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 in some sense. They have applications in deformation theory and rational homotopy theory.
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Definition
A differential graded Lie algebra is a graded vector space
the graded Jacobi identity:
and the graded Leibniz rule:
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:
Alternative equivalent definitions of a differential graded Lie algebra include:
- a Lie algebra object internal to the category of chain complexes;
- a strict
L ∞
A morphism of differential graded Lie algebras is a graded linear map
Products and coproducts
The product of two differential graded Lie algebras,
The coproduct of two differential graded Lie algebras,
Connection to deformation theory
The main application is to the deformation theory over fields of 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 differential graded Lie algebra.A Maurer-Cartan element is a degree -1 element,