Homotopy Lie algebra

In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or $L_{\infty }$ -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. Therefore, a differential graded Lie algebra can be seen as a homotopy Lie algebra where the Jacobi identity holds on the nose.

Homotopy Lie algebras have applications within mathematics and mathematical physics; they are linked, for instance, to the BV formalism much like differential graded Lie algebras are.

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 $V=\bigoplus V_{i}$ is a continuous derivation of order $>1$ that squares to zero $m$ on the formal manifold ${\hat {S}}\Sigma V^{*}$ . Here ${\hat {S}}$ is the completed symmetric algebra, $\Sigma$ 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 ${\hat {S}}\Sigma V^{*}$ 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 $f:(V,m_{V})\to (W,m_{W})$ as a morphism $f:{\hat {S}}\Sigma V^{*}\to {\hat {S}}\Sigma W^{*}$ of their representing commutative differential graded algebras that commutes with the vector field, i.e. $f\circ m_{V}=m_{W}\circ f$ . 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 $V=\bigoplus V_{i}$ is a collection of symmetric multi-linear maps $l_{n}:V^{\otimes n}\to V$ of degree $n-2$ , sometimes called the $n$ -ary bracket, for each $n\in \mathbb {N}$ . Moreover, the maps $l_{n}$ satisfy the generalised Jacobi identity:

\Sigma _{i+j=n+1}\Sigma _{\sigma \in {\text{UnShuff}}(i,n-i)}\chi (\sigma ,v_{1},\dots ,v_{n})(-1)^{i(j-1)}l_{j}(l_{i}(v_{\sigma (1)},\dots ,v_{\sigma (i)}),v_{\sigma (i+1)},\dots ,v_{\sigma (n)})=0,

for each n. Here the inner sum runs over $(i,j)$ -unshuffles and $\chi$ 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 $n\geq 3$ 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 $f_{n}:V^{\otimes n}\to W$ 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 $L_{\infty }$ operad.

(Quasi) isomorphisms and minimal models

A morphism of homotopy Lie algebras is said to be a (quasi) isomorphism if its linear component $f:V\to W$ 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.