Homotopy Lie algebra

Definition

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 V = ⨁ V i is a continuous derivation of order > 1 that squares to zero m on the formal manifold S ^ Σ V ∗ . Here S ^ is the completed symmetric algebra, Σ is the suspension of a graded vector space, and V ∗ denotes the linear dual. Typically one describes ( V , m ) as the homotopy Lie algebra and S ^ Σ V ∗ with the differential m as its representing commutative differential graded algebra.

Using this definition of a homotopy Lie algebra, one defines a morphism of homotopy Lie algebras f : ( V , m V ) → ( W , m W ) as a morphism f : S ^ Σ V ∗ → S ^ Σ W ∗ of their representing commutative differential graded algebras that commutes with the vector field, i.e. f ∘ m V = m W ∘ f . 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 V = ⨁ V i is a collection of symmetric multi-linear maps l n : V ⊗ n → V of degree n − 2 , sometimes called the n -ary bracket, for each n ∈ N . Moreover, the maps l n satisfy the generalised Jacobi identity:

for each n. Here the inner sum runs over ( i , j ) -unshuffles and χ is the signature of the permutation. The above formula have meaningful interpretations for low values of n; for instance, when n = 1 it is saying that l 1 squares to zero (i.e. it is a differential on V), when n = 2 it is saying that l 1 is a derivation of l 2 , and when n = 3 it is saying that l 2 satisfies the Jacobi identity up to an exact term of l 3 (i.e. it holds up to homotopy). Notice that when the higher brackets l n for n ≥ 3 vanish, the definition of a differential graded Lie algebra on V 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 f n : V ⊗ n → W which satisfy certain conditions.

Definition via operads

There also exists a more abstract definition of a homotopy algebra using the theory of operads: that is, an homotopy Lie algebra is an algebra over an operad in the category of chain complexes over the L ∞ operad.

(Quasi) isomorphisms and minimal models

A morphism of homotopy Lie algebras is said to be a (quasi) isomorphism if its linear component f : V → W is a (quasi) isomorphism where the differentials of V and W are just the linear components of m V and m W .

An important special class of homotopy Lie algebras are the so-called minimal homotopy Lie algebras. These are those where the linear component l 1 vanishes. 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.