The Structure And Representation Of Lie Conformal Superalgebras

In this thesis,we aim to study the structure and representation theories of Lie conformal superalgebras and Hom-Lie conformal algebras.It mainly contains derivation,generalized derivation,biderivation,semidirect product,extending structure,cohomology and deformation.And it is divided into four parts as follows:Part 1 is concerned with the derivation theory of Lie conformal superalgebras.Firstly,we discuss different kinds of generalized derivations of Lie conformal superalgebras as a generalization of derivations,such as GDer(R),QDer(R),C(R),QC(R),ZDer(R),then further study their relations and properties.Secondly,we define the biderivation of Lie conformal superalgebras,and compute biderivations of the Loop Virasoro Lie conformal algebra,Loop W(a,b)Lie conformal algebra and Loop Virasoro Lie conformal superalgebra.Especially,all biderivations on the Virasoro Lie conformal algebra and Neveu-Schwarz Lie conformal superalgebra are inner biderivations.Part 2 is dedicated to the representation theory of Lie conformal superalgebras.Firstly,we review modules of a Lie conformal superalgebra,construct the semidirect product of a Lie conformal superalgebra with its module,and give a sufficient and necessary condition to determine a conformal derivation of this semidirect product.Secondly,we extend the cohomology theory of Lie conformal algebras to the super case and introduce the Nijenhuis operator of Lie conformal superalgebras.As an application of the cohomology theory,we give the definition of a deformation,construct a deformation by a Nijenhuis operator and obtain that the deformation is a 2-cocycle.Then we construct some new algebraic structures of the Lie conformal superalgebra.Part 3 is devoted to the extending structure of Lie conformal superalgebras.In this setion,we generalize the C[(?)]-split extending structures problem of Lie conformal algebras to the super case.Firstly,we introduce a unified product of a given Lie conformal superalgebra R and a given Z2-graded C[(?)]-module Q.By unified products,we give a theoretical answer to the C[(?)]-split extending structures problem:Describe and classify all Lie conformal superalgebra structures on E=R(?)Q up to isomorphism such that R.is a subalgebra of E,where the direct sum is the sum of Z2-graded C[(?)]-modules.Secondly,we use the general theory to study unified products when R.is a free C[(?)]-module and Q is a free C[(?)]-module of rank I.Finally,unified products include some other products such as the twisted product,crossed product,and bicrossed product.Using these products,we describe and classify all Lie conformal superalgebra structures on E=R(?)Q when R and Q are subalgebras or ideals of the Lie conformal superalgebra E.Part 4 is committed to the study of Hom-Lie conformal algebras.Firstly,we define an ?k-derivation of a multiplicative Hom-Lie conformal algebra and obtain that the set of all ?k-derivations can construct a multiplicative Hom-Lie conformal algebra.In particular,any ?-derivation gives rise to a derivation extension of a multiplicative Hom-Lie conformal algebra.Secondly?we discuss different kinds of generalized derivations of Hom-Lie conformal algebras:GDer(R),QDer(R),C(R),QC(R),ZDer(R),then further discuss their relations and properties.Finally,we define the cohomology of Hom-Lie conformal algebras.As an application of the cohomology in the deformation theory,we introduce the Nijenhuis operator,and construct a deformation by the Nijenhuis operator.Especially,the deformation is a 2-cocycle.Then we get a series of new algebraic structures on the previous Hom-Lie conformal algebra.