Deformation Quantization of Nonassociative Algebras

We investigate formal deformations of certain classes of nonassociative algebras including classes of K[Σ3]-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 algebras (A,μ) a type of algebras (A,μ,ψ) such that formal deformations of (A,μ) appear as quantizations of (A,μ,ψ). The process of polarization/depolarization associates to each nonassociative algebra a couple of algebras which products are respectively commutative and skew-symmetric and it is linked with the algebra obtained from the formal deformation. The anti-associative case is developed with a link with the Jacobi–Jordan algebras.