Structures And Representations Of A Class Of Non-simple Lie Conformal Superalgebras

In this thesis,we study the representation theory of a class of infinite rank non-simple Lie conformal superalgebra S(a,b)and the structure and representation theory of its loop Lie conformal superalgebra S(a,b),where a,b are complex parameters.In Chapter 1,we introduce the research background and the main results of this thesis.In Chapter 2,we introduce the concepts and theoretical knowledge of Lie conformal superalgebra.We review some basic definitions of Lie conformal superalgebras.We also review the results of the representation theory of Virasoro conformal algebra.In Chapter 3,we mainly study the representations of S(a,b).Since S(a,b)contains Virasoro subalgebra,S(a,b)can be regarded as super deformation of Heisenberg-Virasoro type Lie conformal algebras.In this chapter,we use the results of Virasoro conformal algebra to roughly classify the free rank(1+1)conformal modules and finite irreducible conformal modules over S(a,b).In our proof,we employ a Cheng-Kac’s lemma and an important known conclusion:the conformal module of a Lie conformal superalgebra can be considered as the conformal module of its extended annihilation algebra.In Chapter 4,we study conformal derivations and representations of a class of loop Lie conformal superalgebra S(a,b),where a,b are complex parameters.First,we determine conformal derivations of S(a,b),and show that S(a,b)admits outer derivations if and only a=1.Then,we completely classify the conformal modules of rank(1+1)over S(a,b).Finally,we classify the Z-graded free intermediate series modules over S(a,b)for a≠1.In our proof,we mainly use technical results on polynomial equations in Chapter 3.In Chapter 5,we discuss the content that deserves further research.(1)When a=1,we can study the classification of Z-graded free intermediate series modules over S(a,b).(2)We can consider the structure and representations theory of Schrodinger-Virasoro Lie conformal superalgebras and their loop Lie conformal superalgebras.(3)We can make a corresponding study on the deformation of S(a,b).There are totaly 8 tables and 54 references in this thesis.