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

SIGMA 12 (2016), 053, 20 pages      arXiv:1509.09032      https://doi.org/10.3842/SIGMA.2016.053

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

Abstract
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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References

  1. Akman F., On some generalizations of Batalin-Vilkovisky algebras, J. Pure Appl. Algebra 120 (1997), 105-141, q-alg/9506027.
  2. Alfaro J., Damgaard P.H., Non-abelian antibrackets, Phys. Lett. B 369 (1996), 289-294, hep-th/9511066.
  3. Aschenbrenner M., Logarithms of iteration matrices, and proof of a conjecture by Shadrin and Zvonkine, J. Combin. Theory Ser. A 119 (2012), 627-654, arXiv:1009.5518.
  4. Bandiera R., Nonabelian higher derived brackets, J. Pure Appl. Algebra 219 (2015), 3292-3313, arXiv:1304.4097.
  5. Bandiera R., Formality of Kapranov's brackets in Kähler geometry via pre-Lie deformation theory, Int. Math. Res. Not., to appear, arXiv:1307.8066.
  6. Batalin I., Marnelius R., Quantum antibrackets, Phys. Lett. B 434 (1998), 312-320, hep-th/9805084.
  7. Batalin I.A., Vilkovisky G.A., Gauge algebra and quantization, Phys. Lett. B 102 (1981), 27-31.
  8. Batalin I.A., Vilkovisky G.A., Closure of the gauge algebra, generalized Lie equations and Feynman rules, Nuclear Phys. B 234 (1984), 106-124.
  9. Batalin I.A., Vilkovisky G.A., Existence theorem for gauge algebra, J. Math. Phys. 26 (1985), 172-184.
  10. Becchi C., Rouet A., Stora R., The abelian Higgs-Kibble model, unitarity of the $S$-operator, Phys. Lett. B 52 (1974), 344-346.
  11. Becchi C., Rouet A., Stora R., Renormalization of the abelian Higgs-Kibble model, Comm. Math. Phys. 42 (1975), 127-162.
  12. Becchi C., Rouet A., Stora R., Renormalization of gauge theories, Ann. Physics 98 (1976), 287-321.
  13. Bering K., Non-commutative Batalin-Vilkovisky algebras, homotopy Lie algebras and the Courant bracket, Comm. Math. Phys. 274 (2007), 297-341, hep-th/0603116.
  14. Bering K., Damgaard P.H., Alfaro J., Algebra of higher antibrackets, Nuclear Phys. B 478 (1996), 459-503, hep-th/9604027.
  15. Dotsenko V., Shadrin S., Vallette B., Pre-Lie deformation theory, Mosc. Math. J., to appear, arXiv:1502.03280.
  16. Ecalle J., Théorie des invariants holomorphes, Publication Mathématiques d'Orsay, 1974, available at http://sites.mathdoc.fr/PMO/PDF/E_ECALLE_67_74_09.pdf.
  17. Fiorenza D., Manetti M., Formality of Koszul brackets and deformations of holomorphic Poisson manifolds, Homology Homotopy Appl. 14 (2012), 63-75, arXiv:1109.4309.
  18. Getzler E., Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994), 265-285, hep-th/9212043.
  19. Hanna P.D., Sequence A180609, The On-line Encyclopedia of Integer Sequences, 2010, available at http://oeis.org.
  20. Koszul J.L., Crochet de Schouten-Nijenhuis et cohomologie, Astérisque (1985), Numero Hors Serie, 257-271.
  21. Kravchenko O., Deformations of Batalin-Vilkovisky algebras, in Poisson Geometry (Warsaw, 1998), Banach Center Publ., Vol. 51, Polish Acad. Sci., Warsaw, 2000, 131-139, math.QA/9903191.
  22. Lada T., Markl M., Strongly homotopy Lie algebras, Comm. Algebra 23 (1995), 2147-2161, hep-th/9406095.
  23. Lada T., Stasheff J., Introduction to SH Lie algebras for physicists, Internat. J. Theoret. Phys. 32 (1993), 1087-1103, hep-th/9209099.
  24. Manetti M., Uniqueness and intrinsic properties of non-commutative Koszul brackets, J. Homotopy Relat. Struct., to appear, arXiv:1512.05480.
  25. Markl M., Higher braces via formal (non)commutative geometry, in Geometric Methods in Physics, Trends in Mathematics, Springer International Publishing, 2015, 67-81, arXiv:1411.6964.
  26. Markl M., On the origin of higher braces and higher-order derivations, J. Homotopy Relat. Struct. 10 (2015), 637-667, arXiv:1309.7744.
  27. Nijenhuis A., Richardson Jr. R.W., Deformations of Lie algebra structures, J. Math. Mech. 17 (1967), 89-105.
  28. Shadrin S., Zvonkine D., Changes of variables in ELSV-type formulas, Michigan Math. J. 55 (2007), 209-228, math.AG/0602457.
  29. Sloane N.A.J., Sequences A134242 and A134243, The On-line Encyclopedia of Integer Sequences, 2010, available at http://oeis.org.
  30. Tyutin I.V., Gauge invariance in field theory and statistical physics in operator formalism, arXiv:0812.0580.

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