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

TL;DR

This paper explores advanced algebraic structures, introducing non-commutative higher brackets, their homotopy Lie algebra properties, and extending the Courant bracket to a broader algebraic context with explicit formulas.

Contribution

It presents a novel non-commutative generalization of higher Koszul brackets and extends the concept of homotopy Lie algebras to include these structures, also providing formulas for higher Courant brackets.

Findings

Higher Koszul brackets form a homotopy Lie algebra

Derived brackets satisfy a generalized Jacobi identity with Bernoulli number coefficients

Extended Courant bracket to the big bracket with explicit formulas

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.