  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.
