Some Structures And Representations Of Infinite Dimensional Lie Superalgebras

This paper is devoted to studying structures and representations of some infinite di-mensional Lie superalgebras.A Lie superalgebra is a generalization of a Lie algebra to include a Z₂-grading.Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry.Lie superalgebras have widely used in mathematics and theoretical physics.At present,the research on the structures and representations of the infinite dimensional Lie superalgebras and related superalgebras are in the rise.In partic-ular,there are many important problems still unsolved in the representation theory,such as the classification of irreducible quasi-finite weight modules,the construction and clas-sification of non-weight modules,and so on.It is very important to study these questions in the representation theory of infinite dimensional Lie superalgebras.In terms of structural theory,we study superbiderivations and 2-local superderiva-tions of several classes of Lie superalgebras.It is well known that any biderivation can be decomposed into a skew-symmetric biderivation and a symmetric biderivation.Skew-symmetric biderivations on many Lie（super）algebras,which are related to the Virasoro algebra can be determined.However,there is no any sufficient tool to determine symmet-ric biderivations on some Lie algebras and Lie superalgebras,up to now.Motivated by the idea in calculating cohomology groups of the Virasoro algebra,we develop a general method to determine all symmetric biderivations on some Lie algebras and Lie superalge-bras related to the Virasoro algebra in this paper.Thus,superbiderivations over the super Virasoro algebra,the Heisenberg-Virasoro superalgebra,the super W（2,2）algebra,the N=1 super BMS₃algebra and the Ramond N=2 algebras are determined.As an applica-tion,commutative post-Lie algebra structures on these Lie superalgebras are obtained.In addition,we study 2-local superderivations of the super Virasoro algebra and super W（2,2）algebra.It is proved that all 2-local superderivations on the super Virasoro algebra as well as the super W（2,2）algebra are（global）superderivations.In terms of representation theory,we study classification and characterization of weight modules and non-weight modules over some infinite dimensional Lie superalge-bras and their related algebras.In terms of weight modules,we get theΩ-operators for the Ovsienko-Roger superalgebras and then use it to classify all simple Cuspidal mod-ules over the Z-graded and₂¹Z-graded Ovsienko-Roger superalgebras.By this result,we can easily classify all simple Harish-Chandra modules over some related Lie superalge-bras,including the N=1 super-BMS₃algebra,the super W（2,2）,etc.We study the tensor products of weight modules over the Heisenberg-Virasoro superalgebra.Specifically,we compute Kac determinant for the Heisenberg-Virasoro superalgebra in order to determine the irreducibility of Verma module of the quotient algebra;We give the intermediate se-ries modules over the the Heisenberg-Virasoro superalgebra and give the irreducibility criterion for the tensor products of Verma modules with intermediate series modules over the Heisenberg-Virasoro superalgebra and also give the necessary and sufficient condi-tions for any two such tensor products to be isomorphic by using“shifting technique”established for the Heisenberg-Virasoro superalgebra.In terms of non-weight modules,this paper is devoted to studying classification and characterization of Whittaker modules and high order Whittaker modules for the N=1 super-BMS₃algebras;We also study the U（h）-free modules over the Lie algebra of differential operators on the circle by the result of two subalgebras—Virasoro algebra and the Heisenberg-Virasoro algebra.Then we determine the necessary and sufficient conditions for the tensor products of quasi-finite highest weight modules and U（h）-free modules to be irreducible,and obtain that any two such tensor products are isomorphic if and only if the corresponding highest weight mod-ules and U（h）-free modules are isomorphic.By the way,we can extend such result to the Lie algebras of differential operators in the general case.Moreover,U（h）-free modules over the Heisenberg-Virasoro superalgebra are also studied.