View source for Differential graded Lie algebra

{{Redirect|dgla|the fatty acid|Dihomo-γ-linolenic acid}}

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 applications in [[deformation theory]]<ref name="Hinich1998">{{cite journal|last=Hinich |first=Vladimir |title=DG coalgebras as formal stacks|journal = [[Journal of Pure and Applied Algebra]] | volume= 162| year=2001|issue =2–3 | pages=209–250| mr= 1843805| doi=10.1016/S0022-4049(00)00121-3 |arxiv=math/9812034|s2cid=15720862 }}</ref> and [[rational homotopy theory]].

== Definition ==

A '''differential graded Lie algebra''' is a graded vector space <math>L = \bigoplus L_i</math> over a [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] zero together with a bilinear map <math>[\cdot,\cdot]\colon L_i \otimes L_j \to L_{i+j}</math> and a differential <math>d: L_i \to L_{i-1}</math> satisfying

:<math>[x,y] = (-1)^{|x||y|+1}[y,x],</math>

the graded [[Jacobi identity]]:

:<math>(-1)^{|x||z|}[x,[y,z]] +(-1)^{|y||x|}[y,[z,x]] +(-1)^{|z||y|}[z,[x,y]] = 0,</math>

and the graded [[product rule|Leibniz rule]]:

:<math>d [x,y] = [d x,y] + (-1)^{|x|}[x, d y]</math>

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 [[homology (mathematics)|homologically]] graded. If instead the differential raised degree the differential graded Lie algebra is said to be [[cohomology|cohomologically]] graded (usually to reinforce this point the grading is written in superscript: <math>L^i</math>). 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 <math>L^i=L_{-i}</math>.

Alternative equivalent definitions of a differential graded Lie algebra include:

# a Lie algebra object internal to the [[Chain_complex#Category_of_chain_complexes|category of chain complexes]];
# a strict <math>L_\infty</math>-algebra.

A [[morphism]] of differential graded Lie algebras is a [[graded linear map]] <math>f:L\to L^\prime</math> that commutes with the bracket and the differential, i.e., <math>f [x,y]_{L} = [f(x),f(y)]_{L^\prime}</math> and <math>f (d_L x) = d_{L^\prime} f (x)</math>. Differential graded Lie algebras and their morphisms define a [[category (mathematics)|category]].

== Products and coproducts ==

The [[product (category theory)|product]] of two differential graded Lie algebras, <math>L\times L^\prime</math>, is defined as follows: take the [[direct sum]] of the two graded vector spaces <math>L\oplus L^\prime </math>, and equip it with the bracket <math>[(x,x^\prime),(y,y^\prime)]=([x,y],[x^\prime,y^\prime])</math> and differential <math>D(x,x^\prime ) = (dx,d^\prime x^\prime)</math>.

The [[coproduct]] of two differential graded Lie algebras, <math>L*L^\prime</math>, is often called the free product. It is defined as the free graded Lie algebra on the two underlying [[vector space]]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]] over [[Field (mathematics)|fields]] of characteristic zero (in particular over the [[complex number]]s.) The idea goes back to [[Daniel Quillen]]'s work on [[rational homotopy theory]]. One way to formulate this thesis  (due to [[Vladimir Drinfeld]], [[Boris Feigin]], [[Pierre Deligne]], [[Maxim Kontsevich]], and others) might be:<ref name="Hinich1998"/>
: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, <math> x\in L_{-1}</math>, that is a solution to the [[Maurer–Cartan form#Maurer–Cartan form on a homogeneous space|Maurer–Cartan equation]]:

:<math> dx +\frac{1}{2}[x,x]=0.</math>

== See also ==
*[[Differential graded algebra]] (DGA)
*[[Simplicial Lie algebra]]
*[[Homotopy Lie algebra]]

== References ==
{{reflist}}
* {{citation|mr=0258031|first=Daniel |last=Quillen|authorlink=Daniel Quillen| title=Rational homotopy theory|journal=[[Annals of Mathematics]]| volume=90|year=1969|pages=205–295| doi=10.2307/1970725|jstor=1970725|issue=2}}

== Further reading ==
* [[Jacob Lurie]], [http://www.math.harvard.edu/~lurie/papers/DAG-X.pdf Formal moduli problems], section 2.1

== External links ==
*{{nlab |id=differential+graded+Lie+algebra |title=differential graded Lie algebra}}
*{{nlab |id=model+structure+on+dg-Lie+algebras |title=model structure on dg Lie algebras}}

[[Category:Differential algebra]]
[[Category:Lie algebras]]