Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 053, 20 pages      arXiv:1509.09032

Universal Lie Formulas for Higher Antibrackets

Marco Manetti a and Giulia Ricciardi bc
a) Dipartimento di Matematica ''Guido Castelnuovo'', Università degli studi di Roma La Sapienza, P. le Aldo Moro 5, I-00185 Roma, Italy
b) Dipartimento di Fisica ''E. Pancini'', Università degli studi di Napoli Federico II, Complesso Universitario di Monte Sant'Angelo, Via Cintia, I-80126 Napoli, Italy
c) INFN, Sezione di Napoli, Complesso Universitario di Monte Sant'Angelo, Via Cintia, I-80126 Napoli, Italy

Received November 17, 2015, in final form May 31, 2016; Published online June 06, 2016

We prove that the hierarchy of higher antibrackets (aka higher Koszul brackets, aka Koszul braces) of a linear operator $\Delta$ on a commutative superalgebra can be defined by some universal formulas involving iterated Nijenhuis-Richardson brackets having as arguments $\Delta$ and the multiplication operators. As a byproduct, we can immediately extend higher antibrackets to noncommutative algebras in a way preserving the validity of generalized Jacobi identities.

Key words: Lie superalgebras; higher brackets.

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