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 $$n\in\N$$. 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.

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. pg 28 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 $$n$$. pg 42 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 $$V_0= (\R^3,\times)$$ where $$\times$$ is the cross-product of vectors and $$V_1=\R$$ 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 pg 45.

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{align}

& 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_nw \text{ for } n \geq 3 \end{align},$$ where $$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.

Introduction

 * Deformation Theory (lecture notes) - gives an excellent overview of homotopy Lie algebras and their relation to deformation theory and deformation quantization

In physics

 * — Towards classification of perturbative gauge invariant classical fields.
 * — Towards classification of perturbative gauge invariant classical fields.

Related ideas

 * (Lie algebras in the derived category of coherent sheaves.)