Classical Lie Superalgebras In Prime Characteristic And Their Representations

In this thesis,We mainly study the basic classical Lie superalgebras over an algebraically closed field of prime characteristic and their representations. The main contents are listed below:The first main content is to determine the Lie superalgebras of reduced algebraic supergroups.The Lie superalgebras Lie(G) of reduced algebraic supergroups G can be defined as the tangent vector space,but not the same as the Lie algebra of an algebraic group case,which can be identified with the Lie algebra of left invariant derivations on the coordinate ring with the Lie product as the usual Lie bracket of derivations.In chapterâ…¡,we introduce a subalgebra of the Lie superalgebra of left invariant derivations which is subject to the condition of "admissible" so that the tangent vector at the identity can be uniquely determined by the admissible derivation for reduced groups.Then the Lie superalgebra of a reduced supergroup turns out to be identified with the Lie superalgebra of all admissible left invariant derivations.The second main content displays our on-going research work for the center of the universal enveloping superalgebra of basic classical Lie superalgebra. We apply the method of geometry quotient to analyze the structure of the center Z.Assume g = Lie(G),where G is the classical algebraic supergroup. First we get that each element in Z has even degree.In some cases,we have proved that Z has no zero-divisors in U(g) and the fraction ring D(g) of U(g) over the integral ring Z is simple superalgebra,which we conjecture true for basic classical Lie superalgebras.If the center Z has no zero-divisors and D(g) is simple,we obtain the quotient field of Z coincides with that of the subalgebra generated by the G_ev-invariant of Z and the p-center.The third part is to study the restricted representations of the general linear Lie superalgebra.By the tool of the restricted Kac-module and restriced baby Verma-module which are different highest weight modules,we obtain some properties of irreducible modules,and compute the Cartan invariants. Since the category of restricted representations and that of the 1-th Frobenius kernel of algebraic supergroup are equivalent.We discuss the weight block of restricted representation in the representation category of the distribution algebra.In the last chapter,we compute a series of center for C(n) and B(0|1) type Lie superalgebras.