Modular Representations of Lie Superalgebras

Over the last decades, the representation theory of Lie algebras in both characteristic 0 and characteristic p>0 and the representation theory of Lie superalgebras over the complex numbers have been studied extensively. In this dissertation, we initiate and develop systematically the modular (that is, over a field of prime characteristic p>2) representation theory of Lie superalgebras.;We formulate a superalgebra generalization of the celebrated Kac-Weisfeiler conjecture which exhibits a mixture of p-power and 2-power divisibilities of dimensions of modules. We then establish the Super Kac-Weisfeiler conjecture for basic classical Lie superalgebras with arbitrary p -characters as well as for the queer Lie superalgebra with nilpotent p-characters. We provide an irreducibility criterion for baby Verma modules as well as a semisimplicity criterion for the reduced enveloping superalgebras associated with semisimple p-characters for both basic classical and queer Lie super-algebras. For a type I basic classical Lie superalgebra, we construct an equivalence between typical blocks of the module category of the Lie superalgebra and that of its even subalgebra.