In
mathematics, in particular
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
and
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a homotopy Lie algebra (or
-algebra) is a generalisation of the concept of a
differential graded Lie algebra
In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible. Such objects have appl ...
. To be a little more specific, the
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
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. This was later extended to all characteristics by Jonathan Pridham.
Homotopy Lie algebras have applications within mathematics and
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
; 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 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
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
.
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 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
In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible. Such objects have appl ...
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
There also exists a more abstract definition of a homotopy algebra using the theory of operad
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one define ...
s: 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
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
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
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
In degrees 0 and 1
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. 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_,
showing it is a higher Lie bracket. In fact, some authors write the maps as ,
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
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
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to p ...
. 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 .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
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
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
of vectors
:
It can be checked the differential of this -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 -algebras is a complex problem. For example, 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 Lie algebras. Furthermore, structures on graded vector spaces whose underlying vector space is two dimensional have been completely classified.
See also
*Homotopy associative algebra In mathematics, an algebra such as (\R,+,\cdot) has multiplication \cdot whose associativity is well-defined on the nose. This means for any real numbers a,b,c\in \R we have
:a\cdot(b\cdot c) - (a\cdot b)\cdot c = 0.
But, there are algebras R which ...
* Differential graded algebra
* BV formalism
*Simplicial Lie algebra In algebra, a simplicial Lie algebra is a simplicial object in the category of Lie algebras. In particular, it is a simplicial abelian group, and thus is subject to the Dold–Kan correspondence
In mathematics, more precisely, in the theory of si ...
*Hochschild homology
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, ...
* 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
*
In physics
*
* — Towards classification of perturbative gauge invariant classical fields.
In deformation and string theory
*
*
*
Related ideas
* (Lie algebras in the derived category of coherent sheaves.)
External links
*{{cite web , url=https://www.mpim-bonn.mpg.de/node/8756 , title=Learning seminar on deformation theory , year=2018 , publisher=Max Planck Institute for Mathematics Discusses deformation theory in the context of -algebras.
Homotopical algebra
Differential algebra
Lie algebras