Sandwich operators and Einstein deformations of compact symmetric spaces related to Jordan algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929679391260672 |
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| author | Hall, Stuart James Schwahn, Paul Semmelmann, Uwe |
| author_facet | Hall, Stuart James Schwahn, Paul Semmelmann, Uwe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08770 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sandwich operators and Einstein deformations of compact symmetric spaces related to Jordan algebras Hall, Stuart James Schwahn, Paul Semmelmann, Uwe Differential Geometry Rings and Algebras 17B20, 17B60, 53C24, 53C25, 53C35 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. |
| title | Sandwich operators and Einstein deformations of compact symmetric spaces related to Jordan algebras |
| topic | Differential Geometry Rings and Algebras 17B20, 17B60, 53C24, 53C25, 53C35 |
| url | https://arxiv.org/abs/2412.08770 |