Triality for Homogeneous Polynomials

TL;DR

This paper explores the triality symmetry of SO(8,C) to analyze three interconnected bases of invariant polynomials in Lie algebras, revealing their relations and automorphisms, with implications for Hitchin fibrations.

Contribution

It introduces a novel application of triality to explicitly relate three invariant polynomial bases and their automorphisms in the context of Lie algebra Hitchin fibrations.

Findings

Identification of three interrelated bases of invariant polynomials

Explicit automorphisms relating the bases

Connection to Hitchin fibrations

Abstract

Through the triality of ${\rm SO}(8,\mathbb{C})$, we study three interrelated homogeneous basis of the ring of invariant polynomials of Lie algebras, which give the basis of three Hitchin fibrations, and identify the explicit automorphisms that relate them.