Extensions Of Conformal Modules Over Lie Conformal Algebras Related To Virasoro Algebra

Lie conformal algebra,as a special branch of vertex operator algebra,was introduced by Kac when he axiomatised the operator product of the chiral field in the conformal field theory.It is a frontier of researches in Lie theory.Lie conformal algebra has applications in many areas such as quantum mechanics,string theory and conformal field theory recently.Representation theory of Lie conformal algebras is an important tool to study infinite dimensional Lie algebras which satisfy local properties.Theory related to Virasoro conformal algebra,which is the simplest nontrivial conformal algebra,has become a focus of recent researches.In this thesis,we mainly study extensions of conformal modules over HeisenbergVirasoro conformal algebra,W(b)conformal algebra,Schr(?)dinger-Virasoro conformal algebra and extended Schr(?)dinger-Virasoro conformal algebra.We also study the structure of finite irreducible conformal modules over extended Schr(?)dinger-Virasoro conformal algebra.Firstly,we study extensions of conformal modules over Heisenberg-Virasoro conformal algebra.Specifically,we study extensions of nontrivial irreducible conformal modules through one dimensional conformal modules,extensions of one dimensional conformal modules through nontrivial irreducible conformal modules and extensions between nontrivial irreducible conformal modules over Heisenberg-Virasoro conformal algebra,respectively.We obtain conditions for existences of extensions of nontrivial conformal modules over Heisenberg-Virasoro conformal algebra and characterize extensions of corresponding nontrivial conformal modules.Secondly,we study extensions of conformal modules over W(b)conformal algebra.W(b)conformal algebra represents a family of conformal algebra,the index b can take any complex number.When b = 0,W(b)conformal algebra is indeed HeisenbergVirasoro conformal algebra.When b?0,we study extensions of modules over W(b)conformal algebra.We obtain conditions for existences of extensions of nontrivial conformal modules over W(b)conformal algebra.We characterize extensions of nontrivial conformal modules over W(b)conformal algebra.Finally,we study extensions of conformal modules over Schr(?)dinger-Virasoro conformal algebra and extended Schr(?)dinger-Virasoro conformal algebra.For extensionsof nontrivial conformal modules over Schr(?)dinger-Virasoro conformal algebra,we obtain conditions for existences of extensions of nontrivial conformal modules over Schr(?)dingerVirasoro conformal algebra.For structure of conformal modules over extended Schr(?)dinger-Virasoro conformal algebra,we obtain structures of finite irreducible conformal modules over extended Schr(?)dinger-Virasoro conformal algebra.For extensions of nontrivial conformal modules over extended Schr(?)dinger-Virasoro conformal algebra,we obtain conditions for existences of extensions of nontrivial conformal modules over extended Schr(?)dinger-Virasoro conformal algebra,and characterize extensions of nontrivial modules.