The moduli spaces of left-invariant statistical structures on Lie groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917419110367232 |
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| author | Kobayashi, Hikozo Ohno, Yu Okuda, Takayuki Tamaru, Hiroshi |
| author_facet | Kobayashi, Hikozo Ohno, Yu Okuda, Takayuki Tamaru, Hiroshi |
| contents | 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 $α$-connections on the Takano Gaussian space is also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04442 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The moduli spaces of left-invariant statistical structures on Lie groups Kobayashi, Hikozo Ohno, Yu Okuda, Takayuki Tamaru, Hiroshi Differential Geometry Primary 53C30, Secondary 53B12, 22E25, 53A15 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 $α$-connections on the Takano Gaussian space is also given. |
| title | The moduli spaces of left-invariant statistical structures on Lie groups |
| topic | Differential Geometry Primary 53C30, Secondary 53B12, 22E25, 53A15 |
| url | https://arxiv.org/abs/2510.04442 |