Sandwich operators and Einstein deformations of compact symmetric spaces related to Jordan algebras
Stuart James Hall, Paul Schwahn, Uwe Semmelmann
TL;DR
The paper calculates second-order obstructions to deforming Einstein metrics on irreducible compact symmetric spaces, focusing on the two families SU(n)/SO(n) and SU(2n)/Sp(n). It introduces sandwich operators for compact Lie algebras and ties obstructions to invariant cubic polynomials, leveraging a Jordan-algebra perspective to explain infinitesimal Einstein deformations. The main result shows the Koiso obstruction Ψ is typically a nontrivial multiple of the cubic invariant, yielding rigidity for odd n and nonlinear instability for most deformable spaces, including the two studied families, with connections to Perelman's ν-entropy and Ricci soliton dynamics. The approach provides explicit constants and norms for the obstruction, unifying the Lie-algebraic and Jordan-algebraic structures that govern deformation behavior in these 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.