LEFT-INVARIANT AFFINE STRUCTURES ON REDUCTIVE LIE-GROUPS

We describe left-invariant affine structures (that is, left-invariant flat torsion-free affine connections del) on reductive linear Lie grou ps G. They correspond bijectively to LSA-structures on the Lie algebra g of G. Here LSA stands for left-symmetric algebra. If g has trivial or one-dimensional center z then the affine representation alpha = lam bda + 1 of g, induced by any LSA-structure g(lambda) on g is radiant, i.e., the radiance obstruction c(alpha) is an element of H-1(g, g(lamb da)) vanishes. If dim z = 1 we prove that g = s + z, where s is split simple, admits LSA-structures if and only if s is of type A(1), that i s, g = gl(n). Here we have the associative LSA-structure given by ordi nary matrix multiplication corresponding to the bi-invariant affine st ructure on GL(n), which was believed to be essentially the only possib le LSA-structure on gl(n). We exhibit interesting LSA-structures diffe rent from the associative one. They arise as certain deformations of t he matrix algebra. Then we classify all LSA-structures on gl(n) using a result of Baues. For n = 2 we compute all structures explicitly over the complex numbers. (C) 1996 Academic Press, Inc.