Quantizations of Transposed Poisson algebras by Novikov deformations

Abstract

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.