tor^{U\mathfrak{g}}_*(U(\mathfrak{g}_{Lie}),A) ,
\end{displaymath}
\begin{displaymath}
HL^*(\mathfrak{g},M) = Ext_{U\mathfrak{g}}^*(U(\mathfrak{g}_{Lie}),A)
\end{displaymath}
where $U(\mathfrak{g}_{Lie})$ is the universal enveloping of the maximal Lie algebra quotient $\mathfrak{g}_{Lie}$ of $\mathfrak{g}$ and $U\mathfrak{g}$ is the universal enveloping of a Leibniz algebra $\mathfrak{g}$.

Fopr $n\geq 0$, the $n$-cocycles are elements in $C^n(\mathfrak{g}, M) = Hom_k(\mathfrak{g}^{\otimes n}, M)$, satisfying the corresponding abelian cocycle condition determined by the differential

\begin{displaymath}
d^n : C^n(\mathfrak{g}, M)\to C^{n+1}(\mathfrak{g}, M)
\end{displaymath}
\begin{displaymath}
(d^n f) (x_1, . . . , x_{n+1}) = [x_1,f(x_2,\ldots,x_{n+1})] +\sum_{n+1}^{i=2} (-1)^i [f(x_1,\ldots, \hat{x}_i, \ldots, x_{n+1}), x_i]
\end{displaymath}
Notice a difference from the Lie algebra cocycles where instead of a tensor power we have an external power. Then $HL^*(\mathfrak{g},M) = H^*(C^*(\mathfrak{g}, M),d^*)$.

There are standard interpretations of cocycles in low dimensions. For example for $n=0$, $HL^0(\mathfrak{g}, M)$ is the submodule of invariants. For $n=1$ there is a natural projection $Der(\mathfrak{g},M)\to HL^1(\mathfrak{g},M)$ whose kernel is generated by inner derivations.

\hypertarget{the_related_kinds_of_algebras}{}\subsection*{{The related kinds of algebras}}\label{the_related_kinds_of_algebras}

The Leibniz operad is quadratic Koszul algebra whose Koszul dual operad is called the operad of dual Leibniz algebras or of [[Zinbiel algebra]]s, see there.

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

\begin{itemize}%
\item [[Jean-Louis Loday]], [[Teimuraz Pirashvili]], \emph{Universal enveloping algebras of Leibniz algebras and (co)homology}, Math. Ann. \textbf{296}, 139-158 (1993), \href{http://www-irma.u-strasbg.fr/~loday/PAPERS/93LodayPira%28Leibniz%29.pdf}{pdf}
\item J-L. Loday, \emph{Algebraic K-theory and the conjectural Leibniz K-theory}, K-Theory 09/2003; 30(2):105-127, \href{http://www-irma.u-strasbg.fr/~loday/PAPERS/2003Loday%28LeibnizConj%29.pdf}{pdf} \href{http://dx.doi.org/10.1023/B:KTHE.0000018382.90150.ce}{doi}
\item [[Jean-Louis Loday]], [[Teimuraz Pirashvili]], \emph{The tensor category of linear maps}, Georg. Math. J. vol. 5, n.3 (1998) 263--276.
\item Jerry M. Lodder, \emph{Leibniz homology, characteristic classes and K-theory, \href{http://www.math.uiuc.edu/K-theory/0493}{K-theory archive/0493};}Leibniz cohomology and the calculus of variations\_, \href{http://arxiv.org/abs/math/9808036}{arXiv:math/9808036}

\end{itemize}
A generalization of [[Lie integration]] to conjectural Leibniz groups has been conjectured by [[J-L. Loday]]. A local version via local Lie [[rack]]s has been proposed in

\begin{itemize}%
\item Simon Covez, \emph{The local integration of Leibniz algebras}, \href{http://arxiv.org/abs/1011.4112}{arXiv:1011.4112}; \emph{On the conjectural cohomology for groups}, \href{http://arxiv.org/abs/1202.2269}{arXiv:1202.2269}; \emph{L'int\'e{}gration locale des alg\`e{}bres de Leibniz}, Thesis (2010), \href{http://tel.archives-ouvertes.fr/docs/00/49/54/69/PDF/THESE_Simon_Covez.pdf}{pdf}

\end{itemize}
This is partly based on earlier insights of Kinyon and Weinstein:

\begin{itemize}%
\item Michael K. Kinyon, \emph{Leibniz algebras, Lie racks, and digroups}, J. Lie Theory \textbf{17}:1 (2007) 099--114, \href{http://arxiv.org/abs/math/0403509}{arxiv:math.GR/0403509}


\item Simon Covez, \emph{On the conjectural Leibniz cohomology for groups}, Journal of K-theory \textbf{10}:03, Dec 2012, pp 519-563 \href{http://dx.doi.org/10.1017/is011011011jkt195}{doi}



\end{itemize}
[[!redirects Leibniz algebras]] [[!redirects Loday algebra]] [[!redirects Leibniz rule]] [[!redirects Leibniz rules]] [[!redirects Leibniz identity]] [[!redirects Leibniz identities]]



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