128.220.159.17: correcting typo

2013-11-11T16:01:30Z

correcting typo

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In algebra, a '''differential graded Lie algebra''', or '''dg Lie algebra''', is a [[graded vector space]] <math>L = \bigoplus L_i</math> over a field of characteristic zero together with a bilinear map <math>[,]: 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 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''.

The main application is to the deformation theory in the "characteristic zero" (in particular over the complex numbers.) The idea goes back to Quillen's work on [[rational homotopy theory]]. One way to formulate this thesis might be (due to Drinfeld, Feigin, Deligne, Kontsevich, et al.):<ref>Hinich, DG coalgebras as formal stacks {{arxiv|math|9812034}}</ref>
:Any reasonable formal deformation problem in characteristic zero can be described by Maurer–Cartan elements of an appropriate dg Lie algebra.

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

== References ==
{{reflist}}
*Daniel Quillen, ''Rational Homotopy Theory''

== Further reading ==
*J. 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}}

{{algebra-stub}}
[[Category:Differential algebra]]
[[Category:Lie algebras]]

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128.220.159.17: correcting typo