Research On Two Classes Of Algebras Related To Virasoro Algebras

Generalized loop planar Galilean conformal algebra and super BMS₃algebras are algebras generated by Virasoro algebras.A centreless Virasoro algebra and a class of primitives generate a Heisenberg Virasoro algebra by a semi-direct product,and then a Heisenberg-Virasoro algebra and two classes of primitives generate a planar Galilean con-formal algebra by a semi-direct product.We change the corner value of planar Galilean conformal algebra from integer to algebraic closed field of characteristic zero,which be-comes generalized planar Galilean conformal algebra,and the generalized loop planar Galilean conformal algebra is a tensor product of the generalized planar Galilean confor-mal algebra and Laurent polynomial algebra.W（2,2）algebras are also generated from centreless Virasoro algebras and a class of primitives.This is the even part of BMS₃al-gebra.Planar Galilean conformal algebra is closely related to non relativity.Affine Lie algebra plays an important role in string theory and two-dimensional algebra,and loop algebra is closely related to affine Lie algebra.A centerless Super BMS₃algebra is a kind of infinite dimensional Lie superalgebra,whose even part is a centreless W（2,2）algebra.Super BMS₃is introduced to study the classification of vertex operator algebras generated by vectors with weight 2.There are two forms:Ramond type and Neveu Schwarz type.In recent years,some researchers have studied its Lie superalgebra structure and repre-sentation theory.In this dissertation,we mainly investigate the structure,representation,Lie bialgebra and quantization of the generalized loop planar Galilean conformal algebra,and we also research quantization of super BMS₃algebra.For the structure of generalized loop planar Galilean conformal algebra,we describe their specific forms of its derivation,second-order cohomology group and automorphism group.We prove that its non-zero derivative is an inner derivative,then give its 0 deriva-tive by algebraic properties,and finally characterize its specific form.Then,we obtain that the expression of second cohomology group and its second cohomology group can be divided into three categories.In addition,through some conclusions of the automor-phism group of the generalized loop Virasoro algebra,we prove that its automorphism group can be expressed as the direct product（and semi-direct product）of some groups.We also obtain that the structure of Lie bialgebras of the generalized loop planar Galilean conformal algebra and super BMS₃are all triangular coboundary.For representation of generalized loop planar Galilean conformal algebra,including its two types of weighted modules and non weighted modules,mainly Harish Chandra modules,Verma modules,Whittaker modules and free modules,we obtain that the Har-ish Chandra module on generalized loop planar Galilean conformal algebra is the highest weight module or the lowest weight module or the intermediate sequence module,and then characterize its intermediate sequence module.In addition,we also discuss the re-ducibility of the Verma module of generalized loop planar Galilean conformal algebra,give some properties of the Whittaker module of generalized loop planar Galilean confor-mal algebra,and some conclusions of the free module of generalized loop planar Galilean conformal algebra.The quantization of generalized loop planar Galilean conformal algebras and super BMS₃algebras are also studied.It is proved that there exists a noncommutative and noncocommutative Hopf algebra structure and that this Hopf algebra structure keeps the multiplication and counit unchanged.We also characterize the specific forms of product and counit.