Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction

Information Geometry, Jordan Algebras,
and a Coadjoint Orbit-Like Construction††This paper is a contribution to the
Special Issue on Differential Geometry Inspired by Mathematical Physics in
honor of Jean-Pierre Bourguignon for his 75th birthday. The full collection is
available at https://www.emis.de/journals/SIGMA/Bourguignon.html

Florio M. CIAGLIA a, Jürgen JOST bcd and Lorenz J. SCHWACHHÖFER e

F.M. Ciaglia, J. Jost and L.J. Schwachhöfer

a) Department of Mathematics, Universidad Carlos III de Madrid, Leganés,
Madrid, Spain<EMAIL_ADDRESS>https://researchportal.uc3m.es/display/inv48159

b) Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
<EMAIL_ADDRESS>
c) Center for Scalable Dynamical Systems, Leipzig University, Germany
https://www.mis.mpg.de/jjost/juergen-jost.html

d) Santa Fe Institute for the Sciences of Complexity, New Mexico, USA

e) Department of Mathematics, TU Dortmund University, Dortmund, Germany
<EMAIL_ADDRESS>http://riemann.mathematik.tu-
dortmund.de/~lschwach/

Received April 12, 2023, in final form October 09, 2023; Published online
October 20, 2023

Jordan algebras arise naturally in (quantum) information geometry, and we want
to understand their role and their structure within that framework. Inspired
by Kirillov’s discussion of the symplectic structure on coadjoint orbits, we
provide a similar construction in the case of real Jordan algebras. Given a
real, finite-dimensional, formally real Jordan algebra $\mathcal{J}$, we
exploit the generalized distribution determined by the Jordan product on the
dual $\mathcal{J}^{\star}$ to induce a pseudo-Riemannian metric tensor on the
leaves of the distribution. In particular, these leaves are the orbits of a
Lie group, which is the structure group of $\mathcal{J}$, in clear analogy
with what happens for coadjoint orbits. However, this time in contrast with
the Lie-algebraic case, we prove that not all points in $\mathcal{J}^{*}$ lie
on a leaf of the canonical Jordan distribution. When the leaves are contained
in the cone of positive linear functionals on $\mathcal{J}$, the pseudo-
Riemannian structure becomes Riemannian and, for appropriate choices of
$\mathcal{J}$, it coincides with the Fisher–Rao metric on non-normalized
probability distributions on a finite sample space, or with the Bures–Helstrom
metric for non-normalized, faithful quantum states of a finite-level quantum
system, thus showing a direct link between the mathematics of Jordan algebras
and both classical and quantum information geometry.

information geometry; Jordan algebras; Lie algebras; Kirillov orbit method;
Fisher–Rao metric; Bures–Helstrom metric

17C20; 17C27; 17B60; 53B12

Dedicated to Jean-Pierre Bourguignon
on the occasion of his 75th birthday

Personal note of Jürgen Jost:
I have known Jean-Pierre Bourguignon for many decades, as a dedicated and
inspiring differential geometer, as a generous friend, and as an efficient and
competent, and above all, always fair, administrator and leader. I am
therefore happy that we can dedicate this paper to him on the occasion of his
75th birthday, with great respect for all his achievements.

##  1 Introduction

Since their introduction in [46] in the context of the foundations of quantum
mechanics, Jordan algebras proved to be extremely versatile in both
mathematics and physics. For instance, we mention the link between Jordan
algebras, symmetric and harmonic analysis [18, 19, 30, 52, 78], the connection
between Jordan algebras and quantum theories [7, 27, 29], and the role Jordan
algebras play in the reconstruction of quantum theories [62, 79]. We also
refer to [43] for an extensive discussion of different fields of application
of Jordan algebras in both mathematics and physics.

Motivated by classical and quantum information geometry, we want to present
here another point of view from which to explore Jordan algebras and their
mathematics.

Information geometry is a multidisciplinary field of research in which
different aspects of mathematics, physics, statistics, and information theory
coexist and mutually influence each other. From a mathematical point of view,
information geometry explores the mathematical structures living on suitable
manifolds of classical probability distributions or quantum states, known as
parametric models, and their relation with information-theoretic and
statistical tasks [3, 4]. In particular, we have the fundamental concept of an
information metric (classical or quantum), a Riemannian metric tensor on a
manifold $M$ that parameterizes classical probability distributions or quantum
states. The first example of such an information metric is the so-called
Fisher–Rao metric tensor, whose introduction in the classical context traces
back to the work of Fisher [31], Mahalanobis [59], and Rao [66]. The
appearance of the Fisher–Rao metric tensor in different applied contexts like
population genetics and statistical inference is partially explained by
Cencov’s theorem [16] which states that, for finite outcome spaces, the
Fisher–Rao metric tensor is the only metric tensor which is invariant with
respect to the class of Markov kernels which are the most general type of maps
respecting the convex geometry of the space of probability distributions.
Cencov’s theorem has recently been extended to continuous outcome spaces [5,
6, 8, 33].

In the quantum case, the situation is not so simple. The first type of quantum
information metric appeared in the context of parameter estimation and quantum
metrology already at the end of the sixties in the work of Helstrom [36, 37,
38]. This Riemannian metric tensor is known as the Bures–Helstrom because it
was proved by Uhlmann [76, 77] that it is the “infinitesimal” version of the
distance function among states of a von Neumann algebra introduced by Bures
[15] to generalize a result by Kakutani [47]. Later, it has been proved by
Petz [64] – building on previous works by Cencov and Morozowa [61] – that
uniqueness is lost in the finite-dimensional quantum case, that is, there is
no analogue of Cencov’s theorem mentioned above. This led to an intensive
study of the so-called monotone quantum metric tensors generalizing the
Fisher–Rao metric tensor to the quantum case [21, 28, 34, 35, 56, 60] and of
their information-theoretic and statistical applications [58, 63, 73, 74, 80].
However, the Bures–Helstrom metric tensor remains the “best one” when it comes
to quantum metrology because it leads to the tightest version of the quantum
Cramer–Rao bound [32, 63].

A common framework to deal with classical and quantum information geometry
simultaneously is provided by $W^{*}$-algebras [22, 23, 24, 44, 45, 54]. In
this context, it has been recently argued that the Fisher–Rao metric tensor
and the Bures–Helstrom metric tensor are related through the Jordan algebra
associated with a $W^{*}$-algebra [22, 23, 24]. These types of Jordan algebras
allow for a unification of the classical and quantum states in terms of normal
states (i.e., the normalized and normal positive linear functionals) on the
algebra. When the Jordan algebra is the associative Jordan algebra of self-
adjoint elements in the commutative $W^{*}$-algebra
$\mathcal{L}^{\infty}(\mathcal{X},\mu)$ of complex-valued, $\mu$-essentially
bounded (equivalence classes of) functions on the measurable space
$(\mathcal{X},\Sigma)$, the normal states are probability measures on
$\mathcal{X}$ which are absolutely continuous with respect to $\mu$, and we
recover the classical case. The quantum case is recovered considering the
Jordan algebra of self-adjoint elements in the $W^{*}$-algebra
$\mathcal{B}(\mathcal{H})$ of bounded linear operators on the complex Hilbert
space $\mathcal{H}$, so that the normal states are density operators on
$\mathcal{H}$ (i.e., trace-class, positive semidefinite operators with unit
trace) which identify quantum states. In this context, the Jordan product
induces a Riemannian metric tensor on suitable parametric models of normal
states, which becomes the Fisher–Rao metric tensor when the algebra is
$\mathcal{L}^{\infty}(\mathcal{X},\mu)$ and the Bures–Helstrom metric tensor
when the algebra is $\mathcal{B}(\mathcal{H})$.

In the finite-dimensional case, it has been observed that the metric tensor on
normal states may be obtained as a kind of inverse of a contravariant tensor
on the dual space of the Jordan algebra determined by the Jordan product
itself [23].

Let us recall that setting so that we can subsequently explain how to abstract
and generalize it in order to develop new insight into classical and quantum
information geometry and at the same time into the structure of Jordan
algebras from that of Lie algebras operating on them. Thus, we let
$\mathscr{A}$ be a finite-dimensional, unital $C^{*}$-algebra, and let
$\mathscr{A}_{\rm sa}$ be the self-adjoint part of $\mathscr{A}$ and
$\mathscr{V}$ be the self-adjoint part of the Banach dual $\mathscr{A}^{*}$ of
$\mathscr{A}$. Of course, there is some duality here, as
$\mathbf{a}\in\mathscr{A}_{\rm sa}$ can be identified with a real-valued,
linear function $l_{\mathbf{a}}$ on $\mathscr{V}$, via

$l_{\mathbf{a}}(\xi):=\xi(\mathbf{a}).$

Since $\mathscr{V}$ is a finite-dimensional Banach space, $\mathbf{a}\mapsto
l_{\mathbf{a}}$ yields an isomorphism between $\mathscr{A}_{\rm sa}$ and
$\mathscr{V}^{*}=\mathscr{A}_{\rm sa}^{**}$. In other words, the differentials
of the linear functions on $\mathscr{V}$ associated with elements in
$\mathscr{A}_{\rm sa}$ generate the cotangent space $T_{\xi}^{*}\mathscr{V}$
at each $\xi$.

The associative product of $\mathscr{A}$ leads to a commutative product
$\\{,\\}$ and to a non-commutative product $[[\,,\,]]$ on $\mathscr{A}_{\rm
sa}$, making $\mathscr{A}_{\rm sa}$ into a _Banach–Lie–Jordan algebra_ [1];
they are

$\displaystyle\\{\mathbf{a},\mathbf{b}\\}:=\frac{1}{2}(\mathbf{ab}+\mathbf{ba}),\qquad[[\mathbf{a},\mathbf{b}]]:=\frac{1}{2\sqrt{-1}}(\mathbf{ab}-\mathbf{ba}).$

We can then exploit these products to introduce a symmetric and an
antisymmetric tensor on the dual of $\mathscr{A}_{\rm sa}$. Both these tensors
play a role in the context of the geometry of finite-level quantum systems and
their open quantum dynamical evolutions [17, 20].

Specifically, we define

$\left(\mathcal{R}(\mathrm{d}l_{\mathbf{a}},\mathrm{d}l_{\mathbf{b}})\right)(\xi):=l_{\\{\mathbf{a},\mathbf{b}\\}}(\xi)=\xi(\\{\mathbf{a},\mathbf{b}\\}),$

and, of course, extend it by linearity to the entire cotangent space,
obtaining a symmetric contravariant tensor associated with the Jordan product.

In the same way that the Jordan structure induces the symmetric tensor
$\mathcal{R}$, the Lie structure also induces an antisymmetric tensor

$\left(\Lambda(\mathrm{d}f_{1},\mathrm{d}f_{2})\right)(\xi):=\xi\left([[\mathrm{d}f_{1}(\xi),\mathrm{d}f_{2}(\xi)]]\right).$

Since the Lie algebra of the group $\mathscr{U}$ of unitary elements in
$\mathscr{A}$ may be identified with the space $\mathscr{A}_{\rm sa}$ of self-
adjoint elements in $\mathscr{A}$, the tensor $\Lambda$ may be interpreted as
the Kostant–Kirillov–Souriau Poisson tensor associated with the coadjoint
action of the unitary group $\mathscr{U}$. Importantly, for our purposes, also
$\mathcal{R}$ can be studied with the help of the action of this Lie algebra.

Returning to the covariant symmetric tensor $\mathcal{R}$, it happens that, on
certain orbits of the Banach–Lie group of invertible elements of $\mathscr{A}$
(of which $\mathscr{U}$ is a subgroup), it makes sense to consider the
covariant tensor $\mathrm{G}:=\mathcal{R}^{-1}$, which becomes a Riemannian
metric tensor when the above-mentioned orbits lie in the space of positive
linear functionals on $\mathscr{A}$. The result is a Riemannian metric tensor,
which is precisely that used in classical and quantum information geometry.

The observation in [23] that we have just explained only hinted at the
possibility of obtaining Riemannian metric tensors with a coadjoint orbit-like
procedure, and the main purpose of this work is to systematically explore this
possibility in the more general context of finite-dimensional, formally real
Jordan algebras.

In fact, the action of the Lie algebra of $\mathscr{U}$ is reminiscent of what
happens in the case of Lie groups and Lie algebras, where the Lie algebra
induces a contravariant tensor on its dual that can be inverted on the so-
called coadjoint orbits in order to equip them with a symplectic structure
known as the Kirillov–Kostant–Souriau (KKS) symplectic form [48, 49, 53, 71].

Therefore, in this paper, we develop a more abstract perspective, moving from
the Jordan algebras arising from $C^{*}$-algebras to general Jordan algebras.
In particular, this will clarify how much of the above construction actually
depends on the ambient $C^{*}$-algebras. In fact, we shall find that, at least
when we deal with finite-dimensional formally real Jordan algebras, the
ambient $C^{*}$-algebras play no role. On one hand, this will provide a deeper
explanation of the Fisher–Rao and Bures–Helstrom tensors and their
generalizations. On the other hand, from this more general perspective, we
obtain Riemannian metric tensors on suitable models of positive linear
functionals on Jordan algebras that do not necessarily arise in the context of
classical and quantum information geometry as the spin factors that are
recently being employed in a new formulation of color perception theory [10,
11, 12, 65, 67].

As anticipated above, our guiding question is to what extent we can develop a
coadjoint orbit-like construction for Jordan algebras that is analogous to
that of Lie algebras, at least in finite dimensions. Specifically, we start
from Kirillov’s fundamental observation that a coadjoint orbit
$\mathcal{O}\subset{\mathfrak{g}}^{\star}$ of the Lie group $\mathrm{G}$
carries a natural homogeneous symplectic structure [48, 49]. Here,
${\mathfrak{g}}$ and ${\mathfrak{g}}^{\star}$ denote the Lie algebra of
$\mathrm{G}$ and its dual. Kirillov’s observation relates algebraic structures
to differential geometry and mathematical physics in a deep and very
productive way, led to spectacular results in representation theory, classical
and quantum mechanics, and it is closely related to geometric quantization
[50].

As far as we know, no analogue of the coadjoint orbit construction for Lie
algebras has been investigated in the case of Jordan algebras, and the main
purpose of this work is precisely to fill this gap. To understand the picture
from a more abstract perspective, we need to go somewhat beyond the setting
sketched above. We shall start by considering a general, finite-dimensional
algebra $\mathcal{A}$, i.e., a finite-dimensional (real or complex) vector
space with a bilinear product
$\bullet\colon\mathcal{A}\times\mathcal{A}\to\mathcal{A}$. At this moment, in
contrast to the setting above, no further conditions like associativity or
identities of Jacobi/Jordan type are assumed. We denote with
$\mathcal{A}^{\star}$ the dual space of $\mathcal{A}$, and the corresponding
pairing is denoted by $\langle\cdot,\cdot\rangle$. Due to the finite-
dimensionality, we have the identification
$\mathcal{A}^{\star\star}\cong\mathcal{A}$, and for the tangent and cotangent
spaces of $\mathcal{A}^{\star}$, we then have
$T_{\xi}\mathcal{A}^{\star}\cong\mathcal{A}^{\star}$ and
$T^{\ast}_{\xi}\mathcal{A}^{\star}\cong\mathcal{A}^{\star\star}\cong\mathcal{A}$.
We may then represent the product $\bullet$ via

$\langle\xi,a\bullet b\rangle\qquad\text{for }\xi\in\mathcal{A}^{\star},\
a,b\in\mathcal{A}.$ (1.1)

Furthermore, this induces a multiplication on
$C^{\infty}(\mathcal{A}^{\star})$ via

$f\bullet g(\xi):=\langle\xi,d_{\xi}f\bullet d_{\xi}g\rangle,$ (1.2)

and for $f\in C^{\infty}(\mathcal{A}^{\star})$, we may define its
$\mathcal{A}$-dual vector field of $f$ as

$\big{(}\nabla^{\mathcal{A}}(f)\big{)}_{\xi}(g):=\\{f,g\\}_{\mathcal{A}}(\xi).$

Note that, in general, this is not a gradient because $\bullet$ need not be
symmetric, but we use the symbol $\nabla$ because it satisfies many of the
formal identities of gradients.

The multiplication in equation (1.2) is of course compatible with the product
$\bullet$ in the sense that, identifying $a\in\mathcal{A}$ with the linear
functional $f_{a}(\xi):=\langle\xi,a\rangle$, the inclusion
$\mathcal{A}\hookrightarrow C^{\infty}(\mathcal{A}^{\star})$ is an algebra
monomomorphism

$f_{a}\bullet f_{b}=f_{a\bullet b},$

justifying the symbol $\bullet$ to denote the multiplication on both
$\mathcal{A}$ and $C^{\infty}(\mathcal{A}^{\star})\supset\mathcal{A}$. The
automorphisms of $\mathcal{A}$, i.e., the linear isomorphisms
$g\colon\mathcal{A}\to\mathcal{A}$ with $g(a\bullet b)=(ga)\bullet(gb)$, form
a Lie group. The Lie algebra of that group consists of the derivations, i.e.,
the linear maps $d\in{\mathfrak{gl}}(\mathcal{A})$ with $d(a\bullet
b)=(da)\bullet b+a\bullet(db)$. Moreover, we have the structure Lie group
$\mathrm{G}(\mathcal{A})$ whose Lie algebra is generated by left
multiplications, i.e., by the maps $l_{a}\colon(b\mapsto a\bullet
b)\in{\mathfrak{gl}}(\mathcal{A})$.

Therefore, even though we do not assume $\mathcal{A}$ to be a Lie algebra,
there is a Lie algebra that is naturally associated to $\mathcal{A}$, and we
may hope to use its theory to gain insight about $\mathcal{A}$ itself. This
works to some extent, but a problem arises from the fact that the
$\mathcal{A}$-dual distribution $\mathcal{H}^{\mathcal{A}}$ on
$\mathcal{A}^{\star}$ defined by

$\mathcal{H}^{\mathcal{A}}_{\xi}:=\big{\\{}\big{(}\nabla^{\mathcal{A}}(f)\big{)}_{\xi}\mid
f\in C^{\infty}(\mathcal{A}^{\star})\big{\\}}\subset
T_{\xi}\mathcal{A}^{\star}.$

is in general not integrable. We recall that, in the Lie algebra case,
$\mathcal{H}^{\mathcal{A}}$ is integrable, and its leaves are precisely the
coadjoint orbits which carry a symplectic structure induced by equation (1.2).
Because $\mathcal{H}^{\mathcal{A}}$ is not integrable for general algebras, we
cannot expect the same level of generality as in Lie algebras.

However, the results become stronger if we also assume some additional
structures on $\mathcal{A}$. Associativity already gives us some leverage, but
the more specific case that we are interested in here is when $\mathcal{A}$ is
a Jordan algebra. Our strategy then is to combine this Jordan structure, and
the identities resulting from it, with the Lie algebra structure that we just
have identified.

Thus, we consider a finite-dimensional real Jordan algebra
$\mathcal{A}=\mathcal{J}$. In this case, we can also define an extended
structure Lie algebra $\hat{\mathfrak{g}}(\mathcal{J})$ that maps surjectively
onto ${\mathfrak{g}}(\mathcal{J})$ [13, 52, 55]. This algebra is the direct
sum $\mathcal{D}{\rm er}_{0}(\mathcal{J})\oplus\mathcal{J}$ of the inner
derivations (those generated by left multiplications $l_{x}$ with algebra
elements $x$) and $\mathcal{J}$ itself. The Lie bracket on $\mathcal{D}{\rm
er}_{0}(\mathcal{J})$ is the commutator, while, for $x,y\in\mathcal{J}$, it is
$[x,y]:=[l_{x},l_{y}]\in\mathcal{D}{\rm er}_{0}(\mathcal{J})$, and, for
$d\in\mathcal{D}{\rm er}_{0}(\mathcal{J})$ and $x\in\mathcal{J}$, it is
$[d,x]=-[x,d]:=d(x)\in\mathcal{J}$. Therefore, putting
${\mathfrak{k}}=\mathcal{D}{\rm er}_{0}(\mathcal{J})$,
${\mathfrak{m}}_{\mathcal{J}}=\mathcal{J}$, we have
$[{\mathfrak{k}},{\mathfrak{k}}]\subset{\mathfrak{k}}$,
$[{\mathfrak{k}},{\mathfrak{m}}_{\mathcal{J}}]\subset{\mathfrak{m}}_{\mathcal{J}}$,
$[{\mathfrak{m}}_{\mathcal{J}},{\mathfrak{m}}_{\mathcal{J}}]={\mathfrak{k}}$,
i.e., we have a transvective symmetric pair (see Definition 2.5) which
provides us with further structure to work with.

The generalized distribution $\mathcal{H}^{\mathcal{J}}$ on
$\mathcal{J}^{\star}$ is still not integrable in general, but the bilinear
form $\mathcal{G}_{\xi}$ induced by equation (1.1) on
$\mathcal{H}^{\mathcal{J}}_{\xi}$ is symmetric, and therefore it defines a
pseudo-Riemannian metric on the ${\mathfrak{m}}_{\mathcal{J}}$-regular part of
each $\mathrm{G}(\mathcal{J})$-orbit $\mathcal{O}^{\rm
reg}_{{\mathfrak{m}}_{\mathcal{J}}}\subset\mathcal{J}^{\star}$ (see Definition
2.3).

On $\mathcal{J}$ there is the symmetric, bilinear form
$\tau(x,y):=\operatorname{tr}l_{\\{x,y\\}}$, which is also associative with
respect to the Jordan product, and we can use it to decompose $\mathcal{J}$
[52, p. 59]. Specifically, in the case of positive Jordan algebras, $\tau$ is
positive definite so that there is a canonical identification
${\mathcal{J}\cong\mathcal{J}^{\star}}$, and we can also derive additional
properties from the above-mentioned decomposition. Each
$\xi\in\mathcal{J}^{\star}$ has a spectral decomposition associating with
$\xi$ its spectral coefficients $(\lambda_{i})_{i=1}^{r}\in{\mathbb{R}}^{r}$,
where $r$ denotes the rank of $\mathcal{J}$. The pair $(n_{+},n_{-})$ counting
the number of positive and negative spectral coefficients is called the
spectral signature of $\xi$. We then show the following:

######  Theorem A (cf. Theorem 4.7).

If $\mathcal{J}$ is a positive simple real Jordan algebra, then the orbits of
the structure group $\mathrm{G}(\mathcal{J})$ consist of all elements with the
same spectral signature.

We also characterize the regular points (i.e., those where the generalized
distribution $\mathcal{H}^{\mathcal{J}}$ is integrable) in such a
$\mathrm{G}(\mathcal{J})$-orbit in Theorem 4.8 and describe the pseudo-
Riemannian metric $\mathcal{G}_{\xi}$ at each regular point
$\xi\in\mathcal{J}^{\star}$ in Proposition 4.9. Specifically, let
$\Omega_{\mathcal{J}}$ denote the cone of squares of $\mathcal{J}$, i.e., the
interior of the set $\big{\\{}x^{2}\mid x\in\mathcal{J}\big{\\}}$. Then
$\Omega_{\mathcal{J}}$ is the $\mathrm{G}(\mathcal{J})$-orbit of the identity
${\bf 1}_{\mathcal{J}}$. The characterizations in Theorem 4.8, Proposition
3.3, and Proposition 4.9 lead to the following remarkable description:

######  Theorem B.

Let $\mathcal{J}$ be a positive simple real Jordan algebra. Then all points of
a $\mathrm{G}(\mathcal{J})$-orbit $\mathcal{O}\subset\mathcal{J}^{\star}$ are
${\mathfrak{m}}_{\mathcal{J}}$-regular iff
$\mathcal{O}\subset\overline{\Omega}_{\mathcal{J}}$ or
$\mathcal{O}\subset-\overline{\Omega}_{\mathcal{J}}$. The form $\mathcal{G}$
on $\mathcal{O}$ is positive definite in the first and negative definite in
the second case, thus defining a Riemannian metric on $\mathcal{O}$ which is
invariant with respect to the action of the automorphism group of
$\mathcal{J}$.

For all regular $\xi\notin\pm\overline{\Omega}_{\mathcal{J}}$ the form
$\mathcal{G}_{\xi}$ is indefinite, so the definiteness of $\mathcal{G}_{\xi}$
gives a new characterization of $\Omega_{\mathcal{J}}$. We provide
descriptions of the orbits $\mathcal{O}$ and the metric $\mathcal{G}$ for the
standard examples of positive simple real algebras. Moreover, the above
results easily generalize to the case of non-simple positive Jordan algebras,
as these algebras are direct sums of positive simple Jordan algebras.

This work is structured as follows. In Section 2, we set the notation and
recall some standard results on generalized distributions and group actions on
manifolds that are needed. In Section 3, we discuss the structure group and
the generalized distributions on an arbitrary finite-dimensional algebra
$\mathcal{A}$. In Section 4, we focus on Jordan algebras, prove the main
results, and discuss relevant examples. Let us stress that some results
recalled in Section 4 are well known to researchers working with Jordan
algebras, but we decided to recall them nonetheless in order to make the work
as self-contained as possible for readers, perhaps coming from information
geometry, who are not familiar with Jordan algebras. Finally, in Section 5, we
discuss our results in relation with some important established results on the
mathematics and geometry of Jordan algebras.

##  2 Preliminary material

###  2.1 Notational conventions

For a finite-dimensional (real or complex) vector space $V$, we denote by
${\mathrm{Gl}}(V)$ the Lie group of linear automorphisms, whose Lie algebra
${\mathfrak{gl}}(V)$ consists of all linear endomorphisms of $V$.

For finite-dimensional vector spaces $V$, $W$ and $U$, we define the
contraction map

$\langle\cdot,\cdot\rangle_{V}\colon\ V^{\star}\otimes V\otimes
U\longrightarrow U,\qquad\langle\alpha,v\otimes u\rangle_{V}:=\alpha(v)u,$
(2.1)

where $V^{\star}$ denotes the dual space of $V$, and we shall usually omit the
subscript if this causes no ambiguity. The notation (2.1) can also be used to
denote maps

$V^{\star}\otimes\Lambda^{k}V^{\star}\otimes
U\longrightarrow\Lambda^{k-1}V^{\star}\otimes U,\qquad V^{\star}\otimes
S^{k}V^{\star}\otimes U\longrightarrow S^{k-1}V^{\star}\otimes U,$

denoting by $S^{k}V^{\star}$ the symmetric $k$-forms, since
$\Lambda^{k}V^{\star}\hookrightarrow V\otimes\Lambda^{k-1}V^{\star}$ and
$S^{k}V^{\star}\hookrightarrow V\otimes S^{k-1}V^{\star}$ are canonically
included.

Finally, the dual of a linear map $\phi\colon V\to W$ is denoted as

$\phi^{\star}\colon\ W^{\star}\longrightarrow
V^{\star},\qquad\langle\theta,\phi(v)\rangle_{W}=\langle\phi^{\star}\theta,v\rangle_{V}.$

###  2.2 Generalized distributions

Let $M$ be a finite-dimensional, real smooth manifold. A generalized
distribution on $M$ is a family $\mathcal{D}=(\mathcal{D}_{p})_{p\in M}$ of
subspaces $\mathcal{D}_{p}\subset T_{p}M$. We let $\Gamma(\mathcal{D})$ be the
set of (smooth) vector fields $X$ on $M$ with $X_{p}\in\mathcal{D}_{p}$ for
all $p$, and we call $\mathcal{D}$ smooth if for each $v\in\mathcal{D}_{p}$
there is a vector field $X\in\Gamma(\mathcal{D})$ with $X_{p}=v$.

Given a smooth generalized distribution $\mathcal{D}$, we define the Frobenius
tensor ${\mathcal{F}}_{p}$ at $p\in M$ as

${\mathcal{F}}_{p}\colon\ \Lambda^{2}\mathcal{D}_{p}\longrightarrow
T_{p}M/\mathcal{D}_{p},\qquad(X_{p},Y_{p})\longmapsto[X,Y]_{p}\mod\mathcal{D}_{p}$

for $X,Y\in\Gamma(\mathcal{D})$. It is straightforward to verify that
${\mathcal{F}}_{p}(X,Y)$ depends on $X_{p}$ and $Y_{p}$ only, i.e.,
${\mathcal{F}}=({\mathcal{F}}_{p})_{p\in M}$ is a well-defined tensor field.
We also define the generalized distribution

${}[\mathcal{D},\mathcal{D}]_{p}:=\mathcal{D}_{p}+\\{[X,Y]_{p}\mid
X,Y\in\Gamma(\mathcal{D})\\},$

so that the image of ${\mathcal{F}}_{p}$ is
$[\mathcal{D},\mathcal{D}]_{p}/\mathcal{D}_{p}$.

######  Definition 2.1.

We call a smooth generalized distribution $\mathcal{D}$ involutive at $p\in
M$, if ${\mathcal{F}}_{p}=0$ or, equivalently, if
$[\mathcal{D},\mathcal{D}]_{p}=\mathcal{D}_{p}$, and we call it involutive, if
this holds for every $p$.

Furthermore, an (immersed) submanifold $N\subset M$ with
$T_{p}N=\mathcal{D}_{p}$ for all $p\in N$ is called an integral leaf of
$\mathcal{D}$. If there is an integral leaf of $\mathcal{D}$ containing $p$,
then we call $\mathcal{D}$ integrable at $p\in M$ and call $p$ an integral
point of $\mathcal{D}$; if this is the case for each $p\in M$, then we call
$\mathcal{D}$ integrable.

Clearly, if $\mathcal{D}$ is integrable (at $p$), then it is also involutive
(at $p$); according to Frobenius’ theorem, the converse of this statement
holds if $\mathcal{D}$ has constant rank. However, if the rank of
$\mathcal{D}$ is non-constant, then the converse may fail to hold [72].

###  2.3 $\boldsymbol{G}$-manifolds

Let $G$ be a finite-dimensional, real Lie group with identity element $e$, Lie
algebra ${\mathfrak{g}}\cong T_{e}G$, and let $M$ be a $G$-manifold, i.e., a
finite-dimensional, real smooth manifold with a smooth left action $\pi\colon
G\times M\to M$, $(g,p)\mapsto g\cdot p$. For $p\in M$ we define the
stabilizer of $p$ to be the subgroup

$H_{p}:=\\{g\in G\mid g\cdot p=p\\}\subset G\qquad\text{with Lie
algebra}\quad{\mathfrak{h}}_{p}\subset{\mathfrak{g}}.$

Evidently, $H_{g\cdot p}=gHg^{-1}$, and ${\mathfrak{h}}_{g\cdot
p}=\operatorname{Ad}_{g}({\mathfrak{h}}_{p})$, so that the stabilizer on each
$G$-orbit is unique up to conjugation.

We define the orbit distribution on $M$ by

$\mathcal{D}^{\mathfrak{g}}_{p}:=\\{(X_{\ast})_{p}\mid X\in{\mathfrak{g}}\\},$

where $X_{\ast}$ denotes the action field on $M$. Evidently,
$\mathcal{D}^{\mathfrak{g}}$ is integrable, as the $G$-orbits are integral
leaves of $\mathcal{D}^{\mathfrak{g}}$. Moreover,

$X\in{\mathfrak{h}}_{p}\quad\Leftrightarrow\quad(X_{\ast})_{p}=0.$

Also recall that the map $X\mapsto X_{\ast}$ is a anti-homomorphism of Lie
algebras, i.e.,

${}[X,Y]_{\ast}=-[X_{\ast},Y_{\ast}].$ (2.2)

For any linear subspace ${\mathfrak{m}}\subset{\mathfrak{g}}$, we define the
smooth generalized distribution $\mathcal{D}^{\mathfrak{m}}$ by

$\mathcal{D}^{\mathfrak{m}}_{p}:=\\{(X_{\ast})_{p}\mid
X\in{\mathfrak{m}}\\}\subset\mathcal{D}^{\mathfrak{g}}_{p}=T_{p}(G\cdot p),$
(2.3)

and evidently,

$(X_{\ast})_{p}\in\mathcal{D}^{\mathfrak{m}}_{p}\quad\Leftrightarrow\quad
X\in{\mathfrak{m}}+{\mathfrak{h}}_{p}.$ (2.4)

######  Lemma 2.2.

Let ${\mathfrak{m}}\subset{\mathfrak{g}}$ be a linear subspace. Then the
following are equivalent:

  1. $(1)$

$\mathcal{D}^{\mathfrak{m}}$ is involutive at $p\in\mathcal{O}$,

  2. $(2)$

$[{\mathfrak{m}},{\mathfrak{m}}]\subset{\mathfrak{m}}+{\mathfrak{h}}_{p}$,

  3. $(3)$

$\mathcal{D}^{[{\mathfrak{m}},{\mathfrak{m}}]}_{p}\subset\mathcal{D}^{\mathfrak{m}}_{p}$.

###### Proof.

For $X,Y\in{\mathfrak{m}}$, ${\mathcal{F}}_{p}(X_{p},Y_{p})=0$ iff
$[X_{\ast},Y_{\ast}]_{p}\in\mathcal{D}^{\mathfrak{m}}_{p}$, which, by equation
(2.2), is the case iff $([X,Y]_{\ast})_{p}\in\mathcal{D}^{\mathfrak{m}}_{p}$,
and, by equation (2.4), this is the case iff
$[X,Y]\in{\mathfrak{m}}+{\mathfrak{h}}_{p}$, showing the equivalence of the
first two conditions.

The second condition is equivalent to saying that for each
$X\in[{\mathfrak{m}},{\mathfrak{m}}]$ there is a $Y\in{\mathfrak{m}}$ such
that $X-Y\in{\mathfrak{h}}_{p}$ or, equivalently, that for each
$X\in[{\mathfrak{m}},{\mathfrak{m}}]$ there is a $Y\in{\mathfrak{m}}$ such
that $(X_{\ast})_{p}=(Y_{\ast})_{p}$, and this is evidently equivalent to the
third condition. ∎

######  Definition 2.3.

Let $M$ be a $G$-manifold and ${\mathfrak{m}}\subset{\mathfrak{g}}$ a linear
subspace. We call $p\in M$ an ${\mathfrak{m}}$-regular point if
$\mathcal{D}^{\mathfrak{m}}_{p}=T_{p}\mathcal{O}$, where $\mathcal{O}\subset
M$ is the $G$-orbit of $p$. The subset of ${\mathfrak{m}}$-regular points in
$\mathcal{O}$ is denoted by $\mathcal{O}^{\rm
reg}_{{\mathfrak{m}}}\subset\mathcal{O}$.

As the rank of $\mathcal{D}^{\mathfrak{m}}_{p}$ is a lower semicontinuous
function in $p$, $\mathcal{O}^{\rm reg}\subset\mathcal{O}$ is open (but
possibly empty). As we shall see in later sections, $\mathcal{O}^{\rm
reg}_{{\mathfrak{m}}}$ may be a proper subset of $\mathcal{O}$ and is not
necessarily connected.

######  Corollary 2.4.

Suppose that ${\mathfrak{m}}\subset{\mathfrak{g}}$ is a linear subspace such
that

${\mathfrak{g}}={\mathfrak{m}}+[{\mathfrak{m}},{\mathfrak{m}}].$ (2.5)

Then for each $p\in M$ the following are equivalent:

  1. $(1)$

$p$ is involutive,

  2. $(2)$

$p$ is integrable,

  3. $(3)$

$p$ is an ${\mathfrak{m}}$-regular point.

In this case, the maximal integral leaf through $p$ is the connected component
of $p$ in

$\mathcal{O}^{\rm reg}_{{\mathfrak{m}}}\subset\mathcal{O}=\mathrm{G}\cdot p.$

###### Proof.

By Lemma 2.2, $p$ is integrable iff
$\mathcal{D}^{[{\mathfrak{m}},{\mathfrak{m}}]}_{p}\subset\mathcal{D}^{\mathfrak{m}}_{p}$,
and as ${\mathfrak{g}}={\mathfrak{m}}+[{\mathfrak{m}},{\mathfrak{m}}]$, this
is the case iff
$\mathcal{D}^{\mathfrak{m}}_{p}=\mathcal{D}^{\mathfrak{g}}_{p}$, i.e., iff $p$
is ${\mathfrak{m}}$-regular. It follows that any integral leaf through $p$
must be (an open subset of) $\mathcal{O}^{\rm
reg}_{{\mathfrak{m}}}\subset\mathcal{O}$, whence the maximal (connected) leaf
through $p$ is its path component in $\mathcal{O}^{\rm reg}_{{\mathfrak{m}}}$.
∎

######  Definition 2.5.

A symmetric pair is a pair $({\mathfrak{g}},{\mathfrak{k}})$ of Lie algebras
with a decomposition ${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{m}}$
satisfying

${}[{\mathfrak{k}},{\mathfrak{k}}]\subset{\mathfrak{k}},\qquad[{\mathfrak{k}},{\mathfrak{m}}]\subset{\mathfrak{m}},\qquad[{\mathfrak{m}},{\mathfrak{m}}]\subset{\mathfrak{k}}.$
(2.6)

We call this pair transvective, if ${\mathfrak{g}}$ is generated by
${\mathfrak{m}}$ as a Lie algebra, i.e., if
$[{\mathfrak{m}},{\mathfrak{m}}]={\mathfrak{k}}$.

Clearly, equation (2.6) is equivalent to saying that the involution
$\sigma\colon{\mathfrak{g}}\to{\mathfrak{g}}$ with ${\mathfrak{k}}$ and
${\mathfrak{m}}$ as the $(+1)$\- and $(-1)$-eigenspace, respectively, is a Lie
algebra automorphism.

##  3 Structure groups and canonical distributions
on duals of algebras

In this section, we shall consider a finite-dimensional algebra $\mathcal{A}$,
by which we simply mean a finite-dimensional (real or complex) vector space
with a bilinear product
$\bullet\colon\mathcal{A}\times\mathcal{A}\to\mathcal{A}$, i.e., a constant
$(2,1)$-tensor. We do not assume any further conditions on this multiplication
such as associativity, Jacobi or Jordan identities, but we shall later discuss
the general definitions in each of these cases.

The dual of $\bullet$ is a map
$\mathcal{A}^{\star}\to\mathcal{A}^{\star}\otimes\mathcal{A}^{\star}$, and as
$T_{\xi}\mathcal{A}^{\star}\cong\mathcal{A}^{\star}$, we may regard this as a
linear bivector field on $\mathcal{A}^{\star}$:

$\mathcal{R}^{\mathcal{A}}\in\Gamma(\mathcal{A}^{\star},T\mathcal{A}^{\star}\otimes
T\mathcal{A}^{\star}),\qquad\mathcal{R}^{\mathcal{A}}_{\xi}(a,b):=\langle\xi,a\bullet
b\rangle$ (3.1)

for all $a,b\in\mathcal{A}$. Therefore, there is an induced multiplication on
the space $C^{\infty}(\mathcal{A}^{\star})$ of real-valued, smooth functions
on $\mathcal{A}^{\star}$ which by abuse of notation we also denote by
$\bullet$, given by

$f\bullet
g(\xi):=\big{(}\mathcal{R}^{\mathcal{A}}\big{)}_{\xi}(d_{\xi}f,d_{\xi}g)=\langle\xi,d_{\xi}f\bullet
d_{\xi}g\rangle$ (3.2)

with the canonical identification $d_{\xi}f,d_{\xi}g\in
T^{\ast}_{\xi}\mathcal{A}^{\star}\cong\mathcal{A}^{\star\star}\cong\mathcal{A}$.
In particular, regarding $\mathcal{A}\subset C^{\infty}(\mathcal{A}^{\star})$
as the set of linear functions $f_{a}(\xi):=\langle\xi,a\rangle$, equation
(3.2) implies

$f_{a}\bullet f_{b}=f_{a\bullet b},$

justifying the ambiguous use of the symbol $\bullet$. Contraction in the first
entry yields a linear map

$\sharp_{\xi}\colon\ T^{\star}_{\xi}\mathcal{A}^{\star}\longrightarrow
T_{\xi}\mathcal{A}^{\star},\qquad\theta\longmapsto\theta\lrcorner\mathcal{R}^{\mathcal{A}}_{\xi},$
(3.3)

whose image we call the $\mathcal{A}$-dual distribution
$\mathcal{H}^{\mathcal{A}}$ on $\mathcal{A}^{\star}$ by

$\mathcal{H}^{\mathcal{A}}_{\xi}:=\sharp_{\xi}(T_{\xi}^{\star}\mathcal{A}^{\star})\subset
T_{\xi}\mathcal{A}^{\star}.$ (3.4)

######  Definition 3.1.

For a finite-dimensional algebra $(\mathcal{A},\bullet)$ we define the
following:

  1. (1)

For $a\in\mathcal{A}$ we let $l_{a}\in{\mathfrak{gl}}(\mathcal{A})$ be the map
$(b\mapsto a\bullet b)\in{\mathfrak{gl}}(\mathcal{A})$.

  2. (2)

The structure Lie algebra of $\mathcal{A}$ is the Lie subalgebra
${\mathfrak{g}}(\mathcal{A})\subset{\mathfrak{gl}}(\mathcal{A})$ generated by
${\mathfrak{m}}_{\mathcal{A}}:=\\{l_{a}\mid a\in\mathcal{A}\\}$.

  3. (3)

The structure Lie group of $\mathcal{A}$ is the connected Lie subgroup
$\mathrm{G}(\mathcal{A})\subset{\mathrm{Gl}}(\mathcal{A})$ with Lie algebra
${\mathfrak{g}}(\mathcal{A})$.

  4. (4)

A derivation of $\mathcal{A}$ is a linear map
$d\in{\mathfrak{gl}}(\mathcal{A})$ with $d(a\bullet b)=(da)\bullet
b+a\bullet(db)$.

  5. (5)

An automorphism of $\mathcal{A}$ is a linear isomorphism
$g\colon\mathcal{A}\to\mathcal{A}$ with $g(a\bullet b)=(ga)\bullet(gb)$.

It is straightforward to verify that the automorphisms and derivations form a
regular Lie subgroup and a Lie subalgebra $\mathcal{A}{\rm
ut}(\mathcal{A})\subset{\mathrm{Gl}}(\mathcal{A})$ and $\mathcal{D}{\rm
er}(\mathcal{A})\subset{\mathfrak{gl}}(\mathcal{A})$, respectively, called the
automorphism group and derivation algebra of $\mathcal{A}$, respectively. In
fact, $\mathcal{D}{\rm er}(\mathcal{A})$ is the Lie algebra of
$\mathcal{A}{\rm ut}(\mathcal{A})$. Moreover, $g\in\mathcal{A}{\rm
ut}(\mathcal{A})$ and $d\in\mathcal{D}{\rm er}(\mathcal{A})$ iff for all
$a\in\mathcal{A}$ we have

$gl_{a}g^{-1}=l_{ga},\qquad[d,l_{a}]=l_{da}.$ (3.5)

That is, the adjoint action of $\mathcal{A}{\rm ut}(\mathcal{A})$ and
$\mathcal{D}{\rm er}(\mathcal{A})$ on ${\mathfrak{gl}}(\mathcal{A})$ preserves
the subspace ${\mathfrak{m}}_{\mathcal{A}}$ and hence the structure Lie
algebra ${\mathfrak{g}}(\mathcal{A})$ and structure Lie group
$\mathrm{G}(\mathcal{A})$.111Actually, it would be more accurate to call
${\mathfrak{g}}(\mathcal{A})$ and $\mathrm{G}(\mathcal{A})$ the left-structure
Lie algebra and group, respectively, and to define the right-structure Lie
algebra and group analogously. However, for simplicity we shall restrict
ourselves to the left-structure case, as the right-structure case can be
treated in complete analogy.

For $\xi\in\mathcal{A}^{\star}$ and $a,b\in\mathcal{A}$, we have

$\langle
l_{a}^{\ast}(\xi),b\rangle=\langle\xi,l_{a}(b)\rangle=\langle\xi,a\bullet
b\rangle\stackrel{{\scriptstyle(\ref{eq:R-bivectorfield})}}{{=}}\big{\langle}a\lrcorner\mathcal{R}^{\mathcal{A}}_{\xi},b\big{\rangle}\stackrel{{\scriptstyle(\ref{eq:sharp-
xi})}}{{=}}\langle\sharp a,b\rangle_{\xi},$

so that $l_{a}^{\ast}=\sharp
a\in\Gamma(\mathcal{A}^{\star},T^{\ast}\mathcal{A}^{\star})$, regarded as a
linear $1$-form on $\mathcal{A}^{\star}$. It follows that

$\mathcal{H}^{\mathcal{A}}_{\xi}=\mathcal{D}^{{\mathfrak{m}}_{\mathcal{A}}}_{\xi}$
(3.6)

with $\mathcal{H}^{\mathcal{A}}_{\xi}$ from equation (3.4) and
${\mathfrak{m}}_{\mathcal{A}}$ from Definition 3.1, regarding
$\mathcal{A}^{\star}$ as a $\mathrm{G}(\mathcal{A})$-manifold via the dual
representation
$\imath\colon\mathrm{G}(\mathcal{A})\to{\mathrm{Gl}}(\mathcal{A}^{\star})$.

More generally, for a smooth function $f\in C^{\infty}(\mathcal{A}^{\star})$,
we define the $\mathcal{A}$-dual vector field of $f$ as

$\nabla^{\mathcal{A}}(f):=\sharp df\in\mathfrak{X}(\mathcal{A}^{\star}).$
(3.7)

Since $\sharp a=\sharp f_{a}=\nabla^{\mathcal{A}}(f_{a})$, it follows that the
$\mathcal{A}$-dual distribution may also be characterized by

$\mathcal{H}^{\mathcal{A}}_{\xi}=\big{\\{}\big{(}\nabla^{\mathcal{A}}(f)\big{)}_{\xi}\mid
f\in C^{\infty}(\mathcal{A}^{\star})\big{\\}},$

and unwinding the definitions, it easily follows that for $f,g\in
C^{\infty}(\mathcal{A}^{\star})$ that

$\big{(}\nabla^{\mathcal{A}}(f)\big{)}(g)=\\{f,g\\}_{\mathcal{A}}.$

######  Remark 3.2.

If the multiplication $\bullet$ is symmetric (e.g., if $\mathcal{A}$ is a
Jordan algebra), the dual vector field $\nabla^{\mathcal{A}}(f)$ is usually
referred to as the gradient vector field of $f$, while in the case of an
antisymmetric multiplication $\bullet$ (e.g., if $\mathcal{A}$ is a Lie
algebra), it is called the Hamiltonian vector field of $f$. That is, the term
$\mathcal{A}$-dual vector field subsumes both cases.

We wish to caution the reader that in case of a skew-symmetric multiplication
$\bullet$, the notation $\nabla^{\mathcal{A}}(f)$ for the Hamiltonian vector
does not match the standard convention. We use it nevertheless to unify our
notation.

If $g\in\mathcal{A}{\rm ut}(\mathcal{A})$ is an automorphism, then by equation
(3.5) the action of $g^{\ast}$ on $\mathcal{A}^{\star}$ preserves the
distribution $\mathcal{H}^{\mathcal{A}}$ and hence permutes integral leaves of
equal dimensions, preserving ${\mathfrak{m}}_{\mathcal{A}}$-regular points.

As it turns out, if the product $\bullet$ is symmetric or skew-symmetric, then
there is a canonical bilinear pairing on $\mathcal{H}^{\mathcal{A}}$.

######  Proposition 3.3.

Let $(\mathcal{A},\bullet)$ be a finite-dimensional real algebra such that
$\bullet$ is symmetric $($skew-symmetric, respectively$)$. Then on
$\mathcal{H}^{\mathcal{A}}_{\xi}=\mathcal{D}^{{\mathfrak{m}}_{\mathcal{A}}}_{\xi}\subset
T_{\xi}\mathcal{A}^{\star}$ there is a canonical non-degenerate symmetric
$($skew-symmetric, respectively$)$ bilinear form, given by

$\mathcal{G}_{\xi}(l_{a}^{\ast}(\xi),l_{b}^{\ast}(\xi)):=\langle\xi,a\bullet
b\rangle.$ (3.8)

Furthermore, $\mathcal{G}$ is preserved by the action of the automorphism
group $\mathcal{A}{\rm ut}(\mathcal{A})$.

###### Proof.

Since $\langle\xi,a\bullet b\rangle=\langle
l_{a}^{\ast}(\xi),b\rangle=\pm\langle l_{b}^{\ast}(\xi),a\rangle$, where the
sign $\pm$ depends on the symmetry or skew-symmetry of $\bullet$, it follows
that $\mathcal{G}$ is indeed well defined and non-degenerate.

Finally, if $g\in\mathcal{A}{\rm ut}(\mathcal{A})$ is an automorphism, then
equation (3.5) implies that

$g^{\ast}l_{a}^{\ast}\big{(}g^{-1}\big{)}^{\ast}=l_{g^{-1}a}^{\ast}$

and from here, the invariance of $\mathcal{G}$ under the action of the
automorphism group follows. ∎

######  Definition 3.4.

Let $\mathcal{A}$ be an algebra. A $\mathrm{G}(\mathcal{A})$-orbit
$\mathcal{O}\subset\mathcal{A}^{\star}$ is called
${\mathfrak{m}}_{\mathcal{A}}$-regular  if $\mathcal{O}^{\rm
reg}_{{\mathfrak{m}}_{\mathcal{A}}}=\mathcal{O}$.

We shall now give classes of examples of these notions.

1\. Lie algebras. Let $(\mathcal{A},\bullet)=({\mathfrak{g}},[\cdot,\cdot])$
be a Lie algebra. Then the induced section
$\Lambda:=\mathcal{R}^{\mathfrak{g}}\in\Gamma\big{(}{\mathfrak{g}}^{\star},\Lambda^{2}T{\mathfrak{g}}^{\star}\big{)}$
from equation (3.1) is a skew-symmetric bi-vector field, and the Jacobi
identity implies that the Schouten–Nijenhuis bracket
$[\Lambda,\Lambda]\in\Gamma({\mathfrak{g}}^{\star},\Lambda^{3}T{\mathfrak{g}}^{\star})$
vanishes [57, 75], so that $\Lambda$ defines a linear Poisson structure
$\\{\cdot,\cdot\\}$ on ${\mathfrak{g}}^{\star}$, also known as the
Kirillov–Kostant–Souriau structure [51].

Comparing our notions with those established for Poisson manifolds, we observe
that for a function $f\in C^{\infty}({\mathfrak{g}}^{\star})$, the
${\mathfrak{g}}$-gradient vector field $\nabla^{\mathcal{A}}(f)$ corresponds
to the Hamiltonian vector field $X_{f}$ for Poisson manifolds, so that the
dual distribution $\mathcal{H}^{\mathcal{A}}_{\xi}$ from equation (3.4) is the
Hamiltonian distribution of the Poisson manifold. It is integrable, as the
Hamiltonian vector fields satisfy the identity

$[X_{f},X_{g}]=-X_{\\{f,g\\}}.$

The Jacobi identity implies that $l_{a}=ad_{a}$ satisfies
$[l_{a},l_{b}]=l_{[a,b]}$, so that ${\mathfrak{m}}_{\mathcal{A}}$ is closed
under the commutator bracket and therefore,
${\mathfrak{g}}(\mathcal{A})={\mathfrak{m}}_{\mathcal{A}}\cong{\mathfrak{g}}/\mathfrak{z}({\mathfrak{g}})$.
That is, the action of the structure group is induced by the coadjoint action
of $G$ on ${\mathfrak{g}}^{\star}$, and the skew-symmetric non-degenerate
bilinear form $\mathcal{G}$ on $\mathcal{H}^{\mathcal{A}}$ from equation (3.8)
coincides with the symplectic form on each coadjoint orbit in
${\mathfrak{g}}^{\star}$. By the Jacobi identity, this action consists of
automorphisms of the Lie algebra structure, whence this symplectic form is
preserved under the coadjoint action.

Therefore, the integral leaves of ${\mathfrak{m}}_{\mathcal{A}}$ are the
coadjoint orbits of ${\mathfrak{g}}^{\star}$, equipped with their canonical
symplectic form, and hence, each orbit is regular in the sense of our
definition.

2\. Associative algebras. The associativity of the product $\bullet$ is
equivalent to saying that $l_{a}l_{b}=l_{a\bullet b}$, so that $\\{l_{a}\mid
a\in\mathcal{A}\\}\subset{\mathfrak{gl}}(\mathcal{A})$ is a subalgebra, which
means that the structure algebra ${\mathfrak{g}}(\mathcal{A})$ equals
${\mathfrak{m}}_{\mathcal{A}}$ with the Lie bracket being the commutator.

Thus, if we regard $\mathcal{A}$ as a Lie algebra with the Lie bracket
$[a,b]:=a\bullet b-b\bullet a$, then the $\mathrm{G}(\mathcal{A})$-orbits are
the coadjoint orbits on $\mathcal{A}^{\star}$, regarded as the dual of a Lie
algebra and thus described in the preceding paragraph.

Note that by Proposition 3.3 the bilinear form $\mathcal{G}$ on these orbits
only exists if $\bullet$ is symmetric or antisymmetric.

If $\mathcal{A}$ is a commutative and associative algebra, then
$\mathrm{G}(\mathcal{A})$ and ${\mathfrak{g}}(\mathcal{A})$ are abelian Lie
groups, respectively. In this case, the $\mathrm{G}(\mathcal{A})$-orbits of
$\mathcal{A}^{\star}$ are diffeomorphic to the direct product of a torus and
Euclidean space.

In the two preceding cases, ${\mathfrak{m}}_{\mathcal{A}}$ is closed under Lie
brackets, so that it coincides with the structure algebra
${\mathfrak{g}}(\mathcal{A})$. This implies that, by the very definition,
${\mathfrak{m}}_{\mathcal{A}}$ is integrable having the
$\mathrm{G}(\mathcal{A})$-orbits in $\mathcal{A}^{\star}$ as leaves. In
particular, all orbits are ${\mathfrak{m}}_{\mathcal{A}}$-regular.

In contrast, for a Jordan algebra $\mathcal{J}$, it is no longer true that
${\mathfrak{m}}_{\mathcal{J}}$ is a Lie algebra, so that not all
$\mathrm{G}(\mathcal{J})$-orbits on the dual $\mathcal{J}^{\star}$ are
${\mathfrak{m}}_{\mathcal{J}}$-regular in our sense. Since the Jordan product
is symmetric, the non-degenerate form $\mathcal{G}$ from (3.8) defines a
pseudo-Riemannian metric on the regular part $\mathcal{O}^{\rm
reg}_{{\mathfrak{m}}_{\mathcal{J}}}$ of each orbit.

We shall describe these structures on the $\mathrm{G}(\mathcal{J})$-orbits on
$\mathcal{J}^{\star}$ and the pseudo-Riemannian metric $\mathcal{G}$ in more
details, and we will see how, for some specific type of positive Jordan
algebras, and suitable orbits, $\mathcal{G}$ is intimately connected with
either the Fisher–Rao metric tensor or with the Bures–Helstrom metric tensor
used in classical and quantum information geometry, respectively. This result
strengthen the connection between Jordan algebras and information geometry
initially hinted at in [22, 23].

##  4 Jordan algebras and Jordan distributions

Let $\mathcal{J}$ be a real, finite-dimensional Jordan algebra, that is, a
real vector space endowed with a bilinear symmetric product
$\\{\cdot,\cdot\\}$, satisfying for $x,y\in\mathcal{J}$ the Jordan identity

$\\{\\{x,y\\},\\{x,x\\}\\}=\\{x,\\{y,\\{x,x\\}\\}\\}.$

By the notions established in the preceding section, we may associate with a
Jordan algebra the symmetric bivector field
$\mathcal{R}^{\mathcal{J}}\in\Gamma\big{(}\mathcal{J}^{\star},S^{2}(T\mathcal{J}^{\star})\big{)}$
from equation (3.1), the musical operator $\\#_{\mathcal{J}}\colon
T^{\ast}\mathcal{J}^{\star}\to T\mathcal{J}^{\star}$ from equation (3.3), the
$\mathcal{J}$-dual vector field
$\nabla^{\mathcal{J}}f=\\#df\in\mathfrak{X}(\mathcal{J}^{\star})$ from (3.7),
and the induced $\mathcal{J}$-dual distribution
$\mathcal{H}^{\mathcal{J}}\subset T\mathcal{J}^{\star}$ from equation (3.4).

In particular, we have
$\mathcal{H}^{\mathcal{J}}=\mathcal{D}^{{\mathfrak{m}}_{\mathcal{J}}}$ by
equation (3.6), where

${\mathfrak{m}}_{\mathcal{J}}=\\{l_{x}^{\ast}\mid
x\in\mathcal{J}\\}\subset{\mathfrak{gl}}(\mathcal{J}^{\star})$

is the space of (left-)multiplications with elements $x\in\mathcal{J}$, acting
on the dual space $\mathcal{J}^{\star}$. For every
${\mathfrak{m}}_{\mathcal{J}}$-regular point $\xi$, the vector space
$\mathcal{H}^{\mathcal{J}}_{\xi}=\mathcal{D}^{{\mathfrak{m}}_{\mathcal{J}}}_{\xi}$
carries the non-degenerate symmetric bilinear form $\mathcal{G}_{\xi}$ defined
in equation (3.8).

As we pointed out before, the space ${\mathfrak{m}}_{\mathcal{J}}$ of
(left-)multiplication in $\mathcal{J}$ is not closed under Lie brackets in
general. However, the following is known.

######  Lemma 4.1 (cf. [52, Lemma IV.7]).

For $x,y\in\mathcal{J}$, the commutator $[l_{x},l_{y}]$ is a derivation of
$\mathcal{J}$, and for each $d\in\mathcal{D}{\rm er}(\mathcal{J})$

$[d,[l_{x},l_{y}]]=[l_{dx},l_{y}]+[l_{x},l_{dy}].$ (4.1)

We denote by $\mathcal{D}{\rm er}_{0}(\mathcal{J})$ the span of all elements
of the form $[l_{x},l_{y}]$ for $x,y\in\mathcal{J}$. By equation (4.1),
$\mathcal{D}{\rm er}_{0}(\mathcal{J})\subset\mathcal{D}{\rm er}(\mathcal{J})$
is an ideal whose elements are called inner derivations of $\mathcal{J}$. This
fact can be used to describe the structure Lie algebra of $\mathcal{J}$.

######  Definition 4.2.

We define the extended structure Lie algebra $\hat{\mathfrak{g}}(\mathcal{J})$
of $\mathcal{J}$ as follows. As a vector space,
$\hat{\mathfrak{g}}(\mathcal{J})$ is defined by

$\hat{\mathfrak{g}}(\mathcal{J})=\mathcal{D}{\rm
er}_{0}(\mathcal{J})\oplus\mathcal{J}.$

The Lie bracket on $\hat{\mathfrak{g}}(\mathcal{J})$ is defined as follows:

  * •

on $\mathcal{D}{\rm er}_{0}(\mathcal{J})\subset\mathcal{D}{\rm
er}(\mathcal{J})\subset{\mathfrak{gl}}(\mathcal{J})$, the Lie bracket is just
the commutator between linear maps;

  * •

for $d\in\mathcal{D}{\rm er}_{0}(\mathcal{J})$ and $x\in\mathcal{J}$,
$[d,x]=-[x,d]:=d(x)\in\mathcal{J}$;

  * •

for $x,y\in\mathcal{J}$ we set $[x,y]:=[l_{x},l_{y}]\in\mathcal{D}{\rm
er}_{0}(\mathcal{J})$.

In fact, the Jacobi identity for this bracket is easily verified using the
definitions and equation (4.1). By the definition of this Lie bracket, it
follows that ($\hat{\mathfrak{g}}(\mathcal{J}),\mathcal{D}{\rm
er}_{0}(\mathcal{J})$) is a transvective symmetric pair in the sense of
Definition 2.5.

There is a canonical Lie algebra representation of
$\hat{\mathfrak{g}}(\mathcal{J})$ on $\mathcal{J}$, called the standard
representation, defined by

$\phi\colon\
\hat{\mathfrak{g}}(\mathcal{J})\longrightarrow{\mathfrak{gl}}(\mathcal{J}),\qquad\begin{cases}\phi(d):=d&\text{for
$d\in\mathcal{D}{\rm
er}_{0}(\mathcal{J})\subset{\mathfrak{gl}}(\mathcal{J})$},\\\
\phi(x):=l_{x}&\text{for $x\in\mathcal{J}$}.\end{cases}$ (4.2)

Indeed, this defines a Lie algebra homomorphism by the definition of the Lie
bracket on $\hat{\mathfrak{g}}(\mathcal{J})$ and by equation (3.5).

Observe that the image
$\phi(\hat{\mathfrak{g}}(\mathcal{J}))\subset{\mathfrak{gl}}(\mathcal{J})$ is
generated by all $l_{x}$, $x\in\mathcal{J}$, whence equals the structure Lie
algebra ${\mathfrak{g}}(\mathcal{J})$ from Definition 3.1. Thus, there is a
surjective Lie group homomorphism
$\hat{\mathrm{G}}(\mathcal{J})\to\mathrm{G}(\mathcal{J})$ with differential
$\phi$, where $\mathrm{G}(\mathcal{J})\subset{\mathrm{Gl}}(\mathcal{J})$ is
the structure group from Definition 3.1.222Definition 3.1 for a Jordan algebra
$\mathcal{J}$ coincides with the definition of the structure Lie group and the
structure Lie algebra of a Jordan algebra, e.g., in [52, Chapter IV].

In general, $\phi$ may fail to be injective (the kernel of $\phi$ contains the
center of
$\mathfrak{z}(\mathcal{J})\subset\mathcal{J}\subset{\mathfrak{g}}(\mathcal{J})$),
so that the structure algebra ${\mathfrak{g}}(\mathcal{J})$ and the extended
structure algebra $\hat{\mathfrak{g}}(\mathcal{J})$ may not be isomorphic.

Then we obtain the following integrability criterion.

######  Proposition 4.3.

Let $\mathcal{J}$ be a Jordan algebra. Then, for the distribution
$\mathcal{H}^{\mathcal{J}}$ from equation (3.4), the following assertions are
equivalent.

  1. $(1)$

$\mathcal{H}_{\xi}^{\mathcal{J}}$ is involutive at
$\xi\in\mathcal{J}^{\star}$,

  2. $(2)$

$\mathcal{H}_{\xi}^{\mathcal{J}}$ is integrable at
$\xi\in\mathcal{J}^{\star}$,

  3. $(3)$

$\mathcal{D}{\rm er}_{0}(\mathcal{J})\cdot\xi\subset\mathcal{J}\cdot\xi$,
where the multiplication refers to the dual action of $\mathcal{D}{\rm
er}_{0}(\mathcal{J})$, $\mathcal{J}\subset\hat{\mathfrak{g}}(\mathcal{J})$ on
$\mathcal{J}^{\star}$.

If this is the case, then the maximal integral leaf through $\xi$ is the
connected component of $\xi$ in $\mathcal{O}^{\rm
reg}_{{\mathfrak{m}}_{\mathcal{J}}}\subset\mathcal{O}=\mathrm{G}(\mathcal{J})\cdot\xi$.

###### Proof.

The map $\phi$ from equation (4.2) defines an action of
$\hat{\mathrm{G}}(\mathcal{J})$ on $\mathcal{J}^{\star}$ such that, by
equation (3.6), it is
$\mathcal{H}_{\xi}^{\mathcal{J}}=\mathcal{D}_{\xi}^{\mathcal{J}}$. Evidently,
$\hat{\mathrm{G}}(\mathcal{J})\cdot\xi=\mathrm{G}(\mathcal{J})\cdot\xi$. Since
($\hat{\mathfrak{g}}(\mathcal{J}),\mathcal{D}{\rm er}_{0}(\mathcal{J})$) is a
transvective symmetric pair, equation (2.5) is satisfied for
${\mathfrak{m}}:=\mathcal{J}\subset\hat{\mathfrak{g}}(\mathcal{J})$, and the
assertion now follows from Corollary 2.4, as by equation (2.3) it is

$\mathcal{D}^{[\mathcal{J},\mathcal{J}]}_{\xi}=\mathcal{D}^{\mathcal{D}{\rm
er}_{0}(\mathcal{J})}_{\xi}=\mathcal{D}{\rm
er}_{0}(\mathcal{J})\cdot\xi,\qquad\mathcal{D}^{\mathcal{J}}_{\xi}=\mathcal{J}\cdot\xi.$

∎

###  4.1 Jordan frames and the Peirce decomposition

Let $\mathcal{J}$ be a real finite-dimensional Jordan algebra. We define the
symmetric bilinear form $\tau$ on $\mathcal{J}$ by

$\tau(x,y):=\operatorname{tr}l_{\\{x,y\\}}.$ (4.3)

Observe that for $x,y\in\mathcal{J}$

$\tau(gx,gy)=\tau(x,y),\quad g\in\mathcal{A}{\rm
ut}(\mathcal{J}),\qquad\tau(dx,y)+\tau(x,dy)=0,\quad d\in\mathcal{D}{\rm
er}(\mathcal{J}).$

Namely, if $g\in\mathcal{A}{\rm ut}(\mathcal{J})$, then
$\tau(gx,gy)=\operatorname{tr}l_{g\\{x,y\\}}=\operatorname{tr}gl_{\\{x,y\\}}g^{-1}=\tau(x,y)$,
and the second identity follows as $\mathcal{D}{\rm er}(\mathcal{J})$ is the
Lie algebra of $\mathcal{A}{\rm ut}(\mathcal{J})$.

A symmetric bilinear form $\beta$ on $\mathcal{J}$ is called associative, if
for all $x,y,z\in\mathcal{J}$

$\beta(\\{x,y\\},z)=\beta(x,\\{y,z\\}),$ (4.4)

i.e., if all $l_{x}$ are self-adjoint w.r.t. $\beta$. Then the following is
known.

######  Proposition 4.4 ([52, p. 59]).

The bilinear form $\tau$ from equation (4.3) is associative.

An element $c\in\mathcal{J}$ is called an idempotent if $c^{2}:=\\{c,c\\}=c$.
Such an idempotent is called primitive, if there is no decomposition
$c=c_{1}+c_{2}$ with idempotents $c_{1},c_{2}\neq 0$. For an idempotent
$c\in\mathcal{J}$, $l_{c}$ is diagonalizable with eigenvalues in
$\big{\\{}0,\frac{1}{2},1\big{\\}}$ [30, Proposition III.1.2]. Therefore, it
follows that

$\tau(c,c)=\operatorname{tr}l_{\\{c,c\\}}=\operatorname{tr}l_{c}\geq 1,$

as the trace is the sum of the eigenvalues, and $c$ is in the $1$-eigenspace
of $l_{c}$.

If $\mathcal{J}$ has an identity element $\mathbf{1}_{\mathcal{J}}$, then a
Jordan frame of $\mathcal{J}$ is a set $(c_{i})_{i=1}^{r}\subset\mathcal{J}$
of primitive idempotents such that

$\\{c_{i},c_{j}\\}=\delta_{ij}c_{i}\qquad\text{and}\qquad
c_{1}+\dots+c_{r}=\mathbf{1}_{\mathcal{J}}.$ (4.5)

Note that

$\tau(c_{i},c_{j})=\tau(\\{c_{i},c_{i}\\},c_{j})\stackrel{{\scriptstyle(\ref{eq:l-adjoint})}}{{=}}\tau(c_{i},\\{c_{i},c_{j}\\})=0\qquad\text{for
$i\neq j$},$

so that $(c_{i})_{i=1}^{r}$ is an $\tau$-orthogonal system. From this, one can
show that the maps $l_{c_{i}}$ commute pairwise [30, Lemma IV.1.3].

Let ${\mathfrak{c}}:={\mathrm{span}}\\{c_{i}\\}$. Then, as all $l_{c_{i}}$ are
diagonalizable, there is a $\tau$-orthogonal decomposition of $\mathcal{J}$
into the common eigenspace of $l_{c_{i}}$, i.e., into spaces of the form

$\mathcal{J}_{\rho}:=\\{x\in\mathcal{J}\mid
l_{c}(x)=\rho(c)x,c\in{\mathfrak{c}}\\}$

for some $\rho\in{\mathfrak{c}}^{\star}$. Since $c_{i}$ has only eigenvalues
$\big{\\{}0,\frac{1}{2},1\big{\\}}$ and $\rho({\bf 1}_{\mathcal{J}})=1$, it
follows that $\rho=\frac{1}{2}(\theta_{i}+\theta_{j})$, $i\leq j$, where
$(\theta_{i})_{i=1}^{r}\in{\mathfrak{c}}^{\star}$ is the dual basis to
$(c_{i})_{i=1}^{r}$. That is, we have the $\tau$-orthogonal eigenspace
decomposition

$\mathcal{J}=\bigoplus_{i\leq j}\mathcal{J}_{ij},$ (4.6)

where $\mathcal{J}_{ij}:=\mathcal{J}_{\frac{1}{2}(\theta_{i}+\theta_{j})}$.
This is called the Peirce decomposition of $\mathcal{J}$ with respect to the
Jordan frame $(c_{i})_{i=1}^{r}$. For convenience, we let
$\mathcal{J}_{ji}:=\mathcal{J}_{ij}$ for $i<j$.

###  4.2 Semi-simple and positive Jordan algebras

For a Jordan algebra $\mathcal{J}$, we define the radical of $\mathcal{J}$ as
the null space of $\tau$, i.e.,

${\mathfrak{r}}(\mathcal{J}):=\\{a\in\mathcal{J}\mid\tau(a,x)=0\text{ for all
$x\in\mathcal{J}$}\\}.$

Evidently, ${\mathfrak{r}}(\mathcal{J})\subset\mathcal{J}$ is an ideal by
Proposition 4.4.

We call $\mathcal{J}$ semi-simple if ${\mathfrak{r}}(\mathcal{J})=0$, i.e., if
$\tau$ is non-degenerate. Moreover, we call $\mathcal{J}$ positive or formally
real, if $\tau$ is positive definite.

We shall now collect some known results on semi-simple and positive Jordan
algebras.

######  Proposition 4.5.

Let $\mathcal{J}$ be a semi-simple real Jordan algebra. Then the following
hold.

  1. $(1)$

$\mathcal{J}$ has a decomposition
$\mathcal{J}=\mathcal{J}_{1}\oplus\dots\oplus\mathcal{J}_{k}$ into simple
Jordan algebras $\mathcal{J}_{i}$, i.e., such that $\mathcal{J}_{i}$ does not
contain a non-trivial ideal [52, Theorem III.11].

  2. $(2)$

$\mathcal{J}$ has an identity element $\mathbf{1}_{\mathcal{J}}$ [52, Theorem
III.9].

  3. $(3)$

$\mathcal{J}$ is positive iff it admits a positive definite associative
bilinear form $\beta$ [30, p. 61].

  4. $(4)$

If $\mathcal{J}$ is positive, then for every $x\in\mathcal{J}$ there is a
Jordan frame $\\{c_{i}\\}_{i=1}^{r}$ with
$x\in{\mathrm{span}}(\\{c_{i}\\}_{i=1}^{r})$ [30, Theorem III.1.2]. In
particular, $\mathcal{J}$ has Jordan frames.

  5. $(5)$

If $\mathcal{J}$ is simple and positive and $(c_{i})_{i=1}^{r}$ and
$(c_{i}^{\prime})_{i=1}^{r}$ are Jordan frames, then there is an automorphism
$h\in\mathcal{A}{\rm ut}_{0}(\mathcal{J})$ with $h(c_{i})=c_{i}^{\prime}$ for
all $i$ [30, Theorem IV.2.5],333In [30] it is only stated that there exists an
element $h\in\mathcal{A}{\rm ut}(\mathcal{J})$ with the asserted property;
however, $h\in\mathcal{A}{\rm ut}_{0}(\mathcal{J})$ follows from the proof.
where $\mathcal{A}{\rm ut}_{0}(\mathcal{J})\subset\mathcal{A}{\rm
ut}(\mathcal{J})$ is the identity component. In particular, all Jordan frames
have the same number $r$ of elements, and $r$ is called the rank of
$\mathcal{J}$.

  6. $(6)$

If $\mathcal{J}$ is positive, then for the Peirce spaces in (4.6) we have [30,
Theorem IV.2.1]

$\\{\mathcal{J}_{ij},\mathcal{J}_{kl}\\}\subset\begin{cases}0&\text{if
$\\{i,j\\}\cap\\{j,k\\}=\varnothing$,}\\\ \mathcal{J}_{jl}&\text{if $i=k$,
$j\neq l$},\\\ \mathcal{J}_{ii}+\mathcal{J}_{jj}&\text{if
$\\{i,j\\}=\\{k,l\\}$}.\end{cases}$

  7. $(7)$

If $\mathcal{J}$ is positive, then $\mathcal{J}_{ii}={\mathrm{span}}(c_{i})$
is one-dimensional for all $i$.

  8. $(8)$

If $\mathcal{J}$ is positive, simple and of rank $r$, then
$\mathcal{J}_{jk}\neq 0$ for all $1\leq j\leq k\leq r$ [30, Theorem IV.2.3].

###### Proof.

We only need to show point (7), as it appears not to be explicitly stated in
the literature. Note that $\mathcal{J}_{ii}$ is a subalgebra, as
$\\{\mathcal{J}_{ii},\mathcal{J}_{ii}\\}\subset\mathcal{J}_{ii}$ by the
product relations in point (6), and by definition $c_{i}={\bf
1}_{\mathcal{J}_{ii}}$. Since $\tau|_{\mathcal{J}_{ii}}$ is a positive
definite associative bilinear form, it follows from point (3) that
$\mathcal{J}_{ii}$ is a positive Jordan algebra as well. However, since
$c_{i}={\bf 1}_{\mathcal{J}_{ii}}$ is primitive, it follows that each Jordan
frame of $\mathcal{J}_{ii}$ consists of $c_{i}$ only, so that by (4) each
$x\in\mathcal{J}_{ii}$ must be a multiple of $c_{i}={\bf
1}_{\mathcal{J}_{ii}}$. ∎

The fourth of these results is called the spectral theorem of positive Jordan
algebras. It shows that each $x\in\mathcal{J}$ admits a decomposition

$x=\sum_{i=1}^{r}\lambda_{i}c_{i}$ (4.7)

for a Jordan frame $(c_{i})_{i=1}^{r}$, and the decomposition in equation
(4.7) is referred to as the spectral decomposition of $x$. The $\lambda_{i}$’s
are called the spectral coefficients of $x$. Evidently, the tuple
$(\lambda_{i})_{i=1}^{r}$ is defined only up to permutation of the entries.
Furthermore, we call the pair $(n_{+},n_{-})$, where $n_{+}$ and $n_{-}$ are
the number of positive and negative spectral coefficients of $x$ the spectral
signature of $x$.

######  Lemma 4.6.

Let $\mathcal{J}$ be a semi-simple, positive Jordan algebra,
$(c_{i})_{i=1}^{r}$ a Jordan frame of $\mathcal{J}$ and
$x=\sum_{i}\lambda_{i}c_{i}$. Then, it holds

$\displaystyle
l_{x}(\mathcal{J})=\bigg{(}\bigoplus_{\lambda_{a}+\lambda_{b}\neq
0}\mathcal{J}_{ab}\bigg{)}\oplus\bigg{(}\bigoplus_{a,\mu}\mathcal{J}_{a\mu}\bigg{)},$
$\displaystyle\mathcal{D}{\rm er}_{0}(\mathcal{J})\cdot
x=\bigg{(}\bigoplus_{\lambda_{a}-\lambda_{b}\neq
0}\mathcal{J}_{ab}\bigg{)}\oplus\bigg{(}\bigoplus_{a,\mu}\mathcal{J}_{a\mu}\bigg{)},$
$\displaystyle{\mathfrak{g}}(\mathcal{J})\cdot
x=\bigoplus_{a,i}\mathcal{J}_{ai},$ (4.8)

where we use the index convention that $i$, $j$ run over $1,\dots,r$, while
$a$, $b$ run over those indices with $\lambda_{a}\neq 0$, and $\nu$, $\mu$
over those indices with $\lambda_{\nu}=0$.

###### Proof.

By point (7) in Proposition 4.5, the Peirce decomposition in equation (4.6)
reads

$\mathcal{J}={\mathfrak{c}}\oplus\bigoplus_{a<b}\mathcal{J}_{ab}\oplus\bigoplus_{a,\mu}\mathcal{J}_{a\mu}\oplus\bigoplus_{\mu<\nu}\mathcal{J}_{\mu\nu},\qquad{\mathfrak{c}}:={\mathrm{span}}\\{c_{i}\\}.$

As $l_{x}(x_{ij})=\frac{1}{2}(\lambda_{i}+\lambda_{j})x_{ij}$ for
$x_{ij}\in\mathcal{J}_{ij}$, the first equality in equation (4.8) is
immediate.

For the second equality, recall that $\mathcal{D}{\rm er}_{0}(\mathcal{J})$ is
spanned by $[l_{x_{ij}},l_{y_{kl}}]$ for $x_{ij},y_{ij}\in\mathcal{J}_{ij}$,
and we compute

$\displaystyle{}[l_{x_{ij}},l_{y_{kl}}](x)$
$\displaystyle=\left\\{x_{ij},\frac{1}{2}(\lambda_{k}+\lambda_{l})y_{kl}\right\\}-\left\\{y_{kl},\frac{1}{2}(\lambda_{i}+\lambda_{j})x_{ij}\right\\}$
$\displaystyle=\frac{1}{2}(\lambda_{k}+\lambda_{l}-\lambda_{i}-\lambda_{j})\\{x_{ij},y_{kl}\\}.$

Therefore, the relation “$\subset$” in the second equality in equation (4.8)
follows easily from the bracket relation of the Peirce spaces in point (6) of
Proposition 4.5. For the converse inclusion, we compute

$\displaystyle[l_{x},l_{x_{ij}}](x)=\\{x,\\{x_{ij},x\\}\\}-\big{\\{}x_{ij},x^{2}\big{\\}}=\frac{1}{4}(\lambda_{i}+\lambda_{j})^{2}x_{ij}-\frac{1}{2}\big{(}\lambda_{i}^{2}+\lambda_{j}^{2}\big{)}x_{ij}$
$\displaystyle\hphantom{[l_{x},l_{x_{ij}}](x)}{}=-\frac{1}{4}(\lambda_{i}-\lambda_{j})^{2}x_{ij}.$

The third equation then follows as ${\mathfrak{g}}(\mathcal{J})\cdot
x=l_{x}(\mathcal{J})+\mathcal{D}{\rm er}_{0}(\mathcal{J})\cdot x$, and
$\lambda_{a}+\lambda_{b}=\lambda_{a}-\lambda_{b}=0$ cannot both hold for
$\lambda_{a},\lambda_{b}\neq 0$. ∎

######  Theorem 4.7.

For a positive simple Jordan algebra $\mathcal{J}$, the following hold:

  1. $(1)$

The orbits of $\mathcal{A}{\rm ut}_{0}(\mathcal{J})$ are the sets of elements
with equal spectral coefficients.

  2. $(2)$

The orbits of the structure group $\mathrm{G}(\mathcal{J})$ consist of all
elements with equal spectral signature.

###### Proof.

Any automorphism maps (primitive) idempotents to (primitive) idempotents and
fixes ${\bf 1}_{\mathcal{J}}$, whence it maps Jordan frames to Jordan frames.
Thus, if $x=\sum_{i=1}^{r}\lambda_{i}c_{i}$ for a Jordan frame
$(c_{i})_{i=1}^{r}$, it follows that for $h\in\mathcal{A}{\rm
ut}_{0}(\mathcal{J})$

$h(x)=\sum_{i=1}^{r}\lambda_{i}h(c_{i}),$

and $(h(c_{i}))_{i=1}^{r}$ is again a Jordan frame, so that $x$, $h(x)$ have
the same spectral coefficients $(\lambda_{i})_{i=1}^{r}$.

Conversely, if $x$, $y$ have the same spectral coefficients
$(\lambda_{i})_{i=1}^{r}$, then

$x=\sum_{i=1}^{r}\lambda_{i}c_{i},\qquad
y=\sum_{i=1}^{r}\lambda_{i}c_{i}^{\prime}$

for Jordan frames $(c_{i})_{i=1}^{r}$ and $(c_{i}^{\prime})_{i=1}^{r}$. Thus,
by point (5) of Proposition 4.5, there is a $h\in\mathcal{A}{\rm
ut}_{0}(\mathcal{J})$ with $h(c_{i})=c_{i}^{\prime}$ and hence, $h(x)=y$. This
shows the first statement.

Concerning the second statement, we define the following subsets of
$\mathcal{J}$:

$\displaystyle\Sigma^{m}:=\\{x\in\mathcal{J}\mid\text{$m$ spectral
coefficients of $x$ are $\neq 0$}\\},$ $\displaystyle\Sigma^{\leq
m}:=\\{x\in\mathcal{J}\mid\text{{\em at most} $m$ spectral coefficients of $x$
are $\neq 0$}\\},$
$\displaystyle\Sigma_{n_{+},n_{-}}:=\\{x\in\mathcal{J}\mid\text{$x$ has
spectral signature $(n_{+},n_{-})$}\\}.$

Evidently,

$\Sigma^{m}=\dot{\bigcup}_{n_{+}+n_{-}=m}\Sigma_{n_{+},n_{-}}.$ (4.9)

Moreover, by the first assertion, all these sets are $\mathcal{A}{\rm
ut}_{0}(\mathcal{J})$-invariant.

There are continuous (in fact, polynomial) functions
$a_{k}\colon\mathcal{J}\to{\mathbb{R}}$ such that

$f(x,\lambda)=\lambda^{r}+\lambda^{r-1}a_{r-1}(x)+\dots+a_{0}(x)$ (4.10)

is the minimal polynomial of all generic $x\in\mathcal{J}$, i.e., elements
with pairwise distinct spectral coefficients $\lambda_{i}(x)$ [30, Proposition
II.2.1]. If $x=\sum_{i}\lambda_{i}(x)c_{i}$ is the spectral decomposition of
$x$, then $\prod_{i}(x-\lambda_{i}(x)x)=0$ by equation (4.5), and as the roots
$\lambda_{i}(x)$ are pairwise distinct, it follows that

$f(x,\lambda)=\prod_{i=1}^{r}(\lambda-\lambda_{i}(x)),$ (4.11)

and as generic $x$’s are dense in $\mathcal{J}$ [30, Proposition II.2.1], it
follows that equation (4.11) holds for any $x\in\mathcal{J}$. Then
$\Sigma^{\leq m}$ are those elements where $\lambda=0$ is a root of
$f(x,\cdot)$ of multiplicity $\geq r-m$; that is,

$\Sigma^{\leq m}=\\{x\in\mathcal{J}\mid
a_{0}(x)=\dots=a_{r-m-1}(x)=0\\}\subset\mathcal{J}.$ (4.12)

As the spectral coefficients of $x$ are unchanged under the automorphism
group, equation (4.10) and equation (4.11) imply

$a_{k}(h\cdot x)=a_{k}(x),\qquad x\in\mathcal{J},\quad h\in\mathcal{A}{\rm
ut}_{0}(\mathcal{J}).$ (4.13)

We assert that $\Sigma^{\leq m}$ is invariant under $\mathrm{G}(\mathcal{J})$.
For this, fix a Jordan frame $(c_{i})_{i=1}^{r}$ and let
$x=\lambda_{a}c_{a}\in{\mathfrak{c}}\cap\Sigma^{m}$, using the index summation
convention from Lemma 4.6. Define the map

$\Phi_{x}\colon\ \mathcal{A}{\rm
ut}_{0}(\mathcal{J})\times{\mathbb{R}}^{m}\longrightarrow\mathrm{G}(\mathcal{J})\cdot
x,\qquad(h,(t_{i}))\mapsto h\cdot\exp(l_{t_{i}c_{i}})\cdot x.$

Since $l_{t_{i}c_{i}}^{k}x=t_{a}^{k}\lambda_{a}c_{a}$ by equation (4.5), it
follows that

$\Phi_{x}(h,(t_{i}))=h\cdot\exp(l_{t_{i}c_{i}})\cdot
x=h\cdot\big{(}e^{t_{a}}\lambda_{a}c_{a}\big{)},$ (4.14)

and as $e^{t_{a}}\lambda_{a}c_{a}\in\Sigma^{m}$, the $\mathcal{A}{\rm
ut}_{0}(\mathcal{J})$-invariance of $\Sigma^{m}$ implies that
$\operatorname{Im}(\Phi_{x})\subset\Sigma^{m}$. Moreover, it follows that the
image of the differential $d_{(e,0)}\Phi_{x}$ is

$\operatorname{Im}d_{(e,0)}\Phi_{x}={\mathrm{span}}\\{c_{a}\\}\oplus\mathcal{D}{\rm
er}_{0}(\mathcal{J})\cdot x\stackrel{{\scriptstyle(\ref{eq:orbit-
xlambda})}}{{\subset}}{\mathfrak{g}}(\mathcal{J})\cdot
x=T_{x}(\mathrm{G}(\mathcal{J})\cdot x).$

In fact, equation (4.8) implies that
$\operatorname{Im}d_{(e,0)}\Phi_{x}=T_{x}(\mathrm{G}(\mathcal{J})\cdot x)$ if
$x=\lambda_{a}c_{a}\in{\mathfrak{c}}$ is generic in $\Sigma^{m}$, that is, if
$\lambda_{a}\neq\lambda_{b}$ for all $a\neq b$.

This implies that for $x\in{\mathfrak{c}}\cap\Sigma^{m}$ generic, there is an
open neighborhood $U\subset\mathrm{G}(\mathcal{J})$ of the identity such that

$U\cdot x\subset\operatorname{Im}(\Phi_{x})\subset\Sigma^{m}.$

Thus, for $X\in{\mathfrak{g}}(\mathcal{J})$ and
$x\in{\mathfrak{c}}\cap\Sigma^{m}$ generic, equation (4.12) implies

$a_{k}(\exp(tX)\cdot x)=0,\qquad k=0,\dots,r-m-1,$ (4.15)

for $|t|$ small enough such that $\exp(tX)\in U$. As all $a_{k}$ are
polynomials, the expressions in equation (4.15) are real analytic in $t$,
whence their vanishing for $|t|$ small implies that they vanish for all
$t\in{\mathbb{R}}$, in particular for $t=1$. That is, we conclude that

$a_{k}(\exp(X)\cdot x)=0,\qquad k=0,\dots,r-m-1,\qquad
X\in{\mathfrak{g}}(\mathcal{J})$ (4.16)

for $x\in{\mathfrak{c}}\cap\Sigma^{m}$ generic, and taking the closure, it
follows that equation (4.16) holds for all
${x\in{\mathfrak{c}}\cap\Sigma^{\leq m}}$. Moreover, by the first part, each
$x\in\Sigma^{\leq m}$ can be written as $x=h\cdot\tilde{x}$ for
$\tilde{x}\in{\mathfrak{c}}\cap\Sigma^{\leq m}$ and $h\in
Aut_{0}(\mathcal{J})$. Thus, it holds

$a_{k}(\exp(X)\cdot
x)=a_{k}(h\cdot\operatorname{Ad}_{h^{-1}}(X)\tilde{x})\stackrel{{\scriptstyle(\ref{eq:a_k-
invariant})}}{{=}}a_{k}(\operatorname{Ad}_{h^{-1}}(X)\cdot\tilde{x})\stackrel{{\scriptstyle(\ref{eq:a_k-
in-1})}}{{=}}0,$

so that equation (4.16) holds for all $x\in\Sigma^{\leq m}$ and
$X\in{\mathfrak{g}}(\mathcal{J})$. Thus, by equation (4.12) it follows that

$\exp({\mathfrak{g}}(\mathcal{J}))\cdot\Sigma^{\leq m}\subset\Sigma^{\leq m},$

and as the connected group $\mathrm{G}(\mathcal{J})$ is generated by
$\exp({\mathfrak{g}}(\mathcal{J}))$, the asserted
$\mathrm{G}(\mathcal{J})$-invariance of $\Sigma^{\leq m}$ follows.

Since $\Sigma^{m}=\Sigma^{\leq m}\backslash\Sigma^{\leq m-1}$ is the
difference of two $\mathrm{G}(\mathcal{J})$-invariant sets, it follows that
$\Sigma^{m}$ is $\mathrm{G}(\mathcal{J})$-invariant as well.

Next, we assert that $\Sigma_{n_{+},n_{-}}\subset\Sigma^{m}$ is relatively
closed. For if $(x_{k})_{k\in{\mathbb{N}}}\in\Sigma_{n_{+},n_{-}}$ converges
to $x_{0}\in\Sigma^{m}$, then, fixing a Jordan frame $(c_{i})_{i=1}^{r}$, we
find $h_{k}\in\mathcal{A}{\rm ut}_{0}(\mathcal{J})$ such that

$y_{k}:=h_{k}\cdot
x_{k}=\sum_{a}\lambda_{a,k}c_{a},\qquad\lambda_{1,k}\geq\dots\geq\lambda_{m,k}.$

Since $y_{k}\in\Sigma_{n_{+},n_{-}}$ as well, it follows that the signs of
$0\neq\lambda_{a,k}$ are equal for all $k$. As $\mathcal{A}{\rm
ut}_{0}(\mathcal{J})$ is compact, we may pass to a subsequence to assume that
$h_{k}\to h_{0}$, whence $y_{k}\to h_{0}x_{0}$, i.e.,

$h_{0}x_{0}=\sum_{a}\lambda_{a,0}c_{a},\qquad\lambda_{a,0}=\lim_{k\to\infty}\lambda_{a,k}.$

Since $x_{0}$ and hence $h_{0}x_{0}\in\Sigma^{m}$, it follows that
$\lambda_{a,0}\neq 0$ for all $a$, whence $\lambda_{a,0}$ has the same sign as
all $\lambda_{a,k}$, so that $h_{0}x_{0}\in\Sigma_{n_{+},n_{-}}$, i.e.,
$x_{0}\in\Sigma_{n_{+},n_{-}}$.

Thus, equation (4.9) is the disjoint decomposition of $\Sigma^{m}$ into
finitely many relatively closed subsets, and since $\mathrm{G}(\mathcal{J})$
and hence all orbits are connected, it follows that each
$\mathrm{G}(\mathcal{J})$-orbit must be contained in some
$\Sigma_{n_{+},n_{-}}$.

On the other hand, as elements with equal spectral coefficients lie in the
same $\mathcal{A}{\rm ut}_{0}(\mathcal{J})$-orbit, equation (4.14) immediately
implies that $\mathrm{G}(\mathcal{J})$ acts transitively on
$\Sigma_{n_{+},n_{-}}$, which completes the proof. ∎

For a positive Jordan algebra $\mathcal{J}$ we identify $\mathcal{J}$ and
$\mathcal{J}^{\star}$ by the isomorphism

$\displaystyle\flat\colon\
\mathcal{J}\longrightarrow\mathcal{J}^{\star},\qquad x\longmapsto
x^{\flat}:=\tau(x,\cdot),$ $\displaystyle\\#\colon\
\mathcal{J}^{\star}\longrightarrow\mathcal{J},\qquad\\#:=\flat^{-1}.$

By the spectral theorem (cf. point (4) of Proposition 4.5), for each
$\xi\in\mathcal{J}^{\star}$ there is a Jordan frame $(c_{i})_{i=1}^{r}$ on
$\mathcal{J}$ such that

$\xi^{\\#}=\lambda_{i}c_{i},\qquad\mbox{ and
}\qquad\xi=\lambda_{i}c_{i}^{\flat},$

and we define the spectral coefficients $(\lambda_{i})_{i}$ and the spectral
signature $(n_{+},n_{-})$ of $\xi$ to be the spectral coefficients and
signature of $\xi^{\\#}$. We let

$\mathcal{O}_{n_{+},n_{-}}\subset\mathcal{J}^{\star}$

be the set of elements of spectral signature $(n_{+},n_{-})$. Furthermore, we
define the dual of $\tau$ to be the scalar product on $\mathcal{J}^{\star}$
given by

$\tau^{\flat}(\eta_{1},\eta_{2}):=\tau\big{(}\eta_{1}^{\\#},\eta_{2}^{\\#}\big{)}\qquad\mathrm{or}\qquad\tau^{\flat}\big{(}x_{1}^{\flat},x_{2}^{\flat}\big{)}:=\tau(x_{1},x_{2}).$

For $x,y\in\mathcal{J}$ and $\xi\in\mathcal{J}^{\star}$, we have
$(l_{x}^{\ast}\xi)(y)=\xi(l_{x}y)=\tau\big{(}\xi^{\\#},l_{x}y\big{)}=\tau\big{(}l_{x}\xi^{\\#},y\big{)}=\big{(}l_{x}\xi^{\\#}\big{)}^{\flat}(y)$,
so that

$l_{x}^{\ast}\xi=\big{(}l_{x}\xi^{\\#}\big{)}^{\flat}.$ (4.17)

Therefore, by the definition of the dual action, it follows that

$\mathcal{A}{\rm ut}_{0}(\mathcal{J})\cdot\xi=\big{(}\mathcal{A}{\rm
ut}_{0}(\mathcal{J})\cdot\xi^{\\#}\big{)}^{\flat},\qquad\mathrm{G}(\mathcal{J})\cdot\xi=\big{(}\mathrm{G}(\mathcal{J})\cdot\xi^{\\#}\big{)}^{\flat},$
(4.18)

so that, by Theorem 4.7, we obtain that the orbits of the action of
$\mathrm{G}(\mathcal{J})$ on $\mathcal{J}^{\star}$ are the sets
$\mathcal{O}_{n_{+},n_{-}}$.

The open cone of squares in $\mathcal{J}$ is

$\Omega_{\mathcal{J}}:=\mathrm{Int}\big{\\{}x^{2}\mid
x\in\mathcal{J}\big{\\}}.$

Looking at the spectral decomposition in equation (4.7), it follows that
$x\in\Omega_{\mathcal{J}}$ iff all its spectral coefficients are positive iff
$l_{x}$ is positive definite, and the latter description shows that
$\Omega_{\mathcal{J}}$ is indeed a convex cone; in fact, it easily follows
from this characterization that

$\displaystyle\Omega_{\mathcal{J}}=\mathcal{O}_{r,0}={\mathfrak{g}}(\mathcal{J})\cdot{\bf
1}_{\mathcal{J}},\qquad\overline{\Omega}_{\mathcal{J}}=\dot{\bigcup}_{n_{+}\geq
0}\mathcal{O}_{n_{+},0}.$ (4.19)

######  Theorem 4.8.

Let $\mathcal{J}$ be a positive, simple Jordan algebra with structure group
$G(\mathcal{J})\subset{\mathrm{Gl}}(\mathcal{J})$. Then
$\xi\in\mathcal{J}^{\star}$ is ${\mathfrak{m}}_{\mathcal{J}}$-regular iff the
spectral coefficients $(\lambda_{i})$ of $\xi$ satisfy:

$\lambda_{a}+\lambda_{b}\neq 0\qquad\text{whenever
$\lambda_{a},\lambda_{b}\neq 0$}.$ (4.20)

In particular, the $\mathrm{G}(\mathcal{J})$-orbit $\mathcal{O}_{n_{+},n_{-}}$
is ${\mathfrak{m}}_{\mathcal{J}}$-regular iff $n_{+}=0$ or $n_{-}=0$, i.e.,
iff it is contained in $\overline{\Omega}_{\mathcal{J}}$ or
$-\overline{\Omega}_{\mathcal{J}}$.

###### Proof.

Let $x:=\xi^{\\#}\in\mathcal{J}$. By equation (4.18), $\mathcal{A}{\rm
ut}_{0}(\mathcal{J})\cdot\xi\subset\mathcal{J}\cdot\xi$ iff $\mathcal{A}{\rm
ut}_{0}(\mathcal{J})\cdot x\subset\mathcal{J}\cdot x=l_{x}(\mathcal{J})$, and,
recalling point (8) in Proposition 4.5, by equation (4.8) this condition is
satisfied iff equation (4.20) holds. Recalling Proposition 4.3, the first
statement follows.

If $n_{+},n_{-}>0$, then evidently, $\mathcal{O}_{n_{+},n_{-}}$ contains
elements two of whose spectral coefficients satisfy
$\lambda_{a}=-\lambda_{b}\neq 0$, so that $\mathcal{O}_{n_{+},n_{-}}$ is not
${\mathfrak{m}}_{\mathcal{J}}$-regular.

On the other hand, on $\mathcal{O}_{n_{+},0}$ ($\mathcal{O}_{0,n_{-}}$,
respectively) $\lambda_{a},\lambda_{b}>0$ ($<0$, respectively) so that
equation (4.20) holds; whence $\mathcal{O}_{n_{+},0}$ and
$\mathcal{O}_{0,n_{-}}$ are the only ${\mathfrak{m}}_{\mathcal{J}}$-regular
orbits, and by equation (4.19) these are the orbits contained in
$\bar{\Omega}_{\mathcal{J}}$ or $-\bar{\Omega}_{\mathcal{J}}$, respectively. ∎

Let us now describe the pseudo-Riemannian metric $\mathcal{G}$ on
$\mathcal{O}^{\rm reg}_{{\mathfrak{m}}_{\mathcal{J}}}$. Take
$\xi\in\mathcal{J}^{\star}$ with spectral decomposition

$\xi=\lambda_{a}c_{a}^{\flat}\in\mathcal{J}^{\star}\quad\Rightarrow\quad
x:=\xi^{\\#}=\lambda_{a}c_{a}\in\mathcal{J}$ (4.21)

for some Jordan frame $(c_{i})_{i=1}^{r}$, and assume it satisfies equation
(4.20). Then, it holds

$T_{\xi}\mathcal{O}=\mathfrak{g}(\mathcal{J})\cdot\xi=(\mathfrak{g}(\mathcal{J})\cdot
x)^{\flat}\stackrel{{\scriptstyle(\ref{eq:orbit-
xlambda})}}{{=}}\bigoplus_{a,i}\mathcal{J}_{ai}^{\flat},$

and we have the following Proposition.

######  Proposition 4.9.

Let $\xi=\lambda_{a}c_{a}^{\flat}\in\mathcal{J}^{\star}$ be as above. Then, it
holds

$\mathcal{G}_{\xi}=\sum_{a,i}\frac{2}{\lambda_{a}+\lambda_{i}}\tau^{\flat}|_{\mathcal{J}_{ai}^{\flat}},$
(4.22)

which is equivalent to

$\mathcal{G}_{\xi}\big{(}x_{ai}^{\flat},y_{bj}^{\flat}\big{)}=\begin{cases}\dfrac{2}{\lambda_{a}+\lambda_{i}}\tau^{\flat}\big{(}x_{ai}^{\flat},y_{ai}^{\flat}\big{)}&\text{if
$(a,i)=(b,j)$,}\\\\[2.84526pt] 0&\text{else.}\end{cases}$

###### Proof.

For $x_{ij}\in\mathcal{J}_{ij}$, we have

$l_{x_{ij}}^{\ast}(\xi)\stackrel{{\scriptstyle(\ref{eq:l*dual})}}{{=}}\big{(}l_{x_{ij}}\xi^{\\#}\big{)}^{\flat}=\frac{1}{2}(\lambda_{i}+\lambda_{j})x_{ij}^{\flat}.$

Therefore, evaluating both sides of equation (3.8) gives us

$\displaystyle\mathcal{G}_{\xi}(l_{x_{ai}}^{\ast}(\xi),l_{y_{bj}}^{\ast}(\xi))$
$\displaystyle=\frac{1}{4}(\lambda_{a}+\lambda_{i})(\lambda_{b}+\lambda_{j})\mathcal{G}_{\xi}\big{(}x_{ai}^{\flat},y_{bj}^{\flat}\big{)}$
$\displaystyle\mathcal{G}_{\xi}(l_{x_{ai}}^{\ast}(\xi),l_{y_{bj}}^{\ast}(\xi))$
$\displaystyle=\xi(\\{x_{ai},y_{bj}\\})=\tau\big{(}\xi^{\\#},\\{x_{ai},y_{bj}\\}\big{)}$
$\displaystyle=\tau(\\{\xi^{\\#},x_{ai}\\},y_{bj})\stackrel{{\scriptstyle(\ref{eq:define-
xi})}}{{=}}\frac{1}{2}(\lambda_{a}+\lambda_{i})\tau(x_{ai},y_{bj}).$

Since both equations must be equal, (4.22) follows as the Peirce decomposition
${\mathcal{J}=\bigoplus_{ij}\mathcal{J}_{ij}}$ is $\tau$-orthogonal. ∎

######  Remark 4.10.

  1. (1)

Comparing the description of the regular points in $\mathcal{O}$ in Theorem
4.8 and equation (4.22), it follows that $\mathcal{G}_{\xi}$ has a pole of
order $1$ on $\mathcal{O}\backslash\mathcal{O}^{\rm
reg}_{{\mathfrak{m}}_{\mathcal{J}}}$.

  2. (2)

As $\mathcal{G}_{\xi}$ is positive or negative definite on
$\mathcal{J}_{aa}^{\flat}$, depending on the sign of $\lambda_{a}\neq 0$, it
follows that $\mathcal{G}_{\xi}$ is indefinite at any regular point of
spectral signature $(n_{+},n_{-})$ with $n_{+},n_{-}>0$.

That is, $\mathcal{G}$ on $\mathcal{O}$ is definite (and hence defines a
Riemannian metric) iff $\mathcal{O}=\mathrm{G}(\mathcal{J})\cdot\xi$ is a
regular orbit, iff $\mathcal{O}\subset\overline{\Omega}_{\mathcal{J}}$ is
contained in the closure of the cone of squares ($\mathcal{G}>0$) or
$\mathcal{O}\subset-\overline{\Omega}_{\mathcal{J}}$ ($\mathcal{G}<0$).

  3. (3)

It is also evident from the description of regular points in Theorem 4.8 that
for a non-regular orbit $\mathcal{O}_{n_{+},n_{-}}$ with $n_{+},n_{-}>0$ the
regular part $\left(\mathcal{O}_{n_{+},n_{-}}\right)^{\rm
reg}_{{\mathfrak{m}}_{\mathcal{J}}}$ is not path connected.

  4. (4)

Note that the Riemannian metric $\mathcal{G}$ on $\mathcal{O}_{n_{+},0}$ (and,
similarly, $-\mathcal{G}$ on $\mathcal{O}_{0,n_{-}}$) is not complete. Namely,
for $t>0$, the curve

$\alpha(t):=t^{2}(c_{1}+\dots+c_{n_{+}})^{\flat}\in\mathcal{O}_{n_{+},0}$

for a Jordan frame $(c_{i})_{i=1}^{r}$ has constant speed with respect to
$\mathcal{G}$ because

$\mathcal{G}_{\alpha(t)}(\dot{\alpha},\dot{\alpha})=\sum_{a=1}^{n_{+}}\frac{1}{t^{2}}\tau(2tc_{a},2tc_{a})=4\sum_{a=1}^{n_{+}}\tau(c_{a},c_{a}).$

However, $\alpha$ cannot be extended in $\mathcal{O}_{n_{+},0}$ at $t=0$.

###  4.3 Examples

We shall now describe the metric $\mathcal{G}$ for the standard examples of
positive Jordan algebras.

###  4.4 The Fisher–Rao metric for finite sample spaces

We regard $\mathcal{J}:={\mathbb{R}}^{n}$ as a positive Jordan algebra whose
algebraic operations are defined in a component-wise way. Then, it is not
difficult to see that $\Omega_{\mathcal{J}}$ can be identified with the first
orthant ${\mathbb{R}}^{n}_{+}\subset{\mathbb{R}}^{n}\cong\mathcal{J}^{\star}$.
The metric $\mathcal{G}_{\xi}$ at
$\xi=(\xi_{1},\dots,\xi_{n})\in\Omega_{\mathcal{J}}$ is given by

$\mathcal{G}_{\xi}(u,v)=\sum_{i}\dfrac{1}{\xi_{i}}u_{i}v_{i},\qquad
u=(u_{i})_{i=1}^{n},v=(v_{i})_{i=1}^{n}\in{\mathbb{R}}^{n}.$

When interpreting $\Omega_{\mathcal{J}}$ as the set of positive finite
measures on $\mathcal{X}_{n}=\\{1,\dots,n\\}$, it is clear that $\mathcal{G}$
is such its pullback to the submanifold of strictly positive probability
distributions on $\mathcal{X}_{n}$ (i.e., open interior of the unit simplex
inside ${\mathbb{R}}^{n}_{+}$) coincides with the Fisher–Rao metric tensor
which naturally occurs in classical information geometry [2, 6]. As partially
noted in [22, 23], this instance shows that we may look at the non-normalized
Fisher–Rao metric tensor on $\Omega_{\mathcal{J}}={\mathbb{R}}^{n}_{+}$ as the
analogue of the homogeneous symplectic form on co-adjoint orbits in the case
of Lie algebras.

###  4.5 The Jordan algebras $\boldsymbol{M_{n}^{\rm sa}({\mathbb{K}})}$,
$\boldsymbol{{\mathbb{K}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}}}$

Let ${\mathbb{K}}$ denote either the real, complex or quaternionic numbers,
and we define the Jordan algebra of self-adjoint matrices

$M_{n}^{\rm sa}({\mathbb{K}}):=\\{A\in{\mathbb{K}}^{n\times n}\mid
A=A^{\ast}\\},\qquad\\{A,B\\}:=\dfrac{1}{2}(AB+BA).$

For convenience, we replace $\tau$ from equation (4.3) by the associative
inner product

$\hat{\tau}(A,B):=\operatorname{Tr}(AB),$

so that $\tau$ and $\hat{\tau}$ only differ by the multiplicative constant
$\frac{1}{n}\dim_{\mathbb{R}}M_{n}^{\rm sa}({\mathbb{K}})$.

Let $E_{ij}\in{\mathbb{K}}^{n\times n}$ denote the matrix with a $1$ in the
$(i,j)$-entry. Then $\\{E_{11},\dots,E_{nn}\\}$ is a Jordan frame of
$M_{n}^{\rm sa}({\mathbb{K}})$, and the remaining Pierce spaces with respect
to this frame are given as

$(M_{n}^{\rm sa}({\mathbb{K}}))_{ij}=\\{zE_{ij}+\bar{z}E_{ji}\mid
z\in{\mathbb{K}}\\},\qquad i<j.$

For ${\mathbb{K}}={\mathbb{R}},{\mathbb{C}}$ and ${\mathbb{H}}$, the
automorphism group of $M_{n}^{\rm sa}({\mathbb{K}})$ is ${\rm SO}(n)$, ${\rm
U}(n)$ and ${\rm Sp}(n)$, respectively, acting on $M_{n}^{\rm
sa}({\mathbb{K}})$ by conjugation. Thus, in particular, each $A\in M_{n}^{\rm
sa}({\mathbb{K}})$ is diagonalizable by an element in the automorphism group,
so that the spectral coefficients are the eigenvalues of $A$. Thus, by
Proposition 4.9, for $\xi=\lambda_{a}E_{aa}^{\flat}\in(M_{n}^{\rm
sa}({\mathbb{K}}))^{\star}$ the metric $\mathcal{G}_{\xi}$ reads

$\mathcal{G}_{\xi}\big{(}zE_{ai}^{\flat},wE_{bj}^{\flat}\big{)}=\begin{cases}\dfrac{2}{\lambda_{a}+\lambda_{i}}(z\bar{w}+w\bar{z})&\text{if
$(a,i)=(b,j)$,}\\\\[2.84526pt] 0&\text{else.}\end{cases}$

Moreover, because of Proposition 3.3, it follows that $\mathcal{G}$ is
preserved by the automorphism group of $M_{n}^{\rm sa}({\mathbb{K}})$.

As already mentioned in the introduction, and in accordance with the results
put forward in [22, 23], an interesting link between Jordan algebras and
quantum information geometry appears when $\mathbb{K}=\mathbb{C}$. In this
case, we may identify $M_{n}^{\rm sa}({\mathbb{C}})$ with the Jordan algebra
of self-adjoint observables of a finite-level quantum system with Hilbert
space $\mathcal{H}\cong{\mathbb{C}}^{n}$. Then, if we focus on the
${\mathfrak{m}}_{\mathcal{J}}$-regular orbit $\Omega_{\mathcal{J}}$ of
faithful, non-normalized quantum states, which can be identified with the dual
of the orbit of invertible positive matrices in $M_{n}^{\rm
sa}({\mathbb{C}})$, the metric tensor $\mathcal{G}$ is such that its pullback
to the submanifold of faithful quantum states, determined by the condition
$\operatorname{Tr}(A)=1$, coincides with the so-called Bures–Helstrom metric
tensor [9, 25, 26, 36, 37, 38, 39, 68, 69, 70, 77]. Analogously, if we focus
on the ${\mathfrak{m}}_{\mathcal{J}}$-regular orbit through non-normalized
pure states, which are identified with rank-one matrices in $M_{n}^{\rm
sa}({\mathbb{C}})$, the metric tensor $\mathcal{G}$ is such that its pullback
to the submanifold of pure states, determined by the condition
$\operatorname{Tr}(A)=1$, is a multiple of the Fubini–Study metric tensor,
essentially because of its unitary invariance. Accordingly, and in analogy
with the Fisher–Rao metric tensor seen before, we may think of the non-
normalized version of the Bures–Helstrom metric tensor and of the Fubini–Study
metric tensor as the analogue of the Kostant–Kirillov–Souriau symplectic form
in the case of the Jordan algebra $M_{n}^{\rm sa}({\mathbb{C}})$.

###  4.6 The spin-factor Jordan algebra
$\boldsymbol{{\mathcal{J}\mathrm{Spin}}(n)}$

Denoting the standard inner product of ${\mathbb{R}}^{n}$ by
$\langle\cdot,\cdot\rangle$, we let

${\mathcal{J}\mathrm{Spin}}(n):={\mathbb{R}}^{n+1}={\mathbb{R}}{\bf
1}\oplus{\mathbb{R}}^{n},\qquad\\{x,y\\}:=\langle x,y\rangle{\bf 1},\qquad
x,y\in{\mathbb{R}}^{n},$

and where ${\bf 1}$ is the identity element of
${\mathcal{J}\mathrm{Spin}}(n)$. An associative inner product is given by

$\hat{\tau}|_{{\mathbb{R}}^{n}}=\langle\cdot,\cdot\rangle,\qquad\hat{\tau}({\bf
1},{\bf 1}):=1,\qquad\hat{\tau}({\bf 1},{\mathbb{R}}^{n})=0.$

The automorphism group is ${\rm SO}(n)$, acting on ${\mathbb{R}}^{n}$ and
fixing ${\bf 1}$. Every Jordan frame is given by

$\left\\{\dfrac{1}{2}({\bf 1}+e_{0}),\dfrac{1}{2}({\bf 1}-e_{0})\right\\}$

for a fixed unit vector $e_{0}\in{\mathbb{R}}^{n}$, and the Peirce space
complementary to the Jordan frame is

${\mathcal{J}\mathrm{Spin}}(n)_{12}:=e_{0}^{\perp}.$

The two spectral coefficients of an element $X=t{\bf 1}+x$ are

$\lambda_{1}=\frac{1}{2}(t+\|x\|),\qquad\lambda_{2}=\frac{1}{2}(t-\|x\|),$

where $\|\cdot\|$ denotes the norm on ${\mathbb{R}}^{n}$ induced by
$\langle\cdot,\cdot\rangle$. Therefore,
$\xi\in{\mathcal{J}\mathrm{Spin}}(n)^{\star}$ is regular iff
$0\neq\lambda_{1}+\lambda_{2}=\hat{\tau}^{\flat}\big{(}\xi,{\bf
1}^{\flat}\big{)}$. If

$\xi=t_{0}{\bf
1}^{\flat}+s_{0}e_{0}^{\flat}\in{\mathcal{J}\mathrm{Spin}}(n)^{\star},\qquad
t_{0}\neq 0$

is regular, where $e_{0}\in{\mathbb{R}}^{n}$ is a unit vector, then the
tangent vectors $X_{1},X_{2}\in T_{\xi}\mathcal{O}$ are of the form

$X_{i}=t_{i}{\bf 1}^{\flat}+s_{i}e_{0}^{\flat}+x_{i}^{\flat},$

where $x_{i}\in e_{0}^{\perp}$, and where $t_{0}=\pm s_{0}\Rightarrow
t_{i}=\pm s_{i}$. The spectral coefficients of $\xi$ are
$\lambda_{1}=\frac{1}{2}(t_{0}+s_{0})$ and
$\lambda_{2}=\frac{1}{2}(t_{0}-s_{0})$, and

$X_{i}=(t_{i}+s_{i})\frac{1}{2}({\bf
1}+e_{0})^{\flat}+(t_{i}-s_{i})\frac{1}{2}({\bf
1}-e_{0})^{\flat}+x_{i}^{\flat}.$

Therefore,

$\displaystyle\mathcal{G}_{\xi}(X_{1},X_{2})=\dfrac{2}{t_{0}+s_{0}}(t_{1}+s_{1})(t_{2}+s_{2})+\dfrac{2}{t_{0}-s_{0}}(t_{1}-s_{1})(t_{2}-s_{2})+\dfrac{2}{t_{0}}\langle
x_{1},x_{2}\rangle.$

This metric is positive definite if $t_{0}\geq|s_{0}|$ and negative definite
if $t_{0}\leq-|s_{0}|$, as predicted by Proposition 4.9.

######  Remark 4.11.

According to the classification given in [30, Theorem V.3.7], the first two
classes of examples discussed above give a complete list of simple, positive
Jordan algebras up to the Albert algebra, a $27$-dimensional simple Jordan
algebra of rank $3$. Its automorphism group is $F_{4}$ and the structure
algebra is of type $E_{6}$.

While it would be possible but elaborate to calculate the regular points and
the inner product $\mathcal{G}$ on the tangent to the orbit at a regular
point, our results in Theorem 4.8 and Proposition 4.9, allow understanding the
structure without these explicit calculations.

###  4.7 Non-simple, semisimple positive Jordan algebras

By point (1) of Proposition 4.5, each positive, semisimple Jordan algebra
admits a decomposition
$\mathcal{J}=\mathcal{J}_{1}\oplus\dots\oplus\mathcal{J}_{k}$ into positive
simple Jordan algebras, so that both the automorphism and the structure group
of $\mathcal{J}$ are the direct sum of the automorphism and structure group of
the simple factors $\mathcal{J}_{i}$, respectively. Then, applying Theorems
4.7 and 4.8, and Proposition 4.9, it follows that the
$\mathrm{G}(\mathcal{J})$-orbits are of the form

$\mathcal{O}^{1}_{n^{1}_{+},n^{1}_{-}}\times\dots\times\mathcal{O}^{k}_{n^{k}_{+},n^{k}_{-}}\subset\mathcal{J}^{\star}=\mathcal{J}_{1}^{\star}\oplus\dots\oplus\mathcal{J}^{\star}_{k},$

where $\mathcal{O}^{i}_{n^{i}_{+},n^{i}_{-}}\subset\mathcal{J}^{\star}_{i}$
are $\mathrm{G}(\mathcal{J}_{i})$-orbits. In particular, such an orbit is
regular iff $n^{i}_{+}n^{i}_{-}=0$ for all $i$, and the metric $\mathcal{G}$
on the regular part of this orbit is given by (4.22).

##  5 Discussion

As mentioned in Section 1, the mathematics of Jordan algebras has been
intensively studied, and we want to discuss here where our investigation place
itself in this context.

First of all, it is clear that our construction is highly influenced and
inspired by Kirillov’s theory, and the relation between Jordan algebras and
Kirillov’s theory has been already investigated in the literature [40, 41,
42]. However, the approaches and constructions already available are different
from ours, both conceptually and in terms of mathematical structures. For
instance, in the works mentioned above, the emphasis is on the study of
nilpotent coadjoint orbits of convex types, while we do not study the
coadjoint orbits of the structure group $\mathrm{G}(\mathcal{J})$, rather we
try to develop an analogue of coadjoint orbits for a Jordan algebra. In order
to do so, it turns out we need to rely on the structure group
$\mathrm{G}(\mathcal{J})$ and on (some of) its homogeneous spaces. A clear
departure point from ordinary Kirillov theory is then the fact that the orbits
we find do not possess a natural symplectic structure, but rather a pseudo-
Riemannian one. The shift from an antisymmetric tensor to a symmetric one is
clearly related to the fact that we develop Kirillov’s theory for Jordan
algebras, which have a symmetric product. Moreover, as commented in Sections 1
and 4.3, a direct output of our investigation is the fact that some relevant
Riemannian structures that naturally arise in the context of Classical and
quantum information geometry (namely, the Fisher–Rao metric tensor and the
Bures–Helstrom metric tensor) can be thought of as the Jordan-algebraic
analogue of the natural homogeneous symplectic form on the co-adjoint orbits
of a Lie group.

Other important avenues of research around the mathematics of Jordan algebras
concern the relation between symmetric spaces/cones/domains and Jordan
algebras, as well as the differential geometric structures related with the
algebraic structure of Jordan algebras [13, 14, 18, 30, 52, 78].

In all these investigations, the geometrical objects are subsets of a given
Jordan algebra $\mathcal{J}$, while we consider positive linear functionals on
$\mathcal{J}$. This change of perspective is dictated by the role Jordan
algebras play in information geometry, where the relevant objects live in the
dual space of $\mathcal{J}$ (e.g., classical probability distributions,
quantum states). It is true that it is possible to identify $\mathcal{J}$ with
its dual in finite dimensions, and we exploit this technicality in our proofs,
but the focus of our approach is conceptually different from the previous
ones, and this difference is particularly relevant with respect to possible
generalizations to infinite dimensions (a task that we plan to address in the
future).

Again in relation with the above-mentioned references, the symmetric cones in
real Hilbert spaces that turn out to be in one-to-one correspondence with
formally real (Euclidean) Jordan algebras, in the sense that every such
symmetric cone can be realized as the open cone $\Omega_{\mathcal{J}}$ of
invertible positive elements of a suitable Jordan algebra $\mathcal{J}$, are
given from the onset. From our point of view, the open cone
$\Omega_{\mathcal{J}}$ is not an a priori datum of the problem, but it appears
because it is diffeomorphic to an integral manifold of the Jordan distribution
$\mathcal{H}^{\mathcal{J}}$. Therefore, $\Omega_{\mathcal{J}}$ is a byproduct
of our investigation of the integrability properties of the Jordan
distribution canonically associated with the Jordan algebra product. Moreover,
the symmetric cones are also naturally endowed with a Riemannian structure
$g$, which is invariant under the action of the symmetry group
$\mathrm{G}(\mathcal{J})$ of the cone itself, while the Riemannian structure
we obtain is invariant only under the subgroup of $\mathrm{G}(\mathcal{J})$
composed by automorphisms of $\mathcal{J}$. This instance follows from the
fact that we obtain the Riemannian structure by implementing a symmetric
analogue of Kirillov theory, and not by symmetry considerations. In
particular, in the case of the real associative Jordan algebra
$\mathbb{R}^{n}$, the $\mathrm{G}(\mathcal{J})$-invariant Riemannian metric
tensor $g$ on $\Omega_{\mathcal{J}}=\mathbb{R}^{n}_{+}$ can be written as [18,
Theorem 2.3.19]

$g=\sum_{j=1}^{n}\frac{\mathrm{d}p^{j}\otimes\mathrm{d}p^{j}}{(p^{j})^{2}},$

where $\\{p^{j}\\}_{j=1,\dots,n}$ is a Cartesian coordinate system adapted to
$\Omega_{\mathcal{J}}$ in the sense that elements in $\Omega_{\mathcal{J}}$
have strictly positive values of the coordinates. This Riemannian metric
tensor is different from the Fisher–Rao metric tensor appearing in classical
information geometry [6]. However, as shown in Section 4.3 (in accordance with
[23]), the Riemannian structure emerging from our construction is precisely
the Fisher–Rao metric tensor. Moreover, the $\mathrm{G}(\mathcal{J})$
invariance of $g$ implies it is invariant with respect to dilations also when
$\mathcal{J}=\mathcal{B}_{\rm sa}(\mathbb{C}^{n})\cong M_{n}^{\rm
sa}(\mathbb{C})$, so that $g$ can not be the Bures–Helstrom metric tensor
appearing in quantum information geometry [25], while, again referring to
Section 4.3, the Riemannian metric tensor we obtain reduces to the
Bures–Helstrom metric tensor when $\mathcal{J}=\mathcal{B}_{\rm
sa}(\mathbb{C}^{n})\cong M_{n}^{\rm sa}(\mathbb{C})$.

Another relevant point to stress is that our construction relies entirely on
the algebraic Jordan product. This detail contributes to further differentiate
our work from previous ones, where part of the geometrical structures
associated with Jordan algebras necessarily involve the so-called _Jordan
triple product_ , a trilinear map that exists in every Jordan algebra (and
also in more general objects known as _Jordan triple systems_). For instance,
the $\mathrm{G}(\mathcal{J})$-invariant Riemannian metric tensor discussed
above is defined in terms of the Jordan triple product of $\mathcal{J}$ [18,
Theorem 2.3.19], while our Riemannian metric tensor only requires the Jordan
product. Also, the Jordan triple product induces a natural Lie triple system
that leads to a curvature-like tensor [13], and again, this differential
geometric object makes use of the Jordan product only indirectly, highlighting
the difference with our approach. At this point, it is worth noting that we
plan to address the role of the Jordan triple product and its associated Lie
triple system in information geometry because we believe they are connected
with the so-called Amari–Cencov tensor in the classical case, and with a non-
symmetric generalization of that tensor in the quantum case.

Therefore, while and perhaps because our approach is partly extrinsically
motivated by classical and quantum information geometry, and partly
intrinsically by developing an orbit method-like theory, our constructions and
results are different from previous ones and, taken together, they yield a
fuller picture of the structure of Jordan algebras.

### Acknowledgements

F.M.C. acknowledges that this work has been supported by the Madrid Government
(Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the
line of “Research Funds for Beatriz Galindo Fellowships” (C&QIG-BG-CM-UC3M),
and in the context of the V PRICIT (Regional Program of Research and
Technological Innovation). He also wants to thank the incredible support of
the Max Planck Institute for the Mathematics in the Sciences in Leipzig, where
he was formerly employed when this work was initially started and developed.
L.S. acknowledges partial support by grant SCHW893/5-1 of the Deutsche
Forschungsgemeinschaft, and also expresses his gratitude for the hospitality
of the Max Planck Institute for the Mathematics in the Sciences in Leipzig
during numerous visits. This publication is based upon work from COST Action
CaLISTA CA21109 supported by COST (European Cooperation in Science and
Technology, www.cost.eu). We also thank the anonymous referees for their
valuable comments and suggestions which enabled us to significantly improve
this manuscript.

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