Generalised nice sets

TL;DR

This paper introduces and classifies generalized nice sets, a new combinatorial concept, to facilitate the study of graded contractions of exceptional complex Lie algebras with octonion-based gradings.

Contribution

It provides a purely combinatorial classification of generalized nice sets, essential for understanding graded contractions of certain exceptional Lie algebras.

Findings

Classification of generalized nice sets up to collineations of the Fano plane

Application to graded contractions of exceptional complex Lie algebras

Framework for combinatorial analysis of algebraic structures

Abstract

A new combinatorial object, called generalised nice set, is classified up to collineations of the Fano plane. This classification is necessary to find the graded contractions of all the exceptional complex Lie algebras of dimension at least 52, endowed with $\mathbb Z_2^3$-gradings coming from the octonions. Our classification is of purely combinatorial nature.