Geodesic
Jacobi-type identities in algebras and superalgebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 179 (2014) no. 2, pp. 196-206

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We introduce two remarkable identities written in terms of single commutators and anticommutators for any three elements of an arbitrary associative algebra. One is a consequence of the other (fundamental identity). From the fundamental identity, we derive a set of four identities (one of which is the Jacobi identity) represented in terms of double commutators and anticommutators. We establish that two of the four identities are independent and show that if the fundamental identity holds for an algebra, then the multiplication operation in that algebra is associative. We find a generalization of the obtained results to the super case and give a generalization of the fundamental identity in the case of arbitrary elements. For nondegenerate even symplectic (super)manifolds, we discuss analogues of the fundamental identity.
Keywords: associative algebra, associative superalgebra, Jacobi identity, symplectic supermanifold. 
@article{TMF_2014_179_2_a2,
     author = {P. M. Lavrov and O. V. Radchenko and I. V. Tyutin},
     title = {Jacobi-type identities in algebras and superalgebras},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {196--206},
     publisher = {mathdoc},
     volume = {179},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2014_179_2_a2/}
}
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P. M. Lavrov; O. V. Radchenko; I. V. Tyutin. Jacobi-type identities in algebras and superalgebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 179 (2014) no. 2, pp. 196-206. http://geodesic.mathdoc.fr/item/TMF_2014_179_2_a2/