Homotopy Lie algebra

In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or ${\displaystyle L_{\infty }}$-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 ${\displaystyle L_{\infty }}$-algebras.^[1] This was later extended to all characteristics by Jonathan Pridham.^[2]

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 ${\displaystyle V=\bigoplus V_{i}}$ 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.

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^[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.

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 ${\displaystyle L_{\infty }}$ operad.

(Quasi) isomorphisms and minimal models

A morphism of homotopy Lie algebras is said to be a (quasi) isomorphism if its linear component ${\displaystyle f\colon V\to W}$ is a (quasi) isomorphism, where the differentials of ${\displaystyle V}$ and ${\displaystyle W}$ are just the linear components of ${\displaystyle m_{V}}$ and ${\displaystyle m_{W}}$.

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 ${\displaystyle l_{1}}$. 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 ${\displaystyle L_{\infty }}$-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 ${\displaystyle L_{\infty }}$-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.

Two term L_∞ algebras

In degrees 0 and 1

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 degrees 0 and n

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.

Example in degrees 0 and 1

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}.

Finite dimensional example

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]

See also

• Homotopy associative algebra
• Differential graded algebra
• BV formalism
• Simplicial Lie algebra
• Hochschild homology
• Deformation quantization

References

Introduction

• Deformation Theory (lecture notes) - gives an excellent overview of homotopy Lie algebras and their relation to deformation theory and deformation quantization
• Lada, Tom; Stasheff, Jim (1993). "Introduction to sh Lie algebras for physicists". International Journal of Theoretical Physics. 32 (7): 1087–1104. arXiv:hep-th/9209099. Bibcode:1993IJTP...32.1087L. doi:10.1007/BF00671791. S2CID 16456088.

In physics

• Arvanitakis, Alex S. (2019). "The L∞-algebra of the S-matrix". arXiv:1903.05643 [hep-th].
• Hohm, Olaf; Zwiebach, Barton (2017). "L∞ Algebras and Field Theory". Fortschr. Phys. 65 (3–4): 1700014. arXiv:1701.08824. Bibcode:2017ForPh..6500014H. doi:10.1002/prop.201700014. S2CID 90628041. — Towards classification of perturbative gauge invariant classical fields.

In deformation and string theory

• Pridham, Jonathan P. (2015). "Derived deformations of Artin stacks". Communications in Analysis and Geometry. 23 (3): 419–477. arXiv:0805.3130. doi:10.4310/CAG.2015.v23.n3.a1. MR 3310522. S2CID 14505074.
• Pridham, Jonathan P. (2010). "Unifying derived deformation theories". Advances in Mathematics. 224 (3): 772–826. arXiv:0705.0344. doi:10.1016/j.aim.2009.12.009. MR 2628795. S2CID 14136532.
• Hu, Po; Kriz, Igor; Voronov, Alexander A. (2006). "On Kontsevich's Hochschild cohomology conjecture". Compositio Mathematica. 142 (1): 143–168. arXiv:math/0309369. doi:10.1112/S0010437X05001521. MR 2197407. S2CID 15153116.

Related ideas

• Roberts, Justin; Willerton, Simon (2010). "On the Rozansky–Witten weight systems". Algebraic & Geometric Topology. 10 (3): 1455–1519. arXiv:math/0602653. doi:10.2140/agt.2010.10.1455. MR 2661534. S2CID 17829444. (Lie algebras in the derived category of coherent sheaves.)

External links

• "Learning seminar on deformation theory". Max Planck Institute for Mathematics. 2018. Discusses deformation theory in the context of ${\displaystyle L_{\infty }}$-algebras.