On complete affine structures in Lie groups

Abstract

Left invariant affine structures in a Lie group $G$ are in one-to-one correspondence with left-symmetric algebras over its Lie algebra $\mathfrak g=T_eG$ (``over'' means that the commutator $[x,y]=xy-yx$ coincides with the Lie bracket; left-symmetric algebras can be defined as Lie-admissible algebras such that the multiplication by left defines a representation of the underlying Lie algebra). An affine structure (and the corresponding left symmetric algebra) is complete if $G$ is affinely equivalent to $\mathfrak g$. By the main result of this paper, a complete left symmetric algebra admits a canonical decomposition: there is a Cartan subalgebra $\mathfrak h$ such that the root subspaces for the representations $L$ (by left multiplications) and $\ad$ coincide. Then operators $L(x)$ and $\ad(x)$ have equal semisimple parts for all $x\in\mathfrak h$. This decomposition is unique. For simple complete left-symmetric algebras whose canonical decomposition consists of one dimensional spaces we define two types of graphs and prove some their properties. This makes possible to describe, for dimensions less or equal to 5, these graphs and algebras.