Homotopy Lie algebra

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In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or -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 -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

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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

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A homotopy Lie algebra on a graded vector space   is a continuous derivation,  , of order   that squares to zero on the formal manifold  . Here   is the completed symmetric algebra,   is the suspension of a graded vector space, and   denotes the linear dual. Typically one describes   as the homotopy Lie algebra and   with the differential   as its representing commutative differential graded algebra.

Using this definition of a homotopy Lie algebra, one defines a morphism of homotopy Lie algebras,  , as a morphism   of their representing commutative differential graded algebras that commutes with the vector field, i.e.,  . Homotopy Lie algebras and their morphisms define a category.

Definition via multi-linear maps

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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   is a collection of symmetric multi-linear maps   of degree  , sometimes called the  -ary bracket, for each  . Moreover, the maps   satisfy the generalised Jacobi identity:

 

for each n. Here the inner sum runs over  -unshuffles and   is the signature of the permutation. The above formula have meaningful interpretations for low values of  ; for instance, when   it is saying that   squares to zero (i.e., it is a differential on  ), when   it is saying that   is a derivation of  , and when   it is saying that   satisfies the Jacobi identity up to an exact term of   (i.e., it holds up to homotopy). Notice that when the higher brackets   for   vanish, the definition of a differential graded Lie algebra on   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   which satisfy certain conditions.

Definition via operads

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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   operad.

(Quasi) isomorphisms and minimal models

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A morphism of homotopy Lie algebras is said to be a (quasi) isomorphism if its linear component   is a (quasi) isomorphism, where the differentials of   and   are just the linear components of   and  .

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  . 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

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Because  -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

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One of the approachable classes of examples of  -algebras come from the embedding of differential graded Lie algebras into the category of  -algebras. This can be described by   giving the derivation,   the Lie algebra structure, and   for the rest of the maps.

Two term L algebras

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In degrees 0 and 1

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One notable class of examples are  -algebras which only have two nonzero underlying vector spaces  . Then, cranking out the definition for  -algebras this means there is a linear map

 ,

bilinear maps

 , where  ,

and a trilinear map

 

which satisfy a host of identities.[4] pg 28 In particular, the map   on   implies it has a lie algebra structure up to a homotopy. This is given by the differential of   since the gives the  -algebra structure implies

 ,

showing it is a higher Lie bracket. In fact, some authors write the maps   as  , 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   then   has a structure of a Lie algebra from the induced map of  .

In degrees 0 and n

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In this case, for  , there is no differential, so   is a Lie algebra on the nose, but, there is the extra data of a vector space   in degree   and a higher bracket

 

It turns out this higher bracket is in fact a higher cocyle in Lie algebra cohomology. More specifically, if we rewrite   as the Lie algebra   and   and a Lie algebra representation   (given by structure map  ), then there is a bijection of quadruples

  where   is an  -cocycle

and the two-term  -algebras with non-zero vector spaces in degrees   and  .[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  -algebras in degrees   and   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  -algebra which can then be analyzed as before.

Example in degrees 0 and 1

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One simple example of a Lie-2 algebra is given by the  -algebra with   where   is the cross-product of vectors and   is the trivial representation. Then, there is a higher bracket   given by the dot product of vectors

 

It can be checked the differential of this  -algebra is always zero using basic linear algebra[4]pg 45.

Finite dimensional example

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Coming up with simple examples for the sake of studying the nature of  -algebras is a complex problem. For example,[5] given a graded vector space   where   has basis given by the vector   and   has the basis given by the vectors  , there is an  -algebra structure given by the following rules

 

where  . Note that the first few constants are

 

Since   should be of degree  , the axioms imply that  . There are other similar examples for super[6] Lie algebras.[7] Furthermore,   structures on graded vector spaces whose underlying vector space is two dimensional have been completely classified.[3]

See also

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References

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  1. ^ Lurie, Jacob. "Derived Algebraic Geometry X: Formal Moduli Problems" (PDF). p. 31, Theorem 2.0.2.
  2. ^ Pridham, Jonathan Paul (2012). "Derived deformations of schemes". Communications in Analysis and Geometry. 20 (3): 529–563. arXiv:0908.1963. doi:10.4310/CAG.2012.v20.n3.a4. MR 2974205.
  3. ^ a b Daily, Marilyn Elizabeth (2004-04-14).   Structures on Spaces of Low Dimension (PhD). hdl:1840.16/5282.
  4. ^ a b c Baez, John C.; Crans, Alissa S. (2010-01-24). "Higher-Dimensional Algebra VI: Lie 2-Algebras". Theory and Applications of Categories. 12: 492–528. arXiv:math/0307263.
  5. ^ Daily, Marilyn; Lada, Tom (2005). "A finite dimensional   algebra example in gauge theory". Homology, Homotopy and Applications. 7 (2): 87–93. doi:10.4310/HHA.2005.v7.n2.a4. MR 2156308.
  6. ^ Fialowski, Alice; Penkava, Michael (2002). "Examples of infinity and Lie algebras and their versal deformations". Banach Center Publications. 55: 27–42. arXiv:math/0102140. doi:10.4064/bc55-0-2. MR 1911978. S2CID 14082754.
  7. ^ Fialowski, Alice; Penkava, Michael (2005). "Strongly homotopy Lie algebras of one even and two odd dimensions". Journal of Algebra. 283 (1): 125–148. arXiv:math/0308016. doi:10.1016/j.jalgebra.2004.08.023. MR 2102075. S2CID 119142148.

Introduction

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In physics

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In deformation and string theory

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