SOME HOMOLOGICAL PROPERTIES OF CATEGORY FOR LIE SUPERALGEBRAS

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