Left-symmetric algebras and homogeneous improper affine spheres
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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.
KeywordsLeft-symmetric algebra Affine spheres Cayley hypersurface
- 14.Jacobson, N.: Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, vol. 10. Interscience Publishers, New York (1962)Google Scholar