Study On Some Problems In Finite W-algebras And Finite W-superalgebras

In this thesis, We mainly study the finite W-algebras and finite W-superalgebras over complex field and over algebraically closed field of prime characteristic. The main contents are listed below:1. The work on finite W-algebras:The first main content is the study on B2type finite W-algebras over com-plex field. We obtain explicit generators and relations of these algebras by com-putational methods, and the result characterizes the structure of these algebras completely.The second main content is the study on the center of reduced enveloping algebras of restricted Lie algebras sl2over prime characteristic field. We obtain the polynomial realization of reduced enveloping algebra's center of restricted Lie algebra sl2with regular nilpotent p-character χ using the methods of finite W-algebras.The third main content is the study on the center of finite W-algebras and their subalgebras, and the reduced algebras of these centers over prime charac-teristic field. Firstly we prove that U(gk)Gk, the invariant center of the universal enveloping algebras of restricted Lie algebras gk can be embedding to the algebra Z(U(gk, e)), the center of "passage" subalgebra U(gk, e) of the responding finite W-algebras U(gk, e) over prime characteristic field. In [57] Premet introduced the algebra Zx(U(gk)) which is a subalgebra of the center algebra Z(Ux(gk)) of re-duced algebras of restricted Lie algebras gk. Inspired by his idea, We not only define the algebra Zx(U(gk, e)) which is a subalgebra of the center of reduced al-gebras of finite W-algebras(in our article we call them reduced W-algebras), but also prove there is homomorphism between Premet's algebra Zx(U(gk)) and our algebra Zx(U(gk,e)) under projection map. At last we give a conjecture on the center of the "passage" subalgebras for the finite W-algebras (We can see that this algebra completely determine the center of finite W-algebras by Premet's theory on finite W-algebras). We proved the homomorphism just mentioned is actually an isomorphism if the conjecture holds.2. The work on finite W-superalgebras:The forth part is the core content of this article, which is the study on the structure theory of finite W-superalgebras corresponding to basic classical Lie superalgebras both over complex field and over prime characteristic field. The theory of finite W-superalgebras is the development of finite W-algebras with a lot of basic theories needing to be constructed. Following Premet, Weiqiang Wang and Lei Zhao's articles, We firstly give three equivalent definitions of finite W-superalgebras corresponding to the type of whose basic classical Lie superal-gebras (except D(2,1;a)(a∈C\Q)-type) over complex field, then introduce the Kazhdan filtration for finite W-superalgebras, and prove the Skryabin's equiva-lence theorem which is connected to the representation category of Lie superal-gebras. Secondly using the method of modular reduction, We define the finite W-superalgebras and their reduced algebras over prime characteristic field (whose characteristic p>>0), and then prove the Morita's equivalence theorem between the representation category of reduced algebras of finite W-superalgebras (in our article we call them reduced W-superalgebras) and the representation category of reduced enveloping algebras of Lie superalgebras. Thirdly we prove the PBW theorem for the reduced algebras of finite W-superalgebras over prime character-istic field following a series of lemmas, and then prove the PBW theorem for finite W-superalgebras over complex field using the method "admissible reduction" Finally we do some research on the structure of finite W-superalgebras and their subalgebras.The fifth main content is on the study of the lowest dimensional represen-tation theory of finite W-superalgebras and the existence of lowest dimensional representation in restricted Lie superalgebras'super Kac-Weisfeiler property. Ac-cording to the two different situations for the structure of finite W-superalgebras over complex field, at first we give a conjecture about the dimension of their low-est dimensional representation respectively. In certain special case we prove that this conjecture holds. Assuming the conjecture holds, we prove that when the field k whose characteristic p>>0, then for any p-nilpotent character χ∈((?)k/)(?) of restricted Lie superalgebras, there exist irreducible modules whose dimension is the same as the super Kac-Weisfeiler property of the corresponding reduced enveloping algebras Uχ(gk) using the tool of finite W-superalgebras'theory over prime characteristic field. It is noteworthy that in the proof of this theorem, many of the conclusions about finite W-superalgebras are not depend on the conjecture.The sixth main content is on the study of finite W-superalgebras of osp(1/2)-type Lie superalgebra over complex field. We obtain specific generators and relations by computational methods, which completely characterize the structure of these algebras. And then we make use of these explicit formulae obtained to characterize their the lowest dimensional representation. It can be found by the conclusion that the conjecture of lowest dimension of finite W-superalgebras' representation over complex field in the fifth part holds for osp(1/2)-type Lie superalgebras.The seventh main content is on the study of explicit formulae for genera-tors of small Kazhdan degree of finite W-superalgebras and their relations over complex field. We construct explicit formulae for generators of small Kazhdan degree of finite W-superalgebras over complex field, and then discuss the relation between these generators. This work plays an important role for the study in structure theory of general finite W-superalgebras in low rank.