Invariant four-forms and symmetric pairs

TL;DR

This paper establishes criteria for certain Lie algebra representations to be s-representations, classifies specific complex representations, and offers a new conceptual approach to constructing exceptional Lie algebras of compact type.

Contribution

It introduces new criteria for s-representations, classifies particular complex representations, and provides a novel, computation-free method for constructing exceptional Lie algebras.

Findings

Criteria for real, complex, and quaternionic representations as s-representations.

Classification of complex representations with specific properties of their second exterior power.

A conceptual, computation-free construction of exceptional Lie algebras of compact type.

Abstract

We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations whose second exterior power is irreducible or has an irreducible summand of co-dimension one, and we give a conceptual computation-free argument for the construction of the exceptional Lie algebras of compact type.