Deformation quantization of nonassociative algebras

TL;DR

This paper explores formal deformation quantization of various nonassociative algebras, establishing a framework similar to Poisson algebra for associatives, and linking deformations to algebraic structures like Jacobi-Jordan algebras.

Contribution

It introduces a deformation quantization process for nonassociative algebras, extending concepts from associative cases and connecting to Jacobi-Jordan algebra structures.

Findings

Deformation quantization framework for nonassociative algebras established.

Identification of algebraic structures linked to formal deformations.

Connection between anti-associative algebras and Jacobi-Jordan algebras.

Abstract

We investigate formal deformations of certain classes of nonassociative algebras including classes of K[{\Sigma}3]-associative algebras, Lie-admissible algebras and anti-associative algebras. In a process which is similar to Poisson algebra for the associative case we identify for each type of algebra (A, {\mu}), an algebra (A, {\mu}, {\psi}) such that the formal deformation (A[[t]], {\mu}t) is the quantization deformation of (A, {\mu}, {\psi}). The process of polarization/depolarization associate to each nonassociative algebra a couple of algebras which products are respectively commutative and skew-symmetric and is linked with the algebra obtained from the formal deformation. The anti-associative case is developed with a link with the Jacobi-Jordan algebras