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On Frobenius structures in symmetric cones
Authors
Noemie C. Combe
Publication date
8 September 2023
Publisher
View
on
arXiv
Abstract
We prove that in any strictly convex symmetric cone
Ω
\Omega
Ω
there exists a non empty locus where the WDVV equation is satisfied (i.e. there exists a hyperplane being a Frobenius manifold). This result holds over any real division algebra (with a restriction to the rank 3 case if we consider the field
O
\mathbb{O}
O
) but also on their linear combinations. This theorem holds as well in the case of pseudo-Riemannian geometry, in particular for a Lorentz symmetric cone of Anti-de-Sitter type. Our statement can be considered as a generalisation of a result by Ferapontov--Kruglikov--Novikov and Mokhov. Our construction is achieved by merging two different approaches: an algebraic/geometric one and the analytic approach given by Calabi in his investigations on the Monge--Amp\`ere equation for the case of affine hyperspheres
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oai:arXiv.org:2309.04334
Last time updated on 06/10/2023