Structures And Representations Of Lie Conformal Algebras Of Galilean Type

In this thesis,we study the structure and representation theories of Lie conformal algebras of planar Galilean type PG(a,b)and pg(a,b),and Galilean-Virasoro type GD(a,b)and gd(a,b),where a,b are complex parameters.Here,PG(a,b)and GD(a,b)are infinite rank loop Lie conformal algebras,while pg(a,b)and gd(a,b)are finite rank Lie conformal algebras.In Chapter 1,we introduce the research background and the main results of this thesis.The structure of Lie conformal algebra was given by algebraist V.Kac[1]in 1996 when studying conformal fields theory.This new algebra structure can be viewed as a generalization of Lie algebra[2].There exist conformal versions of Lie theorem and Engel theorem of the Lie algebra theory.However,there does not exist conformal version of Levi theorem,which leads that the theory of non-semisimple Lie conformal algebras becomes more complicated.Both the two class of Lie conformal algebras of Galilean type studied in this thesis are non-semisimple,and have close relation with the well-known Virasoro Lie algebra.So,it is very interesting to study their structure and representation theories.In Chapter 2,we mainly study the structure and representation theories of infinite Lie conformal algebras PG(a,b)of planar Galilean type.We first determine the conformal derivations of PG(a,b),and show that all their conformal derivations are inner.Then,we classify the rank one conformal modules over PG(a,b).We also give the results of finite Lie conformal algebras pg(a,b)of planar Galilean type.Finally,we classify the Z-graded free intermediate series modules over PG(a,b).In Chapter 3,we study the representation theory of finite Lie conformal algebras pg(a,b)of planar Galilean type.We first prove a technical lemma on the representations of certain subquotient algebra of the annihilation algebra of pg(a,b).Then,we give a complete classification of finite irreducible conformal modules over pg(a,b)by showing that they must be free of rank one.Finally,we compare with the Euclidean group E(2),and discuss related density modules.In Chapter 4,we mainly study the structure and representation theories of infinite Lie conformal algebras GD(a,b)of Galilean-Virasoro type.We first determine the conformal derivations of GD(a,b),and show that all their conformal derivations are inner.Then,we classify the rank one conformal modules over GD(a,b).We also give the results of finite Lie conformal algebras gd(a,b)of Galilean-Virasoro type.Finally,we classify the Z-graded free intermediate series modules over GD(a,b),and compare our results with those for Heisenberg-Virasoro type Lie conformal algebras and give a summary.In Chapter 5,we study the representation theory of finite Lie conformal algebras gd(a,b)of Galilean-Virasoro type.We first give a complete classification of finite irreducible conformal modules over g d(a,b)by showing that they must be free of rank one.Here the key step is to classify the representations of certain subquotient algebra of the annihilation algebra of gd(a,b).Then,we study the extension problem for the finite irreducible conformal modules over gd(a,b).As a byproduct,new indecomposable conformal modules over gd(a,b)are obtained.In Chapter 6,we discuss some problems for future research.On one hand,based on the classification results on finite irreducible conformal modules over finite Lie conformal algebras pg(a,b)of planar Galilean type in Chapter 3,we will consider the extension problem for these irreducible conformal modules.On the other hand,since the theory of Lie conformal superalgebras are much more complicated and interesting than the theory of Lie conformal algebras,it deserves to construct and investigate super analogues of Lie conformal algebras of Galilean type studied in this thesis.There are totaly 6 tables and 96 references in this thesis.