SOME HOMOLOGICAL PROPERTIES of CATEGORY O for LIE SUPERALGEBRAS

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule Δ(λ) to be such that every nonzero homomorphism from another Verma supermodule to Δ(λ) is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras pe(n) and, furthermore, to reduce the problem of description of Ext1O(L(μ), Δ(λ)) for pe(n) to the similar problem for the Lie algebra gl(n). Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category Op for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra pe(n) and the orthosymplectic Lie superalgebra osp(2|2n).