There exists several different definitions of a homotopy Lie algebra, some particularly suited to certain situations more than others. The most traditional definition is via symmetric multi-linear maps, but there also exists a more succinct geometric definition using the language of formal geometry. Here the blanket assumption that the underlying field is of characteristic zero is made.
Geometric definition
A homotopy Lie algebra on a graded vector space is a continuous derivation,, of order that squares to zero on the formal manifold. Here is the completed symmetric algebra, is the suspension of a graded vector space, and denotes the linear dual. Typically one describes as the homotopy Lie algebra and with the differential as its representing commutative differential graded algebra. Using this definition of a homotopy Lie algebra, one defines a morphism of homotopy Lie algebras,, as a morphism of their representing commutative differential graded algebras that commutes with the vector field, i.e.,. Homotopy Lie algebras and their morphisms define a category.
Definition via multi-linear maps
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. It should be stated that the two definitions are equivalent. A homotopy Lie algebra on a graded vector space is a collection of symmetric multi-linear maps of degree, sometimes called the -ary bracket, for each. Moreover, the maps satisfy the generalised Jacobi identity: for each n. Here the inner sum runs over -unshuffles and is the signature of the permutation. The above formula have meaningful interpretations for low values of ; for instance, when it is saying that squares to zero, when it is saying that is a derivation of, and when it is saying that satisfies the Jacobi identity up to an exact term of . Notice that when the higher brackets for vanish, the definition of a differential graded Lie algebra on is recovered. Using the approach via multi-linear maps, a morphism of homotopy Lie algebras can be defined by a collection of symmetric multi-linear maps which satisfy certain conditions.
A morphism of homotopy Lie algebras is said to be a isomorphism if its linear component is a isomorphism, where the differentials of and are just the linear components of and. 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. This means that any quasi isomorphism of minimal homotopy Lie algebras must be an isomorphism. Any homotopy Lie algebra is quasi-isomorphic to a minimal one, which must be unique up to isomorphism and it is therefore called its minimal model.
Examples
Because -algebras have such a complex structure describing even simple cases can be a non-trivial task in most cases. Fortunately, there are the simple cases coming from differential graded Lie algebras and cases coming from finite dimensional examples.
Differential graded Lie algebras
One of the approachable classes of examples of -algebras come from the embedding of differential graded Lie algebras into the category of -algebras. This can be described by giving the derivation, the Lie algebra structure, and for the rest of the maps.
Coming up with simple examples for the sake of studying the nature of -algebras is a complex problem. For example, given a graded vector space where has basis given by the vector and has the basis given by the vectors, there is an -algebra structure given by the following rules where. Note that the first few constants are Since should be of degree, the axioms imply that. There are other similar examples for super Lie algebras.. Furthermore, structures on graded vector spaces whose underlying vector space is two dimensional have been completely classified.
