Cohomology And Deformation Quantization Of Poisson Conformal Algebras

In this thesis,we first recall the notion of(noncommutative)Poisson conformal algebras and describe some constructions of them.Then we study the formal distribution(noncommutative)Poisson algebras and coefficient(noncommutative)Poisson algebras.Next,we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits.At last,we develop the cohomology theory of noncommutative Poisson conformal algebras and use this cohomology to study their deformations.This thesis consists of five chapters.In Chapter 1,we introduce the backgrounds and recent developments of(noncommutative)Poisson conformal algebras and describe the main results in this thesis.In Chapter 2,we first recall the notion of(noncommutative)Poisson conformal algebras and quadratic Lie conformal algebras.Then we introduce the notion of a quadratic(noncommutative)Poisson conformal algebra.In Chapter 3,we study the formal distribution(noncommutative)Poisson algebras and coefficient(noncommutative)Poisson algebras.The former give some constructions of(noncommutative)Poisson conformal algebras in this thesis,while the later are useful to study infinite-dimensional(noncommutative)Poisson algebras.In Chapter 4,we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits and the ?-adic associative conformal algebras are deformation quantization of Poisson conformal algebras.Furthermore,we study conformal infinitesimal deformations and extensions of conformal n-deformations to conformal(n + 1)-deformations of a commutative associative conformal algebra.In Chapter 5,we study the Flato-Gerstenhaber-Voronov cohomology theory for noncommutative Poisson conformal algebra associated to a Poisson conformal module.And we use cohomology of noncommutative Poisson conformal algebras to study their linear deformations.