Homotopy Lie algebra

A homotopy Lie algebra on a graded vector space ${\displaystyle V=\bigoplus V_{i}}$ is a continuous derivation of order ${\displaystyle >1}$ that squares to zero ${\displaystyle m}$ on the formal manifold ${\displaystyle {\hat {S}}\Sigma V^{*}}$. Here ${\displaystyle {\hat {S}}}$ is the completed symmetric algebra, ${\displaystyle \Sigma }$ is the suspension of a graded vector space, and ${\displaystyle V^{*}}$ denotes the linear dual. Typically one describes ${\displaystyle (V,m)}$ as the homotopy Lie algebra and ${\displaystyle {\hat {S}}\Sigma V^{*}}$ with the differential ${\displaystyle m}$ as its representing commutative differential graded algebra.

Using this definition of a homotopy Lie algebra, one defines a morphism of homotopy Lie algebras ${\displaystyle f\colon (V,m_{V})\to (W,m_{W})}$ as a morphism ${\displaystyle f:{\hat {S}}\Sigma V^{*}\to {\hat {S}}\Sigma W^{*}}$ of their representing commutative differential graded algebras that commutes with the vector field, i.e. ${\displaystyle f\circ m_{V}=m_{W}\circ f}$. 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. It should be stated that the two definitions are equivalent.

A homotopy Lie algebra on a graded vector space ${\displaystyle V=\bigoplus V_{i}}$ is a collection of symmetric multi-linear maps ${\displaystyle l_{n}\colon V^{\otimes n}\to V}$ of degree ${\displaystyle n-2}$, sometimes called the ${\displaystyle n}$-ary bracket, for each ${\displaystyle n\in \mathbb {N} }$. Moreover, the maps ${\displaystyle l_{n}}$ satisfy the generalised Jacobi identity:

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

for each n. Here the inner sum runs over ${\displaystyle (i,j)}$-unshuffles and ${\displaystyle \chi }$ is the signature of the permutation. The above formula have meaningful interpretations for low values of n; for instance, when ${\displaystyle n=1}$ it is saying that ${\displaystyle l_{1}}$ squares to zero (i.e., it is a differential on ${\displaystyle V}$), when ${\displaystyle n=2}$ it is saying that ${\displaystyle l_{1}}$ is a derivation of ${\displaystyle l_{2}}$, and when ${\displaystyle n=3}$ it is saying that ${\displaystyle l_{2}}$ satisfies the Jacobi identity up to an exact term of ${\displaystyle l_{3}}$ (i.e. it holds up to homotopy). Notice that when the higher brackets ${\displaystyle l_{n}}$ for ${\displaystyle n\geq 3}$ 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 ${\displaystyle f_{n}\colon V^{\otimes n}\to W}$ which satisfy certain conditions.

There also exists a more abstract definition of a homotopy algebra using the theory of operads: that is, a homotopy Lie algebra is an algebra over an operad in the category of chain complexes over the ${\displaystyle L_{\infty }}$ operad.