A tensor-triangular property for categories of representations of restricted Lie algebras

TL;DR

This paper introduces a property linking cohomological support and tensor-triangular geometry for restricted Lie algebras, exploring how different group scheme structures affect their representation categories and spectra.

Contribution

It defines a new property for restricted Lie algebras based on their tensor-triangular spectra and analyzes its relation to representation type, including examples and conjectures.

Findings

Restricted Lie algebras satisfy the property if subgroup sets match minimal radical thick tensor-ideals.

No abelian restricted Lie algebra of wild representation type satisfies the property.

Conjecture: the property is equivalent to having finite or tame representation type.

Abstract

We define a property for restricted Lie algebras in terms of cohomological support and tensor-triangular geometry of their categories of representations. By Tannakian reconstruction, the different symmetric tensor category structures on the underlying linear category of representations of a restricted Lie algebra correspond to different cocommutative Hopf algebra structures on the restricted enveloping algebra. In turn this equates together the linear categories of representations for various group scheme structures. The tensor triangular spectrum, for representations of a restricted Lie algebra, is known to be isomorphic to the scheme of 1-parameter subgroups of the infinitesimal group scheme structure associated to the Lie algebra. Points in the spectrum of a tensor triangulated category correspond to minimal radical thick tensor-ideals, provided the spectrum is noetherian, as is known in our case of finite group schemes. When the group scheme structure changes from the Lie algebra structure, a set of subgroups can still yield points of the spectrum, but there may not be enough to cover the spectrum. Restricted Lie algebras satisfy our property if, for each group scheme structure, the remaining set of subgroups correspond to minimal radical thick tensor-ideals having identical Green-ring structure to that of the original Lie algebra. Some small examples of algebras of finite and tame representation type satisfying our property are given. We show that no abelian restricted Lie algebra of wild representation type may have our property. We conjecture that satisfying our property is equivalent to having finite or tame representation type.