Minuscule embeddings

TL;DR

This paper classifies specific embeddings of simple algebraic groups where the module components are minuscule, explores their implications for algebraic structures, and relates these to twisted forms of the groups.

Contribution

It provides a classification of embeddings for J=SL_2 and SL_3 with minuscule module components and links these to algebraic structures and twisted forms of the groups.

Findings

Classified all embeddings for J=SL_2 and SL_3.

Connected embeddings to exceptional algebraic structures.

Related properties to twisted forms of G.

Abstract

We study embeddings $J \rightarrow G$ of simple linear algebraic groups with the following property: the simple components of the $J$ module Lie($G$)/Lie($J$) are all minuscule representations of $J$. One family of examples occurs when the group $G$ has roots of two different lengths and $J$ is the subgroup generated by the long roots. We classify all such embeddings when $J = SL_2$ and $J = SL_3$, show how each embedding implies the existence of exceptional algebraic structures on the graded components of Lie($G$), and relate properties of those structures to the existence of various twisted forms of $G$ with certain relative root systems.