Higher derived brackets and homotopy algebras

By T. Voronov. 2003

344 citations recorded by the source corpus.

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 Δ. Given this, we introduce an infinite sequence of higher brackets on the image of the projector, and explicitly calculate their Jacobiators in terms of Δ2. This allows to control higher Jacobi identities in terms of the “order” of Δ2. Examples include Stasheff's strongly homotopy Lie algebras and variants of homotopy Batalin–Vilkovisky algebras. There is a generalization with Δ replaced by an arbitrary odd derivation. We discuss applications and links with other constructions.

Subfields: Algebra and Number Theory, Geometry and Topology, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/ca2ad5f87bcf3d022ad153e6a05c9b43ad167e8e

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Citations in the corpus

n-ary algebras: a review with applications (2010, background): “13; see also [94] for a unified review and [95] for other aspects.”
Deformations of associative Rota-Baxter operators (2019, methodology): “[31] Th. Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl.”
Relative formality theorem and quantisation of coisotropic submanifolds (2005, background): “-structure whose structure maps are multiderivations, see Theorem 2.2. Algebraically, the P∞-brackets are obtained from the Poisson structure on Mas higher derived brackets in the sense of T. Voronov [Vo]. This L∞-structure induces the Poisson bracket oncohomology. AtthispointtheL∞-machinery canbeap”
Membrane sigma-models and quantization of non-geometric flux backgrounds (2012, methodology): “Following [85], we define the n-th derived bracket of Π as”
Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory (2003, background): “o remain a quasi-Lie bialgebroid. The world of the “quasi” structures which we explore here is certainly nothing but a small part of the realm of homotopy structures, L∞, G∞, etc. See, in particular, [53] and the articles of Stasheﬀ [49], Bangoura [7] and Huebschmann [21]. We hope to show that these”