# ALT

## Seminar

### Atypicality, complexity and module varieties for classical Lie superalgebras

Seminar Room 1, Newton Institute

#### Abstract

Let ${\frak g}={\frak g}_{\bar 0}\oplus {\frak g}_{\bar 1}$ be a classical Lie superalgebra and ${\mathcal F}$ be the category of finite dimensional ${\frak g}$-supermodules which are semisimple over ${\frak g}_{\bar 0}$. 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 ${\frak g}$ 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 ${\frak g}$ has a (strong) duality then we prove that the conditions of being tilting or projective are equivalent.#### Video

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