Atypicality, complexity and module varieties for classical Lie superalgebras

Duration: 56 mins 40 secs

Description:	Nakano, D (Georgia) Tuesday 23 June 2009, 14:00-15:00

Created:

2011-06-01 10:42

Collection:

Algebraic Lie Theory

Publisher:

Isaac Newton Institute

Nakano, D

Language:

eng (English)

Credits:

Author:	Nakano, D
Producer:	Steve Greenham


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

Abstract:

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

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