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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 groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
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 ho ...
, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a
graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be ...
with added
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
and
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
structures that are compatible. Such objects have applications in
deformation theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesi ...
and
rational homotopy theory
In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homo ...
.
Definition
A differential graded Lie algebra is a graded vector space
over a
field of
characteristic zero together with a bilinear map
and a differential
satisfying
:
the graded
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 assoc ...
:
:
and the graded product rule">Leibniz rule
Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following:
* Product rule in differential calculus
* General Leibniz rule, a generalization of the product rule
* Leibniz integral rule
* The alternating series test, al ...
:
:
for any homogeneous elements ''x'', ''y'' and ''z'' in ''L''. Notice here that the differential lowers the degree and so this differential graded Lie algebra is considered to be
homologically graded. If instead the differential raised degree the differential graded Lie algebra is said to be
cohomologically graded (usually to reinforce this point the grading is written in superscript:
). The choice of cohomological grading usually depends upon personal preference or the situation as they are equivalent: a homologically graded space can be made into a cohomological one via setting
.
Alternative equivalent definitions of a differential graded Lie algebra include:
# a Lie algebra object internal to the
category of chain complexes
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
;
# a strict
-algebra.
A
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...
of differential graded Lie algebras is a
graded linear map that commutes with the bracket and the differential, i.e.,
and
. Differential graded 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) ...
.
Products and coproducts
The
product of two differential graded Lie algebras,
, is defined as follows: take the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
of the two graded vector spaces
, and equip it with the bracket
and differential
.
The
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...
of two differential graded Lie algebras,
, is often called the free product. It is defined as the free graded Lie algebra on the two underlying
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s with the unique differential extending the two original ones modulo the relations present in either of the two original Lie algebras.
Connection to deformation theory
The main application is to the
deformation theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesi ...
over
fields of characteristic zero (in particular over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s.) The idea goes back to
Daniel Quillen
Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 19 ...
's work on
rational homotopy theory
In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homo ...
. One way to formulate this thesis (due to
Vladimir Drinfeld,
Boris Feigin
Boris Lvovich Feigin (russian: Бори́с Льво́вич Фе́йгин) (born 20 November 1953) is a Russian mathematician. His research has spanned representation theory, mathematical physics, algebraic geometry, Lie groups and Lie algebras ...
,
Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
,
Maxim Kontsevich
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques a ...
, and others) might be:
:Any reasonable formal deformation problem in characteristic zero can be described by Maurer–Cartan elements of an appropriate differential graded Lie algebra.
A Maurer-Cartan element is a degree −1 element,
, that is a solution to the
Maurer–Cartan equation:
:
See also
*
Differential graded algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
__TOC__
Definition
A differential graded a ...
(DGA)
*
Simplicial Lie algebra
*
Homotopy Lie algebra In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or 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 ...
References
*
Further reading
*
Jacob Lurie
Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. Lurie is a 2014 MacArthur Fellow.
Life
When he was a student in the Science, Mathematics, and Computer Science ...
Formal moduli problems section 2.1
External links
*
*{{nlab , id=model+structure+on+dg-Lie+algebras , title=model structure on dg Lie algebras
Differential algebra
Lie algebras