Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket
| Authors | |
|---|---|
| Year of publication | 2007 |
| Type | Article in Periodical |
| Magazine / Source | Communications in Mathematical Physics |
| MU Faculty or unit | |
| Citation | |
| Web | http://www.arxiv.org/abs/hep-th/0603116 |
| Doi | http://dx.doi.org/10.1007/s00220-007-0278-3 |
| Field | Theoretical physics |
| Keywords | Batalin-Vilkovisky Algebra; Homotopy Lie Algebra; Koszul Bracket; Derived Bracket; Courant Bracket. |
| Description | 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 poly-vectors, and give closed formulas for the higher Courant brackets. |
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