-algebra) is a generalisation of the concept of a differential graded Lie algebra.
To be a little more specific, the Jacobi identity only holds up to homotopy.
[2] Homotopy Lie algebras have applications within mathematics and mathematical physics; they are linked, for instance, to the Batalin–Vilkovisky formalism much like differential graded Lie algebras are.
There exists several different definitions of a homotopy Lie algebra, some particularly suited to certain situations more than others.
Here the blanket assumption that the underlying field is of characteristic zero is made.
A homotopy Lie algebra on a graded vector space
as its representing commutative differential graded algebra.
Homotopy Lie algebras and their morphisms define a category.
The more traditional definition of a homotopy Lie algebra is through an infinite collection of symmetric multi-linear maps that is sometimes referred to as the definition via higher brackets.
A homotopy Lie algebra[3] on a graded vector space
satisfy the generalised Jacobi identity: for each n. Here the inner sum runs over
The above formula have meaningful interpretations for low values of
satisfies the Jacobi identity up to an exact term of
vanish, the definition of a differential graded Lie algebra on
Using the approach via multi-linear maps, a morphism of homotopy Lie algebras can be defined by a collection of symmetric multi-linear maps
A morphism of homotopy Lie algebras is said to be a (quasi) isomorphism if its linear component
An important special class of homotopy Lie algebras are the so-called minimal homotopy Lie algebras, which are characterized by the vanishing of their linear component
Fortunately, there are the simple cases coming from differential graded Lie algebras and cases coming from finite dimensional examples.
-algebras come from the embedding of differential graded Lie algebras into the category of
implies it has a lie algebra structure up to a homotopy.
-algebra structure implies showing it is a higher Lie bracket.
, so the previous equation could be read as showing that the differential of the 3-bracket gives the failure for the 2-bracket to be a Lie algebra structure.
has a structure of a Lie algebra from the induced map of
is a Lie algebra on the nose, but, there is the extra data of a vector space
[4]pg 42 Note this situation is highly analogous to the relation between group cohomology and the structure of n-groups with two non-trivial homotopy groups.
there is a similar relation between Lie algebra cocycles and such higher brackets.
Upon first inspection, it's not an obvious results, but it becomes clear after looking at the homology complex so the differential becomes trivial.
given by the dot product of vectors It can be checked the differential of this
Coming up with simple examples for the sake of studying the nature of
There are other similar examples for super[6] Lie algebras.