Sandwich operators and Einstein deformations of compact symmetric spaces related to Jordan algebras

TL;DR

This paper investigates the deformability of symmetric Einstein metrics on specific compact symmetric spaces, introducing sandwich operators related to Jordan algebras, and demonstrates the nonlinear instability of most such spaces.

Contribution

It develops sandwich operators for Lie algebras, connects Einstein deformations to Jordan algebras, and analyzes the stability of symmetric Einstein metrics.

Findings

Calculated obstruction integrals using invariant polynomials

Linked Einstein deformations to Jordan algebra structures

Proved nonlinear instability for most deformable symmetric spaces

Abstract

We study the deformability of the symmetric Einstein metrics on the spaces $\mathrm{SU}(n)/\mathrm{SO}(n)$ and $\mathrm{SU}(2n)/\mathrm{Sp}(n)$, thereby concluding the problem to second order for all irreducible symmetric spaces. The obstruction integrals are calculated from invariant polynomials on certain Lie algebra representations. To aid the computation, we develop so-called sandwich operators for compact Lie algebras and relate them to quadratic Casimir operators. We also explain the source of the infinitesimal Einstein deformations on irreducible symmetric spaces, except for the complex Grassmannians, by exploring their relation to simple Jordan algebras. As an application we prove the nonlinear instability of most of the infinitesimally deformable irreducible compact symmetric spaces.