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

Abstract:

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.