differential graded Lie algebra

A differential graded Lie algebra is a graded vector space $L\; =\; \backslash bigoplus\; L\_i$ over a field of characteristic zero together with a bilinear map $[\backslash cdot,\backslash cdot]\backslash colon\; L\_i\; \backslash otimes\; L\_j\; \backslash to\; L\_\{i+j\}$ and a differential $d:\; L\_i\; \backslash to\; L\_\{i-1\}$ satisfying

:$[x,y]\; =\; (-1)^$

and the graded Leibniz rule:

:$d\; [x,y]\; =\; [d\; x,y]\; +\; (-1)^$

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: $L^i$). 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 $L^i=L\_\{-i\}$.

Alternative equivalent definitions of a differential graded Lie algebra include:

1. a Lie algebra object internal to the category of chain complexes;
2. a strict $L\_\backslash infty$-algebra.

A morphism of differential graded Lie algebras is a graded linear map $f:L\backslash to\; L^\backslash prime$ that commutes with the bracket and the differential, i.e., $f\; [x,y]\_\{L\}\; =\; [f(x),f(y)]\_\{L^\backslash prime\}$ and $f\; (d\_L\; x)\; =\; d\_\{L^\backslash prime\}\; f\; (x)$. Differential graded Lie algebras and their morphisms define a category.

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

The coproduct of two differential graded Lie algebras, $L*L^\backslash prime$, is often called the free product. It is defined as the free graded Lie algebra on the two underlying vector spaces with the unique differential extending the two original ones modulo the relations present in either of the two original Lie algebras.

The main application is to the deformation theory over fields of characteristic zero (in particular over the complex numbers.) 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:

: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, $x\backslash in\; L\_\{-1\}$, that is a solution to the Maurer–Cartan equation:

:$dx\; +\backslash frac\{1\}\{2\}[x,x]=0.$

• Differential graded algebra (DGA)
• Simplicial Lie algebra
• Homotopy Lie algebra

{{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}}

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

• {{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