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Cohomology, Nilpotent Elements, and Module Invariants

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Rolf Farnsteiner
Universität Kiel
Thu, 22/11/2018 - 10:30 - 11:30
MPIM Lecture Hall

Let (${g}$,$[p]$) be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field $k$ of characteristic $p > 0$. This talk is concerned with the role of the nullcone $\mathcal{N}_p({g}) := \{x \in {g} \ ; \ x^{[p]} = 0\}$ within the representation theory of $({g},[p])$. I will begin by delineating the historical development of the theory of cohomological support varieties and rank varieties (closed conical subsets of  $\mathcal{N}_p({g})$), as expounded by Friedlander-Parshall and Jantzen. The second part of the talk is devoted to subcategories of $({g},[p])$-modules that are defined by properties of the action of elements of $\mathcal{N}_p({g})$. The relevant modules give rise to vector bundles and morphisms to Grassmannians, which in turn can be used to define numerical invariants. Their values are determined by the restrictions of modules to elementary abelian Lie algebras of dimension $>1$.

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