Left-Invariant Affine Structures on Reductive Lie Groups

Abstract

ŽWe describe left-invariant affine structures that is, left-invariant flat torsion-free . affine connections ٌ on reductive linear Lie groups G. They correspond bijectively to LSA-structures on the Lie algebra ᒄ of G. Here LSA stands for left-symmetric algebra. If ᒄ has trivial or one-dimensional center ᒗ then the affine representation ␣ s [ 1 of ᒄ, induced by any LSA-structure ᒄ on ᒄ is radiant, i.e., thewhere ᒐ is split simple, admits LSA-structures if and only if ᒐ is of type A , that l is, ᒄ s ᒄ ᒉ . Here we have the associative LSA-structure given by ordinary matrix n Ž . multiplication corresponding to the bi-invariant affine structure on GL n , which was believed to be essentially the only possible LSA-structure on ᒄ ᒉ . We exhibit n interesting LSA-structures different from the associative one. They arise as certain deformations of the matrix algebra. Then we classify all LSA-structures on ᒄ ᒉ n using a result of Baues. For n s 2 we compute all structures explicitly over the complex numbers.

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

“…Let A be an RSA with Lie algebra g A = gl n (K) over a field K of characteristic zero (see [5], [18], [32]). Then, for k ≥ 1,…”

Classification of Some Simple Graded Pre-Lie Algebras of Growth One

“…This section contains basic material on pre-Lie algebras, some of which can already be found in [Bur96,Bur98], which are concerned with finite dimensional pre-Lie algebras (under the name left-symmetric algebras).…”

Simple left-symmetric algebras with solvable Lie algebra

“…They are classified in [BU2]. Any such structure arises by a deformation of the associative matrix algebra structure.…”