Left-symmetric algebras for ${\mathfrak{gl}}(n)$

We study the classification problem for left-symmetric algebras with commutation Lie algebra ${\mathfrak{gl}}(n)$ in characteristic $0$. The problem is equivalent to the classification of étale affine representations of ${\mathfrak{gl}}(n)$. Algebraic invariant theory is used to characterize those modules for the algebraic group $\operatorname{SL}(n)$ which belong to affine étale representations of ${\mathfrak{gl}}(n)$. From the classification of these modules we obtain the solution of the classification problem for ${\mathfrak{gl}}(n)$. As another application of our approach, we exhibit left-symmetric algebra structures on certain reductive Lie algebras with a one-dimensional center and a non-simple semisimple ideal.