# Dynkin Diagrams and Short Peirce Gradings of Kantor Pairs

## TL;DR

This paper classifies finite dimensional simple short Peirce graded Kantor pairs using Dynkin diagrams and details how to compute their Weyl images, especially for close-to-Jordan pairs, enhancing understanding of their structure.

## Contribution

It provides a classification of simple SP-graded Kantor pairs via Dynkin diagrams and introduces methods to compute Weyl images, advancing the structural theory of these pairs.

## Key findings

- Classification of simple SP-graded Kantor pairs using Dynkin diagrams.
- Method to compute Weyl images from diagrams.
- Construction of reflections for close-to-Jordan Kantor pairs.

## Abstract

In a recent article with Oleg Smirnov, we defined short Peirce (SP) graded Kantor pairs. For any such pair P, we defined a family, parameterized by the Weyl group of type BC_2, consisting of SP-graded Kantor pairs called Weyl images of P. In this article, we classify finite dimensional simple SP-graded Kantor pairs over an algebraically closed field of characteristic 0 in terms of marked Dynkin diagrams, and we show how to compute Weyl images using these diagrams. The theory is particularly attractive for close-to-Jordan Kantor pairs (which are variations of Freudenthal triple systems), and we construct the reflections of such pairs (with nontrivial gradings) starting from Jordan matrix pairs.