# Differential graded Lie algebra

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In algebra, a **differential graded Lie algebra**, or **dg 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:

and the graded Leibniz rule:

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.):^{[1]}

- Any reasonable formal deformation problem in characteristic zero can be described by Maurer–Cartan elements of an appropriate dg Lie algebra.

## See also

## References

- ↑ Hinich, DG coalgebras as formal stacks Template:Arxiv

- Daniel Quillen,
*Rational Homotopy Theory*

## Further reading

- J. Lurie, Formal moduli problems, section 2.1