Weyl groups of fine gradings on matrix algebras, octonions and the Albert algebra

TL;DR

This paper computes the Weyl groups associated with all fine gradings on matrix algebras, octonions, and the Albert algebra, revealing their automorphism structures in algebraic and geometric contexts.

Contribution

It provides a comprehensive calculation of Weyl groups for all fine gradings on key nonassociative algebras over algebraically closed fields.

Findings

Weyl groups for matrix algebra gradings are explicitly determined.

Weyl groups for octonion gradings are classified.

Weyl groups for the Albert algebra are computed.

Abstract

Given a grading $\Gamma: A=\oplus_{g\in G}A_g$ on a nonassociative algebra $A$ by an abelian group $G$, we have two subgroups of the group of automorphisms of $A$: the automorphisms that stabilize each homogeneous component $A_g$ (as a subspace) and the automorphisms that permute the components. By the Weyl group of $\Gamma$ we mean the quotient of the latter subgroup by the former. In the case of a Cartan decomposition of a semisimple complex Lie algebra, this is the automorphism group of the root system, i.e., the so-called extended Weyl group. A grading is called fine if it cannot be refined. We compute the Weyl groups of all fine gradings on matrix algebras, octonions and the Albert algebra over an algebraically closed field (of characteristic different from 2 in the case of the Albert algebra).