Group schemes and their Lie algebras over a symmetric tensor category

TL;DR

This paper develops a theory connecting affine group schemes over symmetric tensor categories with their Lie algebras, highlighting the structure of tangent spaces and illustrating the concepts in a specific characteristic two case.

Contribution

It introduces a framework for understanding the Lie algebra structure of affine group schemes in symmetric tensor categories, including restricted Lie algebras and their interpretations.

Findings

Tangent space at the identity forms a restricted Lie algebra.

The tangent space can be seen as degree one distributions or right invariant derivations.

Application to the symmetric tensor category $ extsf{Ver}_4^+$ in characteristic two.

Abstract

We investigate the theory of affine group schemes over a symmetric tensor category, with particular attention to the tangent space at the identity. We show that this carries the structure of a restricted Lie algebra, and can be viewed as the degree one distributions on the group scheme, or as the right invariant derivations on the coordinate ring. In the second half of the paper, we illustrate the theory in the particular case of the symmetric tensor category $\mathsf{Ver}_4^+$ in characteristic two.