Jacobitype identities in algebras and superalgebras
Abstract
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.
 Publication:

Theoretical and Mathematical Physics
 Pub Date:
 May 2014
 DOI:
 10.1007/s1123201401612
 arXiv:
 arXiv:1304.5050
 Bibcode:
 2014TMP...179..550L
 Keywords:

 associative algebra;
 associative superalgebra;
 Jacobi identity;
 symplectic supermanifold;
 Mathematical Physics;
 High Energy Physics  Theory
 EPrint:
 1+11 pages, no figures, title changed, published version