Annals of Global Analysis and Geometry

, Volume 53, Issue 3, pp 405–443 | Cite as

Left-symmetric algebras and homogeneous improper affine spheres

  • Daniel J. F. Fox


The nonzero level sets in n-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the nth power of the function. The exponentials of the characteristic polynomials of certain left-symmetric algebras yield examples of such functions whose level sets are analogues of the generalized Cayley hypersurface of Eastwood–Ezhov. There are found purely algebraic conditions sufficient for the characteristic polynomial of the left-symmetric algebra to have the desired properties. Precisely, it suffices that the algebra has triangularizable left multiplication operators and the trace of the right multiplication is a Koszul form for which right multiplication by the dual idempotent is projection along its kernel, which equals the derived Lie subalgebra of the left-symmetric algebra.


Left-symmetric algebra Affine spheres Cayley hypersurface 


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Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada a la Ingeniería Industrial, Escuela Técnica Superior de Ingeniería y Diseão IndustrialUniversidad Politécnica de MadridMadridSpain

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