Semisimplification of contragredient Lie algebras

TL;DR

This paper explores the structure of Lie algebras in the Verlinde category obtained through semisimplification of contragredient Lie algebras in characteristic p, introducing new root systems, gradings, and invariant forms.

Contribution

It constructs a root system, lattice grading, and triangular decomposition for semisimplified Lie algebras, providing new examples in the Verlinde category.

Findings

Constructed a root system as a parabolic restriction.

Established a lattice grading with simple components.

Provided new concrete examples of Lie algebras in the Verlinde category.

Abstract

We describe the structure and different features of Lie algebras in the Verlinde category, obtained as semisimplification of contragredient Lie algebras in characteristic $p$ with respect to the adjoint action of a Chevalley generator. In particular, we construct a root system for these algebras that arises as a parabolic restriction of the known root system for the classical Lie algebra. This gives a lattice grading with simple homogeneous components and a triangular decomposition for the semisimplified Lie algebra. We also obtain a non-degenerate invariant form that behaves well with the lattice grading. As an application, we exhibit concrete new examples of Lie algebras in the Verlinde category.