Quantizations of transposed Poisson algebras by Novikov deformations
Siyuan Chen, Chengming Bai
TL;DR
This work develops a deformation-quantization framework for transposed Poisson algebras by introducing Novikov deformations of commutative associative algebras and analyzing their classical limits. It shows that these classical limits lie in a quantizable subclass of transposed Poisson algebras and defines quantizations of transposed Poisson algebras via Novikov deformations, with a focus on Novikov-Poisson type algebras. A key result is that all transposed Poisson algebras of Novikov-Poisson type, including unital ones, admit quantizations, and the authors provide a natural construction ensuring this. The paper culminates with a complete classification of quantizations for 2-dimensional complex transposed Poisson algebras with non-abelian Lie brackets, giving explicit deformation families and equivalence criteria that illuminate the structure of quantizations in low dimensions and guide future higher-dimensional studies.
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.