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<p>It is known that there exists a correspondence between left-invariant affine structures on simply connected Lie group <i>G</i> and left-symmetric structures on the Lie algebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathfrak {g}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation> of <i>G</i>. In this paper, we define an algebraic structure, which we call a generalized left-symmetric structure, and give a correspondence between left-invariant transversely affine foliations of <i>G</i> and generalized left-symmetric structures. Moreover, by using this correspondence, we give an algebraic description of completeness of left-invariant transversely affine foliations. We also give some methods to construct generalized left-symmetric structures.</p>

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Left-invariant transversely affine foliations and a generalization of left-symmetric structures

  • 期刊论文

  • 发表日期：

    2025年10月15日

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  • Naoki Kato

摘要

It is known that there exists a correspondence between left-invariant affine structures on simply connected Lie group G and left-symmetric structures on the Lie algebra \({\mathfrak {g}}\) g of G. In this paper, we define an algebraic structure, which we call a generalized left-symmetric structure, and give a correspondence between left-invariant transversely affine foliations of G and generalized left-symmetric structures. Moreover, by using this correspondence, we give an algebraic description of completeness of left-invariant transversely affine foliations. We also give some methods to construct generalized left-symmetric structures.