Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket

K. Bering

doi:10.1007/s00220-007-0278-3

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arXiv · hep-th/0603116

Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket

K. Bering

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Abstract

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent Δoperator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie brackets with a fixed nilpotent Lie algebra element Q. We find the most general Jacobi-like identity that such a hierarchy satisfies. The numerical coefficients in front of each term in these generalized Jacobi identities are related to the Bernoulli numbers. We suggest that the definition of a homotopy Lie algebra should be enlarged to accommodate this important case. Finally, we consider the Courant bracket as an example of a derived bracket. We extend it to the "big bracket" of exterior forms and multi-vectors, and give closed formulas for the higher Courant brackets.

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BibTeX RIS

K. Bering. 2007-01-30. Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket. https://doi.org/10.1007/s00220-007-0278-3

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