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
is a vector space over
with a
-bilinear
mapping
satisfying
for all
.
This property means that, for each
in
,
the adjoint endomorphism of
,
,
is a derivation of
.
Similarly, by definition, in a right Loday algebra, for each
,
the mapping
is a derivation of
.
A left or right Loday algebra in which the bracket
is skew-symmetric
(or alternating, if
is of characteristic
)
is a
Lie algebra.
Loday algebra structures on a vector space
can be defined as elements of square
with respect to a graded Lie bracket on the vector space of
-valued
multi-linear forms on
[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,
,
of any vector space
can be turned into a Loday algebra such that
,
for
,
.
This is the
free Loday algebra
over
.
Given any
differential Lie algebra
or, more generally, any
differential
left (respectively, right) Loday algebra,
,
define
(respectively,
).
Then
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
with
,
where
denotes that
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
,
over a field of characteristic zero,
is isomorphic to the tensor module of the
Hochschild homology
of
(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
(cf. also
Cyclic cohomology).
Loday (Leibniz) cohomology.
The cohomology can be defined dually to the homology. The
-cochains
on a Loday algebra
,
with coefficients in a representation
of
(see
[a10]
[a13]),
are the
-linear
mappings on
with values in
,
to which the differential
of the
Chevalley–Eilenberg complex
can be lifted. If
is the base field
with the trivial representation, the differential
of an
-cochain
is defined by
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
gives rise to a Loday-algebra structure on
.
The
universal enveloping algebra
of a Loday algebra
[a13]
has the structure of a
di-algebra.