In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or -algebra) is a generalisation of the concept of a differential graded Lie algebra. To be a little more specific, the Jacobi identity only holds up to homotopy. Therefore, a differential graded Lie algebra can be seen as a homotopy Lie algebra where the Jacobi identity holds on the nose. These homotopy algebras are useful in classifying deformation problems over characteristic 0 in deformation theory because deformation functors are classified by quasi-isomorphism classes of -algebras.[1] This was later extended to all characteristics by Jonathan Pridham.[2]
Homotopy Lie algebras have applications within mathematics and mathematical physics; they are linked, for instance, to the Batalin–Vilkovisky formalism much like differential graded Lie algebras are.
^Pridham, Jonathan Paul (2012). "Derived deformations of schemes". Communications in Analysis and Geometry. 20 (3): 529–563. arXiv:0908.1963. doi:10.4310/CAG.2012.v20.n3.a4. MR 2974205.
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