Left-symmetric algebras for 𝔤𝔩(𝔫)

We study the classification problem for left-symmetric algebras with commutation Lie algebra g l ( n ) {\mathfrak {gl}}(n) in characteristic 0 0 . The problem is equivalent to the classification of étale affine representations of g l ( n ) {\mathfrak {gl}}(n) . Algebraic invariant theory is used to characterize those modules for the algebraic group SL ⁡ ( n ) \operatorname {SL}(n) which belong to affine étale representations of g l ( n ) {\mathfrak {gl}}(n) . From the classification of these modules we obtain the solution of the classification problem for g l ( n ) {\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.