Homotopy Lie algebra

In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or formula_1-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 formula_1-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 formula_3 is a continuous derivation, formula_4, of order formula_5 that squares to zero on the formal manifold formula_6. Here formula_7 is the completed symmetric algebra, formula_8 is the suspension of a graded vector space, and formula_9 denotes the linear dual. Typically one describes formula_10 as the homotopy Lie algebra and formula_6 with the differential formula_4 as its representing commutative differential graded algebra.
Using this definition of a homotopy Lie algebra, one defines a morphism of homotopy Lie algebras, formula_13, as a morphism formula_14 of their representing commutative differential graded algebras that commutes with the vector field, i.e., formula_15. 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 formula_3 is a collection of symmetric multi-linear maps formula_17 of degree formula_18, sometimes called the formula_19-ary bracket, for each formula_20. Moreover, the maps formula_21 satisfy the generalised Jacobi identity:
for each n. Here the inner sum runs over formula_23-unshuffles and formula_24 is the signature of the permutation. The above formula have meaningful interpretations for low values of formula_19; for instance, when formula_26 it is saying that formula_27 squares to zero (i.e., it is a differential on formula_28), when formula_29 it is saying that formula_27 is a derivation of formula_31, and when formula_32 it is saying that formula_31 satisfies the Jacobi identity up to an exact term of formula_34 (i.e., it holds up to homotopy). Notice that when the higher brackets formula_21 for formula_36 vanish, the definition of a differential graded Lie algebra on formula_28 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 formula_38 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 formula_1 operad.
(Quasi) isomorphisms and minimal models.
A morphism of homotopy Lie algebras is said to be a (quasi) isomorphism if its linear component formula_40 is a (quasi) isomorphism, where the differentials of formula_28 and formula_42 are just the linear components of formula_43 and formula_44.
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 formula_27. 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 formula_1-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 formula_1-algebras come from the embedding of differential graded Lie algebras into the category of formula_1-algebras. This can be described by formula_27 giving the derivation, formula_31 the Lie algebra structure, and formula_51 for the rest of the maps.
Two term L∞ algebras.
In degrees 0 and 1.
One notable class of examples are formula_1-algebras which only have two nonzero underlying vector spaces formula_53. Then, cranking out the definition for formula_1-algebras this means there is a linear map
bilinear maps
and a trilinear map
which satisfy a host of identities. pg 28 In particular, the map formula_31 on formula_60 implies it is has a lie algebra structure upto a homotopy. This is given by the differential of formula_34 since the gives the formula_1-algebra structure implies
showing it is a higher Lie bracket. In fact, some authors write the maps formula_21 as formula_65, so the previous equation could be read as
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 formula_67 then formula_68 has a structure of a Lie algebra from the induced map of formula_69.
In degrees 0 and n.
In this case, for formula_70, there is no differential, so formula_71 is a Lie algebra on the nose, but, there is the extra data of a vector space formula_72 in degree formula_19 and a higher bracketformula_74It turns out this higher bracket is in fact a higher cocyle in [[Lie algebra cohomology]]. More specifically, if we rewrite formula_71 as the Lie algebra formula_76 and formula_72 and a Lie algebra representation formula_28 (given by structure map formula_79), then there is a bijection of quadruples
and the two-term formula_1-algebras with non-zero vector spaces in degrees formula_84 and formula_19.pg 42 Note this situation is highly analogous to the relation between [[group cohomology]] and the structure of [[N-group (category theory)|n-groups]] with two non-trivial homotopy groups. For the case of term term formula_1-algebras in degrees formula_84 and formula_88 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
so the differential becomes trivial. This gives an equivalent formula_1-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 formula_1-algebra with formula_92 where formula_93 is the cross-product of vectors and formula_94 is the trivial representation. Then, there is a higher bracket formula_34 given by the dot product of vectorsformula_96It can be checked the differential of this formula_1-algebra is always zero using basic linear algebrapg 45.
Finite dimensional example.
Coming up with simple examples for the sake of studying the nature of formula_1-algebras is a complex problem. For example, given a graded vector space formula_99 where formula_71 has basis given by the vector formula_101 and formula_102 has the basis given by the vectors formula_103, there is an formula_1-algebra structure given by the following rules
where formula_106. Note that the first few constants are
Since formula_108 should be of degree formula_109, the axioms imply that formula_110. There are other similar examples for super Lie algebras. Furthermore, formula_1 structures on graded vector spaces whose underlying vector space is two dimensional have been completely classified.
External links.
[[Category:Homotopical algebra]]
[[Category:Differential algebra]]
[[Category:Lie algebras]]