Left-symmetric algebras and homogeneous improper affine spheres

TL;DR

This paper characterizes conditions under which certain functions derived from left-symmetric algebras produce improper affine spheres, linking algebraic properties to geometric structures in affine differential geometry.

Contribution

It introduces algebraic conditions on left-symmetric algebras that ensure their characteristic polynomials generate functions with affine sphere level sets.

Findings

Exponential of characteristic polynomials yields improper affine spheres.

Algebraic conditions involve triangularizability and trace properties of multiplication operators.

Provides purely algebraic criteria for geometric properties of level sets.

Abstract

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 $n$th 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.