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<p>In the context of information geometry, the concept known as left-invariant statistical structure on Lie groups is defined by Furuhata–Inoguchi–Kobayashi&#xa0;(Inf Geom 4(1):177–188, 2021). In this paper, we introduce the notion of the moduli space of left-invariant statistical structures on a Lie group. We study the moduli spaces for three particular Lie groups, each of which has a moduli space of left-invariant Riemannian metrics that is a singleton. As applications, we classify left-invariant conjugate symmetric statistical structures and left-invariant dually flat structures (which are equivalent to left-invariant Hessian structures) on these three Lie groups. A characterization of the Amari–Chentsov <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-connections on the Takano Gaussian space is also given.</p>

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The moduli spaces of left-invariant statistical structures on Lie groups

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  • 发表日期：

    2026年4月13日

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  • Hikozo Kobayashi,
  • Yu Ohno,
  • Takayuki Okuda,
  • Hiroshi Tamaru

摘要

In the context of information geometry, the concept known as left-invariant statistical structure on Lie groups is defined by Furuhata–Inoguchi–Kobayashi (Inf Geom 4(1):177–188, 2021). In this paper, we introduce the notion of the moduli space of left-invariant statistical structures on a Lie group. We study the moduli spaces for three particular Lie groups, each of which has a moduli space of left-invariant Riemannian metrics that is a singleton. As applications, we classify left-invariant conjugate symmetric statistical structures and left-invariant dually flat structures (which are equivalent to left-invariant Hessian structures) on these three Lie groups. A characterization of the Amari–Chentsov \(\alpha \) α -connections on the Takano Gaussian space is also given.