Journal of the
Korean Mathematical Society JKMS

The affine homogeneous hypersurface in $\mathbb R^{n+1}$, which is a graph of a function $F : \mathbb R^n \rightarrow \mathbb R$ with $|\det DdF| = 1$, corresponds to a complete unimodular left symmetric algebra with a nondegenerate Hessian type inner product. We will investigate the condition for the domain over the homogeneous hypersurface to be homogeneous through an extension of the complete unimodular left symmetric algebra, which is called the graph extension.

Keywords: affine homogeneous domain, graph extension, left symmetric algebra, Hessian structure

MSC numbers: Primary 53A15, 53C30, 17D25