Left-Invariant Affine Structures on Reductive Lie Groups Original Research Article

We describe left-invariant affine structures (that is, left-invariant flat torsion-free affine connections backward difference) on reductive linear Lie groupsG. They correspond bijectively to LSA-structures on the Lie algebra image ofG. Here LSA stands for left-symmetric algebra. If image has trivial or one-dimensional center image then the affine representation α=λcircled plus1 of image, induced by any LSA-structure imageλon image isradiant, i.e., the radiance obstructioncαset membership, variantH1(image, imageλ) vanishes. If dim image=1 we prove that image=imagecircled plusimage, where image is split simple, admits LSA-structures if and only if image is of typeAl, that is, image=imageimagen.Here we have the associative LSA-structure given by ordinary matrix multiplication corresponding to the bi-invariant affine structure on GL(n), which was believed to be essentially the only possible LSA-structure on imageimagen. We exhibit interesting LSA-structures different from the associative one. They arise as certain deformations of the matrix algebra. Then we classify all LSA-structures on imageimagenusing a result of Baues. Forn=2 we compute all structures explicitly over the complex numbers.