Quantizations of transposed Poisson algebras by Novikov deformations

TL;DR

This paper introduces Novikov deformations as a form of quantization for transposed Poisson algebras, establishing their relationship with Novikov-Poisson algebras and classifying certain low-dimensional cases.

Contribution

It defines Novikov deformation as a quantization method for transposed Poisson algebras and classifies 2-dimensional non-abelian cases.

Findings

Novikov deformation acts as a quantization of transposed Poisson algebras.

All transposed Poisson algebras of Novikov-Poisson type can be quantized.

Classification of 2-dimensional complex transposed Poisson algebras with non-abelian Lie brackets.

Abstract

The notions of the Novikov deformation of a commutative associative algebra and the corresponding classical limit are introduced. We show such a classical limit belongs to a subclass of transposed Poisson algebras, and hence the Novikov deformation is defined to be the quantization of the corresponding transposed Poisson algebra. As a direct consequence, we revisit the relationship between transposed Poisson algebras and Novikov-Poisson algebras due to the fact that there is a natural Novikov deformation of the commutative associative algebra in a Novikov-Poisson algebra. Hence all transposed Poisson algebras of Novikov-Poisson type, including unital transposed Poisson algebras, can be quantized. Finally, we classify the quantizations of $2$-dimensional complex transposed Poisson algebras in which the Lie brackets are non-abelian up to equivalence.