Loday algebra

Leibniz algebra

Loday algebras were introduced under the name "Leibniz algebras" by J.-L. Loday [a10] [a11] as non-commutative analogues of Lie algebras (cf. also Lie algebra). They are defined by a bilinear bracket which is no longer skew-symmetric. See [a12] for motivations, an overview and additional references. The term "Leibniz algebra" was used in all articles prior to 1996, and in many posterior ones. It had been chosen because, in the generalization of Lie algebras to Loday algebras, it is the derivation property of the adjoint mappings, analogous to the Leibniz rule in elementary calculus, that is preserved, while the skew-symmetry of the bracket is not. However, it has been shown [a1], [a8] that in many instances it is necessary to consider both a bracket and an associative multiplication defined on the same space, and to impose a "Leibniz rule" relating both operations, stating that the adjoint mappings are derivations of the associative multiplication. For this reason, it is preferable to adopt the term "Loday algebra" rather than "Leibniz algebra" when referring to the derivation property of the bracket alone.

A left Loday algebra over a field $k$ is a vector space over $k$ with a $k$-bilinear mapping $[ ., . ] : A \times A \rightarrow A$ satisfying

\begin{equation*} [ a , [ b , c ] ] = [ [ a , b ] , c ] + [ b , [ a , c ] ], \end{equation*}

for all $a , b , c \in A$. This property means that, for each $a$ in $A$, the adjoint endomorphism of $A$, $\operatorname{ad} _ { a } = [ a ,. ]$, is a derivation of $( A , [. ,. ] )$.

Similarly, by definition, in a right Loday algebra, for each $a \in A$, the mapping $x \in A \mapsto [ x , a ] \in A$ is a derivation of $( A , [. ,. ] )$.

A left or right Loday algebra in which the bracket $[ \cdot \ , \ \cdot ]$ is skew-symmetric (or alternating, if $k$ is of characteristic $2$) is a Lie algebra.

Loday algebra structures on a vector space $V$ can be defined as elements of square $0$ with respect to a graded Lie bracket on the vector space of $V$-valued multi-linear forms on $V$ [a3].

A graded version of a left (or right) Loday algebra has been introduced by F. Akman [a1] and further studied in [a8]. The graded Loday algebras generalize the graded Lie algebras (cf. also Lie algebra, graded).

Examples.

The tensor module, $T V = \oplus _ { k \geq 1 } V ^ { \otimes k }$, of any vector space $V$ can be turned into a Loday algebra such that $[ w , v ] = w \otimes v$, for $w \in T V$, $v \in V$. This is the free Loday algebra over $V$.

Given any differential Lie algebra or, more generally, any differential left (respectively, right) Loday algebra, $( A , [ \cdot , \cdot ] , d )$, define $[ x , y ] _ { d } = [ d x , y ]$ (respectively, $[ x , y ] _ { d } = [ x , d y ]$). Then $[ \cdot , \cdot ]_{d}$ is a left (respectively, right) Loday bracket, called the derived bracket [a8]. There is a generalization of this construction to the graded case, and the derived brackets on differential graded Lie algebras, which are graded Loday brackets, have applications in differential and Poisson geometry.

Operads.

The operad associated to the notion of Loday algebra is a Koszul operad [a6]. There is a dual notion, the dual–Loday algebras, which are algebras over the dual operad.

Loday (Leibniz) homology.

This is the homology of the complex $( T V , d )$ with $$ d(x_1 \otimes \dots \otimes x_n) = \sum_{1 \leq i < j \leq n} (-1)^j x_1 \otimes x_2 \otimes x_{i-1} \otimes [x_i, x_j] \otimes x_{i+1} \otimes \dots \otimes \widehat{x_j} \otimes \dots \otimes x_n, $$ where $\widehat{x _ { j }}$ denotes that $x _ { j }$ is omitted. The homology complex of a Loday algebra is a co-algebra in the category of dual–Loday algebras.

The Loday homology of the algebra of matrices over an associative algebra $A$, over a field of characteristic zero, is isomorphic to the tensor module of the Hochschild homology of $A$ (cf. also Extension of an associative algebra) as a group in the category of dual Loday algebras [a4] [a13] [a16]. This is the analogue of the Loday–Quillen–Tsygan theorem relating the Lie-algebra homology of matrices to the graded symmetric algebra over the cyclic homology of $A$ (cf. also Cyclic cohomology).

Loday (Leibniz) cohomology.

The cohomology can be defined dually to the homology. The $n$-cochains on a Loday algebra $A$, with coefficients in a representation $M$ of $A$ (see [a10] [a13]), are the $n$-linear mappings on $A$ with values in $M$, to which the differential of the Chevalley–Eilenberg complex can be lifted. If $M$ is the base field $k$ with the trivial representation, the differential $d \alpha$ of an $n$-cochain $\alpha$ is defined by

\begin{equation*} d \alpha ( x _ { 0 } , \ldots , x _ { n } ) = \sum _ { 0 \leq i < j \leq n } ( - 1 ) ^ { j }\, \alpha ( x_0 , \dots , x _ { i - 1} , [ x _ { i } , x _ { j } ] , x _ { i + 1} , \dots , \widehat{x _ { j }} , \dots , x _ { n } ). \end{equation*}

Di-algebras.

A di-algebra is an algebra with two associative operations satisfying additional axioms [a12]. A di-algebra is a non-commutative analogue of an associative algebra, and any di-algebra structure on a vector space $V$ gives rise to a Loday-algebra structure on $V$. The universal enveloping algebra of a Loday algebra [a13] has the structure of a di-algebra.

References

[a1]	F. Akman, "On some generalizations of Batalin–Vilkovisky algebras" J. Pure Appl. Algebra , 120 (1997) pp. 105–141
[a2]	B. Bakalov, V.G. Kac, A.A. Voronov, "Cohomology of conformal algebras" Comm. Math. Phys. , 200 (1999) pp. 561–598
[a3]	D. Balavoine, "Éléments de carré nul dans les algèbres de Lie graduées" C.R. Acad. Sci. Paris , 321 (1995) pp. 689–694
[a4]	C. Cuvier, "Homologie de Leibniz" Ann. Ecole Norm. Sup. , 27 (1994) pp. 1–45
[a5]	A. Frabetti, "Leibniz homology of dialgebras of matrices" J. Pure Appl. Algebra , 129 (1998) pp. 123–141
[a6]	V. Ginzburg, M.M. Kapranov, "Koszul duality for operads" Duke Math. J. , 76 (1994) pp. 203–272
[a7]	I.V. Kanatchikov, "Novel algebraic structures from the polysymplectic form in field theory" H.D. Doebner (ed.) W. Scherer (ed.) C. Schulte (ed.) , Group 21 , 2 , World Sci. (1997)
[a8]	Y. Kosmann–Schwarzbach, "From Poisson algebras to Gerstenhaber algebras" Ann. Inst. Fourier , 46 (1996) pp. 1243–1274
[a9]	M. Livernet, "Rational homotopy of Leibniz algebras" Manuscripta Math. , 96 (1998) pp. 295–315
[a10]	J.-L. Loday, "Cyclic homology" , Springer (1992) (Second ed.: 1998)
[a11]	J.-L. Loday, "Une version non commutative des algèbres de Lie: les algèbres de Leibniz" Enseign. Math. , 39 (1993) pp. 269–293
[a12]	J.-L. Loday, "Overview on Leibniz algebras, dialgebras and their homology" Fields Inst. Comm. , 17 (1997) pp. 91–102
[a13]	J.-L. Loday, T. Pirashvili, "Universal enveloping algebras of Leibniz algebras and (co)homology" Math. Ann. , 296 (1993) pp. 139–158
[a14]	J.M. Lodder, "Leibniz homology and the Hilton–Milnor theorem" Topology , 36 (1997) pp. 729–743
[a15]	J.M. Lodder, "Leibniz cohomology for differentiable manifolds" Ann. Inst. Fourier , 48 (1998) pp. 73–95
[a16]	J.-M. Oudom, "Coproduct and cogroups in the category of graded dual Leibniz algebras" Contemp. Math. , 202 (1997) pp. 115–135
[a17]	T. Pirashvili, "On Leibniz homology" Ann. Inst. Fourier , 44 (1994) pp. 401–411