Left-Invariant Affine Structures on Reductive Lie Groups

Abstract

We describe left-invariant affine structures (that is, left-invariant flat torsion-free affine connections ∇) on reductive linear Lie groupsG. They correspond bijectively to LSA-structures on the Lie algebra g ofG. Here LSA stands for left-symmetric algebra. If g has trivial or one-dimensional center z then the affine representation α=λ⊕1 of g, induced by any LSA-structure gλon g isradiant, i.e., the radiance obstructioncα∈H1(g,gλ) vanishes. If dimz=1 we prove that g=s⊕z, where s is split simple, admits LSA-structures if and only if s is of typeAl, that is, g=gln.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 gln. 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 glnusing a result of Baues. Forn=2 we compute all structures explicitly over the complex numbers.