Higher derived brackets and homotopy algebras

Details Higher derived brackets and homotopy algebras Higher derived brackets and homotopy algebras

2003-04-03

arxiv.org/abs/math/0304038v3

Mathematics

Quantum Algebra

22 pages; LaTeX 2e. New version included a generalization, the higher derived brackets generated by a not necessarily inner de

Scientific paper

We give a construction of homotopy algebras based on ``higher derived brackets''. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element $\Delta$. Given this, we introduce an infinite sequence of higher brackets on the image of the projector, and explicitly calculate their Jacobiators in terms of $\Delta^2$. This allows to control higher Jacobi identities in terms of the ``order'' of $\Delta^2$. Examples include Stasheff's strongly homotopy Lie algebras and variants of homotopy Batalin--Vilkovisky algebras. There is a generalization with $\D$ replaced by an arbitrary odd derivation. We discuss applications and links with other constructions.

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