Mathematics & Statistics Colloquium: Daniel Nakano

Title: Tensor Triangular Geometry and Applications

Abstract: Tensor triangular geometry as introduced by Paul Balmer is a powerful idea which can be used to extract hidden ambient geometry from a given tensor triangulated category. These geometric structures often arise at the derived/cohomological level and play an important role in understanding the combinatorial property of representations of groups and algebras. 

In this talk  I will first present a general setting for a compactly generated tensor triangulated category that enables one to classify thick tensor ideals and to determine the Balmer Spectrum. Several examples will be presented to illustrate these beautiful connections which includes the stable module category for finite groups and the derived category of bounded perfect complexes for finitely generated \(R\)-modules where \(R\) is a commutative Noetherian ring. 

For a classical Lie superalgebra \({\mathfrak g}={\mathfrak g}_0+{\mathfrak g}_1\), I will later show how to construct a Zariski space from a detecting subalgebra \({\mathfrak f}\) and demonstrate that this topological space governs the tensor triangular geometry for the category of finite dimensional \({\mathfrak g}\)-modules that are semisimple over \({\mathfrak g}_0\). Concrete realizations will be provided for the Lie superalgebra \({\mathfrak gl}(m|n)\).  This answers an old question about finding a geometric object that governs the representation theory for Lie superalgebras. If time permits, I will also discuss other new applications.  

These results represent joint work with B. Boe and J. Kujawa.