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

Source: [Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket](<https://www.sota2.com/research/paper/68fa227e7a2b558e2933c11e>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T04:48:25.638Z

Authors: K. Bering

Published: 2006-03-15T16:28:19.000Z

## Abstract

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent \\Delta operator, 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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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/hep-th/0603116>)

[PDF](<https://arxiv.org/pdf/hep-th/0603116.pdf>)