Odd Reflections In Classical Lie Superalgebras And Their Applications

In this paper,we study some properties of Kac modules and(parabolic) Verma modules over Type Ⅰ classical Lie superalgebras gl(m|n),osp(2|2n)and p(n),applying the principle of odd reflections in classical Lie superalgebras.On the basis of a known result about the socle of the Kac module,we prove again the criterion for simplicity of the Kac modules over gl(m|n) and osp(2|2n),and we get one for p(n).Then we describe the socles of(parabolic) Verma modules over these Lie superalgebras,obtain algorithms for computing the highest weights of the socles of standard Verma modules over gl(m|n)and osp(2|2n)(in integral weight cases),and investigate the degrees of atypicality of these highest weights.In the final part,we give a new proof of the classification theorem of projective-injective modules in category O_λ(λ is integral) for these Lie superalgebras.