# 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. These homotopy algebras are useful in classifying deformation problems over characteristic 0 in deformation theory because deformation functors are classified by quasi-isomorphism classes of $L_{\infty }$-algebras. This was later extended to all characteristics by Jonathan Pridham.
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.

## 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, $m$, of order $>1$ that squares to zero 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, $f\colon (V,m_{V})\to (W,m_{W})$, as a morphism $f\colon {\hat {S}}\Sigma V^{*}\to {\hat {S}}\Sigma W^{*}$ 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 $l_{n}\colon V^{\otimes n}\to V$ 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:
    $\sum _{i+j=n+1}\sum _{\sigma \in \mathrm {UnShuff} (i,n-i)}\chi (\sigma ,v_{1},\dots ,v_{n})(-1)^{i(j-1)}l_{j}(l_{i}(v_{\sigma (1)},\dots ,v_{\sigma (i)}),v_{\sigma (i+1)},\dots ,v_{\sigma (n)})=0,$
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 $f_{n}\colon V^{\otimes n}\to W$ 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, a 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 $f\colon V\to W$ 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, which are characterized by the vanishing of their linear component $l_{1}$. 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 $L_{\infty }$-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 $L_{\infty }$-algebras come from the embedding of differential graded Lie algebras into the category of $L_{\infty }$-algebras. This can be described by $l_{1}$ giving the derivation, $l_{2}$ the Lie algebra structure, and $l_{k}=0$ for the rest of the maps.

### Two term L$_∞$ algebras
#### In degrees 0 and 1

One notable class of examples are $L_{\infty }$-algebras which only have two nonzero underlying vector spaces $V_{0},V_{1}$. Then, cranking out the definition for $L_{\infty }$-algebras this means there is a linear map
    $d\colon V_{1}\to V_{0}$,
bilinear maps
    $l_{2}\colon V_{i}\times V_{j}\to V_{i+j}$, where $0\leq i+j\leq 1$,
and a trilinear map
    $l_{3}\colon V_{0}\times V_{0}\times V_{0}\to V_{1}$
which satisfy a host of identities. $^{pg28}$ In particular, the map $l_{2}$ on $V_{0}\times V_{0}\to V_{0}$ implies it has a lie algebra structure up to a homotopy. This is given by the differential of $l_{3}$ since the gives the $L_{\infty }$-algebra structure implies
    $dl_{3}(a,b,c)=-[[a,b],c]+[[a,c],b]+[a,[b,c]]$,
showing it is a higher Lie bracket. In fact, some authors write the maps $l_{n}$ as $[-,\cdots ,-]_{n}:V_{\bullet }\to V_{\bullet }$, so the previous equation could be read as
    $d[a,b,c]_{3}=-[[a,b],c]+[[a,c],b]+[a,[b,c]]$,
showing that the differential of the 3-bracket gives the failure for the 2-bracket to be a Lie algebra structure. It is only a Lie algebra up to homotopy. If we took the complex $H_{*}(V_{\bullet },d)$ then $H_{0}(V_{\bullet },d)$ has a structure of a Lie algebra from the induced map of $[-,-]_{2}$.

#### In degrees 0 and n

In this case, for $n\geq 2$, there is no differential, so $V_{0}$ is a Lie algebra on the nose, but, there is the extra data of a vector space $V_{n}$ in degree $n$ and a higher bracket
    $l_{n+2}\colon \bigoplus ^{n+2}V_{0}\to V_{n}.$
It turns out this higher bracket is in fact a higher cocyle in Lie algebra cohomology. More specifically, if we rewrite $V_{0}$ as the Lie algebra ${\mathfrak {g}}$ and $V_{n}$ and a Lie algebra representation $V$ (given by structure map $\rho$), then there is a bijection of quadruples
    $({\mathfrak {g}},V,\rho ,l_{n+2})$ where $l_{n+2}\colon {\mathfrak {g}}^{\otimes n+2}\to V$ is an $(n+2)$-cocycle
and the two-term $L_{\infty }$-algebras with non-zero vector spaces in degrees $0$ and $n$.$^{pg42}$ Note this situation is highly analogous to the relation between group cohomology and the structure of n-groups with two non-trivial homotopy groups. For the case of term term $L_{\infty }$-algebras in degrees $0$ and $1$ 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
    $H_{*}(V_{1}\xrightarrow {d} V_{0})$,
so the differential becomes trivial. This gives an equivalent $L_{\infty }$-algebra which can then be analyzed as before.

#### Example in degrees 0 and 1

One simple example of a Lie-2 algebra is given by the $L_{\infty }$-algebra with $V_{0}=(\mathbb {R} ^{3},\times )$ where $\times$ is the cross-product of vectors and $V_{1}=\mathbb {R}$ is the trivial representation. Then, there is a higher bracket $l_{3}$ given by the dot product of vectors
    $l_{3}(a,b,c)=a\cdot (b\times c).$
It can be checked the differential of this $L_{\infty }$-algebra is always zero using basic linear algebra$^{pg45}$.

### Finite dimensional example

Coming up with simple examples for the sake of studying the nature of $L_{\infty }$-algebras is a complex problem. For example, given a graded vector space $V=V_{0}\oplus V_{1}$ where $V_{0}$ has basis given by the vector $w$ and $V_{1}$ has the basis given by the vectors $v_{1},v_{2}$, there is an $L_{\infty }$-algebra structure given by the following rules
    ${\begin{aligned}&l_{1}(v_{1})=l_{1}(v_{2})=w\\&l_{2}(v_{1}\otimes v_{2})=v_{1},l_{2}(v_{1}\otimes w)=w\\&l_{n}(v_{2}\otimes w^{\otimes n-1})=C_{n}w{\text{ for }}n\geq 3\end{aligned}},$
where $C_{n}=(-1)^{n-1}(n-3)C_{n-1},C_{3}=1$. Note that the first few constants are
    ${\begin{matrix}C_{3}&C_{4}&C_{5}&C_{6}\\1&-1&-2&12\end{matrix}}$
Since $l_{1}(w)$ should be of degree $-1$, the axioms imply that $l_{1}(w)=0$. There are other similar examples for super Lie algebras. Furthermore, $L_{\infty }$ structures on graded vector spaces whose underlying vector space is two dimensional have been completely classified.