Homotopy Lie algebra

A homotopy Lie algebra on a graded vector space ${\displaystyle V=\bigoplus V_{i}}$ is a continuous derivation, ${\displaystyle m}$, of order ${\displaystyle >1}$ that squares to zero 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 ${\displaystyle f\colon {\hat {S}}\Sigma V^{*}\to {\hat {S}}\Sigma W^{*}}$ of their representing commutative differential graded algebras that commutes with the vector field, i.e., ${\displaystyle f\circ m_{V}=m_{W}\circ f}$. 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[3] 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:

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

for each n. Here the inner sum runs over ${\displaystyle (i,j)}$-unshuffles and ${\displaystyle \chi }$ is the signature of the permutation. The above formula have meaningful interpretations for low values of ${\displaystyle n}$; 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 ${\displaystyle V}$ 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.

One of the approachable classes of examples of ${\displaystyle L_{\infty }}$-algebras come from the embedding of differential graded Lie algebras into the category of ${\displaystyle L_{\infty }}$-algebras. This can be described by ${\displaystyle l_{1}}$ giving the derivation, ${\displaystyle l_{2}}$ the Lie algebra structure, and ${\displaystyle l_{k}=0}$ for the rest of the maps.

One notable class of examples are ${\displaystyle L_{\infty }}$-algebras which only have two nonzero underlying vector spaces ${\displaystyle V_{0},V_{1}}$. Then, cranking out the definition for ${\displaystyle L_{\infty }}$-algebras this means there is a linear map

${\displaystyle d\colon V_{1}\to V_{0}}$,

bilinear maps

${\displaystyle l_{2}\colon V_{i}\times V_{j}\to V_{i+j}}$, where ${\displaystyle 0\leq i+j\leq 1}$,

and a trilinear map

${\displaystyle l_{3}\colon V_{0}\times V_{0}\times V_{0}\to V_{1}}$

which satisfy a host of identities.[4] ^{pg 28} In particular, the map ${\displaystyle l_{2}}$ on ${\displaystyle V_{0}\times V_{0}\to V_{0}}$ implies it has a lie algebra structure up to a homotopy. This is given by the differential of ${\displaystyle l_{3}}$ since the gives the ${\displaystyle L_{\infty }}$-algebra structure implies

${\displaystyle dl_{3}(a,b,c)=-[[a,b],c]+[[a,c],b]+[a,[b,c]]}$,

showing it is a higher Lie bracket. In fact, some authors write the maps ${\displaystyle l_{n}}$ as ${\displaystyle [-,\cdots ,-]_{n}:V_{\bullet }\to V_{\bullet }}$, so the previous equation could be read as

${\displaystyle d[a,b,c]_{3}=-[[a,b],c]+[[a,c],b]+[a,[b,c]]}$,

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 ${\displaystyle H_{*}(V_{\bullet },d)}$ then ${\displaystyle H_{0}(V_{\bullet },d)}$ has a structure of a Lie algebra from the induced map of ${\displaystyle [-,-]_{2}}$.

In this case, for ${\displaystyle n\geq 2}$, there is no differential, so ${\displaystyle V_{0}}$ is a Lie algebra on the nose, but, there is the extra data of a vector space ${\displaystyle V_{n}}$ in degree ${\displaystyle n}$ and a higher bracket

${\displaystyle l_{n+2}\colon \bigoplus ^{n+2}V_{0}\to V_{n}.}$

It turns out this higher bracket is in fact a higher cocyle in Lie algebra cohomology. More specifically, if we rewrite ${\displaystyle V_{0}}$ as the Lie algebra ${\displaystyle {\mathfrak {g}}}$ and ${\displaystyle V_{n}}$ and a Lie algebra representation ${\displaystyle V}$ (given by structure map ${\displaystyle \rho }$), then there is a bijection of quadruples

${\displaystyle ({\mathfrak {g}},V,\rho ,l_{n+2})}$ where ${\displaystyle l_{n+2}\colon {\mathfrak {g}}^{\otimes n+2}\to V}$ is an ${\displaystyle (n+2)}$-cocycle

and the two-term ${\displaystyle L_{\infty }}$-algebras with non-zero vector spaces in degrees ${\displaystyle 0}$ and ${\displaystyle n}$.[4]^{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 ${\displaystyle L_{\infty }}$-algebras in degrees ${\displaystyle 0}$ and ${\displaystyle 1}$ 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

${\displaystyle H_{*}(V_{1}\xrightarrow {d} V_{0})}$,

so the differential becomes trivial. This gives an equivalent ${\displaystyle L_{\infty }}$-algebra which can then be analyzed as before.

One simple example of a Lie-2 algebra is given by the ${\displaystyle L_{\infty }}$-algebra with ${\displaystyle V_{0}=(\mathbb {R} ^{3},\times )}$ where ${\displaystyle \times }$ is the cross-product of vectors and ${\displaystyle V_{1}=\mathbb {R} }$ is the trivial representation. Then, there is a higher bracket ${\displaystyle l_{3}}$ given by the dot product of vectors

${\displaystyle l_{3}(a,b,c)=a\cdot (b\times c).}$

It can be checked the differential of this ${\displaystyle L_{\infty }}$-algebra is always zero using basic linear algebra[4]^{pg 45}.

Coming up with simple examples for the sake of studying the nature of ${\displaystyle L_{\infty }}$-algebras is a complex problem. For example,[5] given a graded vector space ${\displaystyle V=V_{0}\oplus V_{1}}$ where ${\displaystyle V_{0}}$ has basis given by the vector ${\displaystyle w}$ and ${\displaystyle V_{1}}$ has the basis given by the vectors ${\displaystyle v_{1},v_{2}}$, there is an ${\displaystyle L_{\infty }}$-algebra structure given by the following rules

${\displaystyle {\begin{aligned}&l_{1}(v_{1})=l_{1}(v_{2})=w\\&l_{2}(v_{1}\otimes v_{2})=v_{1},l_{2}(v_{1}\otimes w)=w\\&l_{n}(v_{2}\otimes w^{\otimes n-1})=C_{n}w{\text{ for }}n\geq 3\end{aligned}},}$

where ${\displaystyle C_{n}=(-1)^{n-1}(n-3)C_{n-1},C_{3}=1}$. Note that the first few constants are

${\displaystyle {\begin{matrix}C_{3}&C_{4}&C_{5}&C_{6}\\1&-1&-2&12\end{matrix}}}$

Since ${\displaystyle l_{1}(w)}$ should be of degree ${\displaystyle -1}$, the axioms imply that ${\displaystyle l_{1}(w)=0}$. There are other similar examples for super[6] Lie algebras.[7] Furthermore, ${\displaystyle L_{\infty }}$ structures on graded vector spaces whose underlying vector space is two dimensional have been completely classified.[3]

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