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

Hikozo Kobayashi; Hiroshi Tamaru; Takayuki Okuda; Yu Ohno

arxiv: 2510.04442 · v4 · submitted 2025-10-06 · 🧮 math.DG

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

Hikozo Kobayashi , Yu Ohno , Takayuki Okuda , Hiroshi Tamaru This is my paper

Pith reviewed 2026-05-18 09:49 UTC · model grok-4.3

classification 🧮 math.DG

keywords left-invariant statistical structuresmoduli spacesLie groupsconjugate symmetric structuresdually flat structuresHessian structuresAmari-Chentsov connectionsinformation geometry

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The pith

Moduli spaces of left-invariant statistical structures are determined for three Lie groups that each admit only one left-invariant Riemannian metric.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the moduli space of left-invariant statistical structures on a Lie group as a way to organize these objects up to natural equivalence in the setting of information geometry. It then computes this space explicitly for three specific Lie groups chosen because each has a moduli space of left-invariant Riemannian metrics consisting of a single point. Using this computation, the authors give complete lists of the left-invariant conjugate symmetric statistical structures on the three groups and of the left-invariant dually flat structures, which are equivalent to left-invariant Hessian structures. They further characterize the family of Amari-Chentsov alpha-connections that arise on the Takano Gaussian space.

Core claim

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-connections on the Takano Gaussian space is also given.

What carries the argument

The moduli space of left-invariant statistical structures, which collects equivalence classes of left-invariant Riemannian metrics paired with compatible left-invariant symmetric cubic forms satisfying the statistical condition.

If this is right

All left-invariant conjugate symmetric statistical structures on each of the three groups are now listed.
All left-invariant dually flat structures on each group are identified and shown to coincide with the left-invariant Hessian structures.
The Amari-Chentsov alpha-connections on the Takano Gaussian space receive an explicit geometric description in terms of the moduli space.
These classifications supply concrete examples of statistical structures whose curvature and divergence properties can be computed directly from the Lie algebra data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same moduli-space technique may apply to other low-dimensional Lie groups whose left-invariant metrics form a finite or low-dimensional family.
The classified structures give a supply of homogeneous statistical manifolds whose information-geometric invariants can be compared with those of non-homogeneous models.
If the classifications extend to the full automorphism group rather than just left translations, they would yield global models for certain statistical manifolds.

Load-bearing premise

The three Lie groups under study each have a moduli space of left-invariant Riemannian metrics that reduces to a single point.

What would settle it

An explicit left-invariant statistical structure on one of the three groups that lies outside the listed families of conjugate symmetric or dually flat structures would show the classification is incomplete.

read the original abstract

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 $\alpha$-connections on the Takano Gaussian space is also given.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper introduces moduli spaces for left-invariant statistical structures on Lie groups and classifies them explicitly on three groups that have unique left-invariant metrics.

read the letter

The core contribution is the definition of moduli spaces for left-invariant statistical structures, applied to three Lie groups where the left-invariant Riemannian metric is already known to be unique up to scaling. The authors then classify left-invariant conjugate symmetric statistical structures and dually flat ones on these groups, and they characterize the Amari-Chentsov alpha-connections on the Takano Gaussian space. This sits squarely in the line of work started by the 2021 Furuhata-Inoguchi-Kobayashi paper on left-invariant statistical structures. The new moduli-space notion gives a clean way to organize the possible structures once the metric is fixed, and the classifications come from direct computation with the Lie algebra structure constants. That approach looks internally consistent and produces concrete results rather than abstract existence claims. The work is narrow by design. The authors deliberately pick groups with singleton metric moduli spaces, so the statistical moduli spaces they compute are tied to that choice. This makes the examples clean but limits how far the method immediately generalizes to groups with richer metric families. There are no signs of circularity or post-hoc fitting in the abstract or stress-test description; the classifications appear to follow from the definitions via case-by-case algebra. Readers working in information geometry on homogeneous spaces or Lie groups will find the explicit lists and the Gaussian-space characterization useful. Someone outside that small intersection might see it as incremental. The paper shows clear engagement with the prior definition and delivers verifiable computations, so it is worth a serious referee even if the scope stays specialized. I would send it to peer review.

Referee Report

1 major / 3 minor

Summary. The paper introduces the moduli space of left-invariant statistical structures on Lie groups, extending the definition given by Furuhata-Inoguchi-Kobayashi. It computes this moduli space explicitly for three specific Lie groups, each chosen because its moduli space of left-invariant Riemannian metrics is a singleton. As applications, the authors classify left-invariant conjugate-symmetric statistical structures and left-invariant dually flat structures (shown equivalent to left-invariant Hessian structures) on these groups via direct computation with Lie-algebra structure constants. A characterization of the Amari-Chentsov α-connections on the Takano Gaussian space is also provided.

Significance. If the classifications are complete, the work supplies concrete, computable examples of statistical structures on homogeneous spaces that can serve as test cases for general results in information geometry. The direct, case-by-case approach using structure constants is a strength: it yields explicit, reproducible descriptions rather than abstract existence statements. The paper thereby contributes usable model spaces for studying conjugate symmetry and dual flatness on non-flat Lie groups.

major comments (1)

[§2] §2 (Preliminaries and choice of examples): The three Lie groups are selected on the basis that each has a singleton moduli space of left-invariant Riemannian metrics. While this is asserted, the manuscript should include a brief self-contained verification (or precise citation to the theorem establishing the singleton property) for each group; without it the starting point for the statistical-structure moduli computation is not fully anchored in the text.

minor comments (3)

[Introduction] Introduction: The compatibility condition between the Riemannian metric g and the torsion-free connection ∇ in the definition of a statistical structure is recalled but would benefit from an explicit one-line reminder of the equation ∇g = 0 or its equivalent form.
[§4] §4 (Dually flat structures): The equivalence between left-invariant dually flat structures and left-invariant Hessian structures is invoked; a short paragraph confirming that left-invariance is preserved under this equivalence would improve readability.
Notation: The symbol for the cubic form (or difference tensor) is introduced without a consistent global definition; adding a single displayed equation that fixes the notation for all subsequent sections would eliminate ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We are pleased that the referee recognizes the contributions of our work on moduli spaces of left-invariant statistical structures. We have carefully considered the major comment and will incorporate the suggested improvement in the revised version.

read point-by-point responses

Referee: [§2] §2 (Preliminaries and choice of examples): The three Lie groups are selected on the basis that each has a singleton moduli space of left-invariant Riemannian metrics. While this is asserted, the manuscript should include a brief self-contained verification (or precise citation to the theorem establishing the singleton property) for each group; without it the starting point for the statistical-structure moduli computation is not fully anchored in the text.

Authors: We agree that providing a brief self-contained verification or precise citation for the singleton property of the moduli space of left-invariant Riemannian metrics for each of the three Lie groups would improve the clarity and self-contained nature of the paper. In the revised manuscript, we will add a short paragraph or subsection in §2 that either recalls the relevant theorem with a precise citation or provides a brief verification using the structure constants of the Lie algebra for each group. This will ensure that the choice of examples is fully justified within the text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper cites a 2021 definition of left-invariant statistical structures from prior work involving one coauthor, then introduces the moduli-space notion and classifies structures on three explicitly chosen Lie groups via direct computation of compatible left-invariant cubic forms using Lie-algebra structure constants. No step equates a claimed prediction or classification result to a fitted parameter, self-derived uniqueness theorem, or ansatz smuggled through citation; the singleton-metric-moduli choice is an explicit selection of examples rather than a derived claim. The central results rest on independent case-by-case analysis and are therefore not circular by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the 2021 definition of left-invariant statistical structures and on the known fact that the chosen Lie groups have singleton Riemannian moduli spaces; no free parameters, invented entities, or additional ad-hoc axioms are visible from the abstract.

axioms (2)

domain assumption Left-invariant statistical structure as defined by Furuhata-Inoguchi-Kobayashi (Inf Geom 4(1):177-188, 2021)
The entire study presupposes this prior definition.
domain assumption The three Lie groups each have a singleton moduli space of left-invariant Riemannian metrics
This fact is used to select the examples and is not re-derived in the abstract.

pith-pipeline@v0.9.0 · 5665 in / 1405 out tokens · 38653 ms · 2026-05-18T09:49:46.813704+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

[1]

Proof of Lemma 6.7.By definition, the structure constants ofg RHn with respect to the canonical basis are given bya t 1t = 1,a t t1 =−1 (t≥2), with all othera k ij equal to zero

=− X t≥2 (x11tt +x 1t1t +x t11t), ∇g(x2 1xi) =x 111i − 1 3 X t≥2 (xt1it +x ti1t +x 1tit +x 1itt +x it1t +x i1tt) (fori≥2), ∇g(x1xixj) = 1 3(x11ji +x 1j1i +x j11i +x 11ij +x 1i1j +x i11j) − 1 6 X t≥2 (xtijt +x tjit +x itjt +x ijtt +x jtit +x jitt) (fori, j≥2),and ∇g(xixjxk) = (x1 ⊙x i ⊙x j ⊙x k)− 1 6(xijk1 +x ikj1 +x jik1 +x jki1 +x kij1 +x kji1) (fori, j,...

work page
[2]

Then S3 CS((h3 ⊕g Rn−3)∗, g) ={w 0p1 +p 3 |p 1 ∈S 1(g∗ Rn−3), p 3 ∈S 3(g∗ Rn−3)}, S3 DF((h3 ⊕g Rn−3)∗, g) =∅

∼= Homog2(x1, x2, x3). Then S3 CS((h3 ⊕g Rn−3)∗, g) ={w 0p1 +p 3 |p 1 ∈S 1(g∗ Rn−3), p 3 ∈S 3(g∗ Rn−3)}, S3 DF((h3 ⊕g Rn−3)∗, g) =∅. Moreover, for eachC∈S 3 CS((h3 ⊕g Rn−3)∗, g), we have∇ gC= 0andSect ∇ g (e1, e3)≥1/4>0, where∇is the statistical connection corresponding to(g, C). Furthermore, MLStat(H 3 ×R n−3) ∼= (S(O(2)×O(1))×O(n−3))\Homog 3(x1, . . . ,...

work page
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=− 3 2(x2 1x2)⊗x 3 − 3 2(x2 1x3)⊗x 2, ∇g(x3

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= 3 2(x2 2x1)⊗x 3 + 3 2(x2 2x3)⊗x 1, ∇g(x3

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Gorodski, Topics in polar actions, preprint, arXiv:2208.03577

=− 3 2(x2 3x2)⊗x 1 + 3 2(x2 3x1)⊗x 2, ∇g(x2 1x2) =−(x 1x2x3)⊗x 2 −(x 1x2 2)⊗x 3 + 1 2(x2 1x3)⊗x 1 + 1 2(x3 1)⊗x 3, ∇g(x2 1x3) =−(x 1x2 3)⊗x 2 −(x 1x2x3)⊗x 3 − 1 2(x2 1x2)⊗x 1 + 1 2(x3 1)⊗x 2, ∇g(x2 2x1) = (x1x2x3)⊗x 1 + (x2 1x2)⊗x 3 − 1 2(x2 2x3)⊗x 2 − 1 2(x3 2)⊗x 3, ∇g(x2 2x3) = (x2x2 3)⊗x 1 + (x1x2x3)⊗x 3 − 1 2(x3 2)⊗x 1 + 1 2(x2 2x1)⊗x 2, ∇g(x2 3x1) =−...

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[1] [1]

Proof of Lemma 6.7.By definition, the structure constants ofg RHn with respect to the canonical basis are given bya t 1t = 1,a t t1 =−1 (t≥2), with all othera k ij equal to zero

=− X t≥2 (x11tt +x 1t1t +x t11t), ∇g(x2 1xi) =x 111i − 1 3 X t≥2 (xt1it +x ti1t +x 1tit +x 1itt +x it1t +x i1tt) (fori≥2), ∇g(x1xixj) = 1 3(x11ji +x 1j1i +x j11i +x 11ij +x 1i1j +x i11j) − 1 6 X t≥2 (xtijt +x tjit +x itjt +x ijtt +x jtit +x jitt) (fori, j≥2),and ∇g(xixjxk) = (x1 ⊙x i ⊙x j ⊙x k)− 1 6(xijk1 +x ikj1 +x jik1 +x jki1 +x kij1 +x kji1) (fori, j,...

work page

[2] [2]

Then S3 CS((h3 ⊕g Rn−3)∗, g) ={w 0p1 +p 3 |p 1 ∈S 1(g∗ Rn−3), p 3 ∈S 3(g∗ Rn−3)}, S3 DF((h3 ⊕g Rn−3)∗, g) =∅

∼= Homog2(x1, x2, x3). Then S3 CS((h3 ⊕g Rn−3)∗, g) ={w 0p1 +p 3 |p 1 ∈S 1(g∗ Rn−3), p 3 ∈S 3(g∗ Rn−3)}, S3 DF((h3 ⊕g Rn−3)∗, g) =∅. Moreover, for eachC∈S 3 CS((h3 ⊕g Rn−3)∗, g), we have∇ gC= 0andSect ∇ g (e1, e3)≥1/4>0, where∇is the statistical connection corresponding to(g, C). Furthermore, MLStat(H 3 ×R n−3) ∼= (S(O(2)×O(1))×O(n−3))\Homog 3(x1, . . . ,...

work page

[3] [3]

=− 3 2(x2 1x2)⊗x 3 − 3 2(x2 1x3)⊗x 2, ∇g(x3

work page

[4] [4]

= 3 2(x2 2x1)⊗x 3 + 3 2(x2 2x3)⊗x 1, ∇g(x3

work page

[5] [5]

Gorodski, Topics in polar actions, preprint, arXiv:2208.03577

=− 3 2(x2 3x2)⊗x 1 + 3 2(x2 3x1)⊗x 2, ∇g(x2 1x2) =−(x 1x2x3)⊗x 2 −(x 1x2 2)⊗x 3 + 1 2(x2 1x3)⊗x 1 + 1 2(x3 1)⊗x 3, ∇g(x2 1x3) =−(x 1x2 3)⊗x 2 −(x 1x2x3)⊗x 3 − 1 2(x2 1x2)⊗x 1 + 1 2(x3 1)⊗x 2, ∇g(x2 2x1) = (x1x2x3)⊗x 1 + (x2 1x2)⊗x 3 − 1 2(x2 2x3)⊗x 2 − 1 2(x3 2)⊗x 3, ∇g(x2 2x3) = (x2x2 3)⊗x 1 + (x1x2x3)⊗x 3 − 1 2(x3 2)⊗x 1 + 1 2(x2 2x1)⊗x 2, ∇g(x2 3x1) =−...

work page arXiv 2022

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Amari and H

S. Amari and H. Nagaoka.Methods of Information Geometry, vol. 191. Amer. Math. Soc., 2000. https://doi.org/10.1090/mmono/191

work page doi:10.1090/mmono/191 2000

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Andrada, M

A. Andrada, M. L. Barberis, and I. Dotti. Classification of abelian complex structures on 6-dimensional Lie algebras.J. Lond. Math. Soc. (2), 83(1):232–255, 2011. https://doi.org/10.1112/jlms/jdq071

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