Azumaya Property Of The Center Of The Universal Enveloping Algebra Of A Lie Superalgebra And Irreducibility Of The Nilpotent Cone Of A Restricted Lie Algebra

This thesis is mainly on the study of basic classical Lie superalgebra and Cartan type Lie algebra over an algebraically closed field k of characteristic p>2and their representations. The main results are listed below:The first main result is on the center of the universal enveloping algebra of a basic classical Lie superalgebra. Let g=g0+g1be such a Lie superalgebra over k. There is an algebraic supergroup G satisfying Lie(G)=g, with the purely even subgroup Gev which is a connected reductive group. The center Z(g) of the universal enveloping algebra U(g) turns out to be commutative. Then we prove in two different ways that the center Z(g) is an integral domain, and furthermore prove that U(g)(?)z(g) Frac(Z(g)) is a simple superalgebra over Frac(Z(g)). This enables us to prove the conjecture proposed by Bin Shu and Lisun Zheng about the structure of the center of enveloping algebra. That is, the quotient field of Z(g) coincides with that of the subalgebra generated by the Gev-invariant ring Z(g)Gev of Z(g) and the p-center Zo of U(g0).The second main result is on the representations of Lie superalgebra g=osp(1|2n). As an application of the first main result to this special case, we first show that Z(U(g)) is just generated by Z(g)Gev and Z0. We can also get an algebraic isomorphism between Z(g)Gev and U(η))w, which generalizes the Harish-Chandra theorem in the case of complex semi-simple Lie algebras. Then we investigate the blocks of the underlying even category of Ux(g) for χ∈g*. By equivalence of categories and the property of the anti-center of U(g), we de-scribe precisely the blocks when χ is regular semisimple and regular nilpotent. In particular, when χ is regular nilpotent, we classify the isomorphism classes of irreducible modules; moreover, we show that in this case there is no projective baby Verma module by computing dimensions, generalizing the result of Wang-Zhao for osp(1|2). Next, we give the precise relation between the smooth points of the maximal spectrum Maxspec (Z(g)) and the corresponding irreducible mod- ules for osp(1|2). This result is different from the case of classical Lie algebra. Finally, we count all blocks in the case for osp(1|2).The third result is on the irreducibility of the variety of nilpotent elements in a Lie algebra and the ring of invariant polynomial functions on the Lie algebra. A.A.Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. It is well known that this variety is irreducible for any classical Lie algebra. And then, Premet in [63] confirmed this conjecture for Jacoboson-Witt algebra. In the same paper, he described the ring of invariant polynomial functions on the Jacoboson-Witt algebra by a Chevalley restricted theorem similar to the classical Lie algebras. In the present thesis, we study the the Lie algebra Sn of Cartan type of S type. The conjecture for type S is showed to be true by constructing a dense subset. Moreover, It can be shown that the variety of nilpotent elements in the algebra Sn is a complete intersection. Motivated by the proof of the irreducibility, we show that k[Sn]Aut(Sn)(?)k[T’n]WeylGroup, where T’n is a generic torus in Sn. This result also generalizes the Chevalley restrict theorem of classical Lie algebras. As a byproduct, we also prove that there are infinitely many nilpotent orbits in Sn, which give a positive answer to a conjecture of Yufeng Yao and Bin Shu.