Differential graded Lie algebra

In algebra, a differential graded Lie algebra, or dg Lie algebra, is a graded vector space $L=\bigoplus L_{i}$ over a field of characteristic zero together with a bilinear map $[,]:L_{i}\otimes L_{j}\to L_{i+j}$ and a differential $d:L_{i}\to L_{i-1}$ satisfying

$[x,y]=(-1)^{|x||y|+1}[y,x],$

the graded Jacobi identity:

$(-1)^{|x||z|}[x,[y,z]]+(-1)^{|y||x|}[y,[z,x]]+(-1)^{|z||y|}[z,[x,y]]=0,$

and the graded Leibniz rule:

$d[x,y]=[dx,y]+(-1)^{|x|}[x,dy]$

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.):

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

• Daniel Quillen, Rational Homotopy Theory

Further reading

• J. Lurie, Formal moduli problems, section 2.1

External links

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