Conformal Modules And Extensions Of A Lie Conformal Algebra Of Rank 2

In this thesis,we first construct,from a 2-dimensional Novikov algebra,a rank 2 Lie conformal algebra(?)=C[(?)]L(?)C[(?)]I,which satisfies the followingλ-brackets[LλL]=((?)+2λ)(L+I),[LλI]=((?)+λ)I,[IλL]=λI,[IλI]=0 Then we mainly discuss conformal modules and their extensions over(?),as well as central extensions of(?)Firstly,we aim to classify all the finite nontrivial irreducible conformal modules over(?).To this purpose,we first determine conformal(?)-modules of rank 1 and give sufficient and necessary conditions for irreducible modules.Then we will prove that all finite irreducible conformal modules over(?)are of rank 1 Therefore,all finite irreducible conformal modules over(?)are classifiedSecondly,we discuss extensions of conformal modules over(?).According to the results of classification of finite irreducible conformal(?)-modules,we only need to consider three types of extensions of conformal modules over(?).By definition of extensions of conformal modules of Lie conformal algebras,we will calculate the specific forms of the trivial and nontrivial extensions of conformal(?)modules for each typeFinally,we study central extensions of(?).Based on the theory of cohomology of Lie conformal algebras,we need to compute 2-cocycles over(?).Thus the dimension of cohomology group of(?)is exactly the dimension of central extensions of(?).Because(?)is a perfect Lie conformal algebra,the corresponding central extension is universal and unique.