symmetric monoidal (∞,1)-category of spectra
A Leibniz algebra is like a Lie algebra, but without the condition that the product, often still written as a bracket $[-,-]$, is skew-symmetric. The Jacobi identity however is retained as a condition in its form as the derivation-property of the product over itself. In view of the analogous product law of differentiation (also a derivation-property) attributed to Gottfried Leibniz, this is then called the Leibniz identity which gives Leibniz algebras their modern name (Loday 93, Loday-Pirashvili 93) even though the concept itself is older (Blokh 65).
Leibniz algebras were motivated in Cuvier 91, Loday-Pirashvili 93 as generalizing the relation between Lie algebra cohomology and cyclic homology (Loday-Quillen 84) to one between Leibniz cohomology and Hochschild homology: Where the nilpotency of the differential in the Chevalley-Eilenberg algebras that compute Lie algebra cohomology is equivalent to the Jacobi identity in the corresponding Lie algebra, Leibniz cohomology is defined on non-skew symmetric dg-algebras where now it is the generalization of the Jacobi identity in form of the Leibniz rule (1) which still guarantees the nilpotency of the differential.
More recently, Leibniz algebras have been argued to clarify the nature of the embedding tensor and the resulting tensor hierarchies in gauged supergravity (Lavau 17).
Let $k$ be a commutative ring (typically a field).
(left Leibniz algebra)
A left Leibniz $k$-algebra_ (or: left Loday algebra) is
equipped with
a linear map, the product
such that
the product satisfies the left Leibniz identity, saying that
for all $v_i \in A$.
This says equivalently that the operations $v \cdot (-) \colon A \to A$ of left-multiplication by elements $v \in A$ via the given product is a derivation of the product itself, whence the name (paying tribute to Gottfried Leibniz‘s product rule of differentiation).
Analogously there is the concept of right Leibniz algebras in the evident way.
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.
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:
for $x,y\in\mathfrak{g}$ and for $m\in M$.
For corepresentatons:
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
Abelian extension of right Leibniz algebras is a split short exact sequence of $k$-modules
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
where $f(x_1,x_2)$ satisfy the following 2-cocycle identity:
The extension is split in the category of Leibniz algebras if $f$ is a boundary i.e. there exists a $k$-module map $g:\mathfrak{g}\to M$ such that
As for the Lie algebras, the group of abelian extensions agrees with the 2-cohomology $HL^2(\mathfrak{g},M)$.
A $k$-linear 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:
Such derivations form a $k$-module $Der(\mathfrak{g},M)$.
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:
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
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.
The Leibniz operad is quadratic Koszul algebra whose Koszul dual operad is called the operad of dual Leibniz algebras or of Zinbiel algebras, see there.
In complete analogy to the equivalence between the category of Lie algebras and the category of local Lie groups (Lie's third theorem), the category of Leibniz algebras is equivalent to the category of local pointed augmented Lie racks. See Covez 10.
This equivalence restricts to the equivalence between Lie algebras and local Lie groups.
Here a local pointed augmented Lie rack is a pointed augmented rack object in the category of germs of pointed smooth manifolds. An augmented rack is a triple $(G,X,p\colon X\to G)$, where $G$ is a group, $X$ is a G-set, and $p$ is a map of sets such that $p(g\cdot x)=g p(x)g^{-1}$. An augmented rack is pointed if there is an element $1\in X$ such that $p(1)=1$ and $g\cdot 1=1$ for all $g\in G$.
If $(G,X,p)$ is an augmented rack, then $X$ can be made into a rack as follows: $x\triangleright y = p(x)\cdot y$. If $(G,X,p)$ is a pointed augmented rack, then $X$ is a pointed rack, meaning there is $1\in X$ such that $1\triangleright x=x$ and $x\triangleright 1=1$.
(Leibniz algebra from Lie module with embedding tensor)
Let
$\mathfrak{g}$ be a Lie algebra,
$\mathfrak{g} \otimes V \overset{\rho}{\longrightarrow} V$ be a Lie algebra representation of $\mathfrak{g}$,
$\Theta \colon V \longrightarrow \mathfrak{g}$ be an embedding tensor, hence a linear map such that the “quadratic constraint” is satisfied: for all $v_1, v_2 \in V$ we have
Then $V$ becomes a Leibniz algebra with product defined by
and with respect to this the quadratic constraint (2) becomes the condition that $\Theta$ is a homomorphism of Leibniz algebras.
We directly check the Leibniz rule (1) as follows:
Here the first line is the definition (3), the second line is the Lie action property (here), under the brace we use the quadratic constraint (2) on the embedding tensor, and in the last line we observe again the definition (3).
(Leibniz algebra from dg-Lie algebra)
Let $((V_\bullet, \partial), [-,-])$ be a dg-Lie algebra with underlying chain complex $(V_\bullet, \partial)$ and with super Lie bracket $[-,-]$.
On the graded vector space which is the direct sum
of all the component vector spaces, consider the product given by the formula
Then: Restricted to $V_1 \subset \mathbf{V}$ this product gives a left Leibniz algebra $(V_1, \cdot)$ (Def. ), i.e. satisfies the Leibniz condition (1).
This statement is highlighted in Lavau-Palmkvist 19, 2.1.
We directly compute as follows:
Here the first line is the definition (4), the second line is the graded Jacobi identity, the third line uses the derivation-property and the nilpotency of the differential, and the last line invokes again the definition (4). Over the brace we used the assumption that $v_i \in V_1$.
The construction in Prop. evidently extends to a functor from the category $dgLieAlg$ of dg-Lie algebras to the category $LeibAlg$ of Leibniz algebras (both over the given ground ring/ground field):
Notice the analogy to the evident functor that extract the Lie algebra in degree 0:
The idea of Leibniz algebra, though not by this name, is given already in
The concept was revived (and apparently the name Leibniz algebra was first chosen) in
Jean-Louis Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Volume 44 (1993), Talk no. 5, 25 p. (numdam:RCP25_1993__44__127_0)
Jean-Louis Loday, Teimuraz Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296, 139-158 (1993) (doi:10.1007/BF01445099, pdf)
Early review is in
See also
Relaization of Leibniz algebras as Lie algebra objects in a suitable tensor category:
Relation to Hochschild homology:
Christian Cuvier, Homologie de Leibniz et homologie de Hochschild, C.R. Acad. Sci. Paris, Ser. A-B313, 569-572 (1991)
Jerry M. Lodder, Leibniz homology, characteristic classes and K-theory, K-theory archive/0493;Leibniz cohomology and the calculus of variations_ (arXiv:math/9808036)
Jean-Louis Loday, Algebraic K-theory and the conjectural Leibniz K-theory, K-Theory 09/2003; 30(2):105-127, pdf doi
This is partly based on earlier insights of Kinyon and Weinstein:
Relation of Leibniz algebras to dg-Lie algebras such as the tensor hierarchies in gauged supergravity:
Sylvain Lavau, Tensor hierarchies and Leibniz algebras, J. Geom. Phys. 144:147-189 (2019) (arXiv:1708.07068)
Sylvain Lavau, Jakob Palmkvist, Infinity-enhancing of Leibniz algebras (arXiv:1907.05752)
Sylvain Lavau, Jim Stasheff, $L_\infty$-algebra extensions of Leibniz algebras (arXiv:2003.07838)
A generalization of Lie integration to conjectural Leibniz groups has been conjectured by J-L. Loday. A local version via local Lie racks has been proposed in
Simon Covez, L’intégration locale des algèbres de Leibniz, Thesis (2010) (pdf)
Simon Covez, The local integration of Leibniz algebras, Annales de l’Institut Fourier, Volume 63 (2013) no. 1, p. 1-35 (arXiv:1011.4112, doi:10.5802/aif.2754)
Simon Covez, On the conjectural cohomology for groups, Journal of K-theory 10:03, Dec 2012, pp 519-563 (arXiv:1202.2269, doi:10.1017/is011011011jkt195)
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