In this paper, we are interested in left invariant flat affine structures on Lie groups. These structures has been studied by many authors in different contexts. One of the fundamental questions is the existence of complete affine structures for solvable Lie groups G, raised by Minor . But recently Benoist answered negatively even for the nilpotent case . Also moduli space of such structures for lower dimensional cases has been studied by several authors, sometimes with compatible metrics [5,10,4,12].
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