New Non-naturally reductive Einstein metrics on Exceptional simple Lie groups

TL;DR

This paper constructs new non-naturally reductive Einstein metrics on exceptional simple Lie groups using algebraic decompositions and symbolic computation to solve polynomial systems.

Contribution

It introduces a novel method to find Einstein metrics on exceptional Lie groups via decomposition and Gröbner basis calculations, expanding known metric classifications.

Findings

New Einstein metrics on exceptional Lie groups identified.

Method employs algebraic decomposition and symbolic computation.

Provides explicit solutions to polynomial systems for Einstein metrics.

Abstract

In this article, we achieved several non-naturally reductive Einstein metrics on exceptional simple Lie groups, which are formed by the decomposition arising from general Wallach spaces. By using the decomposition corresponding to the two involutive automorphisms, we calculated the non-zero coefficients in the expression for the components of Ricci tensor with respect to the given metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gr\"{o}bner bases.