Left-invariant affine structures on reductive Lie groups
An affine structure on a manifold \(M\) is equivalent to the existence of a flat torsion-free affine connection \(\nabla\) on the tangent bundle of the manifold. The author investigates the real reductive linear Lie groups \(G\) that admit a left-invariant affine structure. The study of complete left-invariant affine structures leads to solvable groups. By a standard result, there is a one-to-one correspondence between such structures on \(G\) and LSA-structures on \(\mathfrak g\), the Lie algebra of \(G\). Here, LSA stands for left-symmetric algebra. It is equivalent to study, then, the reductive Lie algebras \({\mathfrak g}={\mathfrak z}\oplus{\mathfrak s}\) where \(\mathfrak z\) is the center of \(\mathfrak g\) and \(\mathfrak s\) is the semi-simple part, which will be assumed to be simple. The first result is that the only such Lie algebras with one-dimensional center that admit LSA-structures are those whose simple part is (isomorphic to) \({\mathfrak g}{\mathfrak l}(n)\). (For algebras with a bigger center, there is a greater variety possible.) The second result of the paper is the description of LSA-structures on \({\mathfrak g}{\mathfrak l}(n)\) which are different from the standard one given by ordinary matrix multiplication. The others come as deformations of the matrix algebra. For \(n>2\), these (plus the standard one) exhaust the LSA-structures on \(\mathfrak g\) and, hence, the left-invariant affine structures on \(G\).
scientific article; zbMATH DE number 883791