Deformations and Abelian Extensions on Anti-Pre-Jacobi-Jordan Algebras

Abstract

The underlying algebraic structures of nondegenerate commutative 2-cocycles on Jacobi-Jordan algebras are introduced and are called anti-pre-Jacobi-Jordan algebras. It is proved that they belong to the class of Jacobi-Jordan-admissible algebras. Moreover, it is proved that the negative left multiplication operators of anti-pre-Jacobi-Jordan algebras are representations of the corresponding anticommutator Jacobi-Jordan algebras. Representations of anti-pre-Jacobi-Jordan algebras are also introduced and the second cohomology space of these algebras is given. We introduce the notion of abelian extensions and show that they are classified by the second cohomology space. Next, we study linear deformations of anti-pre-Jacobi-Jordan algebras. The notion of a Nijenhuis operator on an anti-pre-Jacobi-Jordan algebra is introduced and it generates a trivial linear deformation of an anti-pre-Jacobi-Jordan algebra. Moreover, we study formal deformations of anti-pre-Jacobi-Jordan algebras and we show that the infinitesimal of a formal deformation is a 2-cocycle with the coefficients in the regular representation and depends only on its cohomology class. Finally, we prove that, a necessary condition for the rigidity of the anti-pre-Jacobi-Jordan algebra is the nullity of its second cohomology space with coefficients in the regular representation.