Abstract

Let ${mathfrak g}={mathfrak g}oplus {mathfrak g}{
}$ be a classical Lie superalgebra and ${mathcal F}$ be the category of finite dimensional ${mathfrak g}$-supermodules which are semisimple over ${mathfrak g}_{}$.

In this talk we investigate the homological properties of the category ${mathcal F}$. In particular we prove that ${mathcal F}$ is self-injective in the sense that all projective supermodules are injective. We also show that all supermodules in ${mathcal F}$ admit a projective resolution with polynomial rate of growth and, hence, one can study complexity in $mathcal{F}$. If ${mathfrak g}$ is a Type~I Lie superalgebra we introduce support varieties which detect projectivity and are related to the associated varieties of Duflo and Serganova. If in addition $g$ has a (strong) duality then we prove that the conditions of being tilting or projective are equivalent.