Abstract

In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.

Highlights

  • The notion of a left-symmetric algebra appeared for the first time in the work of Koszul [1] and Vinberg [2] concerning bounded homogeneous domains and convex homogeneous cones, respectively

  • Over the field of real numbers, left-symmetric algebras are of special interest because of their role in the differential geometry of affine manifolds, and in the representation theory of Lie groups [3,4]

  • Recall that if K is a left-symmetric algebra and V is a vector space, we say that V is a K -bimodule if there exist two linear maps λ, ρ : K → End (V ) which satisfy the conditions (i) and (ii) stated above

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Summary

Introduction

The notion of a left-symmetric algebra appeared for the first time in the work of Koszul [1] and Vinberg [2] concerning bounded homogeneous domains and convex homogeneous cones, respectively. For a given connected Lie group G with Lie algebra , the left-invariant affine structures on are in one-to-one correspondence with the left-symmetric structures on G compatible with the Lie structure [5]. The classification of left-invariant affine structures on a given Lie group G is reduced to the classification of compatible left-symmetric products on the Lie algebra of G It has been proved [6] that a connected Lie group G which acts transitively on n. For a given connected Lie group G with Lie algebra , the complete left-invariant affine structures on G are in one-to-one correspondence with the complete left-symmetric structures on compatible with the Lie structure. It is known that an n-dimensional connected Lie group admits a complete leftinvariant affine structure if and only if it acts transitively on n transformations [9]

Lie group which is acting simply transitively on n
As in the case of
Assume first that
It is not hard to prove the following
Assume now that
Assume finally
By applying formula
Assume now
We set
Lie algebra Remarks
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