Information geometry, Jordan algebras, and a coadjoint orbit-like construction

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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 L ∞ (X , µ) of complex-valued, µ-essentially bounded (equivalence classes of) functions on the measurable space (X , Σ), the normal states are probability measures on X which are absolutely continuous with respect to µ, and we recover the classical case.The quantum case is recovered considering the Jordan algebra of self-adjoint elements in the W * -algebra B(H) of bounded linear operators on the complex Hilbert space H, so that the normal states are density operators on 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 L ∞ (X , µ) and the Bures-Helstrom metric tensor when the algebra is B(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 A be a finite-dimensional, unital C * -algebra, and let A sa be the self-adjoint part of A and V be the self-adjoint part of the Banach dual A * of A .Of course, there is some duality here, as a ∈ A sa can be identified with a real-valued, linear function l a on V , via Since V is a finite-dimensional Banach space, a → l a yields an isomorphism between A sa and V * = A * * sa .In other words, the differentials of the linear functions on V associated with elements in A sa generate the cotangent space T * ξ V at each ξ.The associative product of A leads to a commutative product {, } and to a non-commutative product [[ , ]] on A sa , making A sa into a Banach-Lie-Jordan algebra [1]; they are We can then exploit these products to introduce a symmetric and an antisymmetric tensor on the dual of A 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 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 R, the Lie structure also induces an antisymmetric tensor Since the Lie algebra of the group U of unitary elements in A may be identified with the space A sa of self-adjoint elements in A , the tensor Λ may be interpreted as the Kostant-Kirillov-Souriau Poisson tensor associated with the coadjoint action of the unitary group U .Importantly, for our purposes, also R can be studied with the help of the action of this Lie algebra.
Returning to the covariant symmetric tensor R, it happens that, on certain orbits of the Banach-Lie group of invertible elements of A (of which U is a subgroup), it makes sense to consider the covariant tensor G := R −1 , which becomes a Riemannian metric tensor when the above-mentioned orbits lie in the space of positive linear functionals on 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 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 O ⊂ g ⋆ of the Lie group G carries a natural homogeneous symplectic structure [48,49].Here, g and g ⋆ denote the Lie algebra of 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, finitedimensional algebra A, i.e., a finite-dimensional (real or complex) vector space with a bilinear product • : A × A → 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 A ⋆ the dual space of A, and the corresponding pairing is denoted by ⟨•, •⟩.Due to the finite-dimensionality, we have the identification A ⋆⋆ ∼ = A, and for the tangent and cotangent spaces of A ⋆ , we then have Furthermore, this induces a multiplication on C ∞ (A ⋆ ) via and for f ∈ C ∞ (A ⋆ ), we may define its A-dual vector field of f as Note that, in general, this is not a gradient because • need not be symmetric, but we use the symbol ∇ because it satisfies many of the formal identities of gradients.The multiplication in equation (1.2) is of course compatible with the product • in the sense that, identifying a ∈ A with the linear functional f a (ξ) := ⟨ξ, a⟩, the inclusion A → C ∞ (A ⋆ ) is an algebra monomomorphism justifying the symbol • to denote the multiplication on both A and C ∞ (A ⋆ ) ⊃ A. The automorphisms of A, i.e., the linear isomorphisms g : A → A with g(a • b) = (ga) • (gb), form a Lie group.The Lie algebra of that group consists of the derivations, i.e., the linear maps d ∈ gl(A) with d(a • b) = (da) • b + a • (db).Moreover, we have the structure Lie group G(A) whose Lie algebra is generated by left multiplications, i.e., by the maps l a : (b → a • b) ∈ gl(A).
Therefore, even though we do not assume A to be a Lie algebra, there is a Lie algebra that is naturally associated to A, and we may hope to use its theory to gain insight about A itself.This works to some extent, but a problem arises from the fact that the A-dual distribution H A on A ⋆ defined by is in general not integrable.We recall that, in the Lie algebra case, H A is integrable, and its leaves are precisely the coadjoint orbits which carry a symplectic structure induced by equation (1.2).Because H 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 A. Associativity already gives us some leverage, but the more specific case that we are interested in here is when 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 A = J .In this case, we can also define an extended structure Lie algebra ĝ(J ) that maps surjectively onto g(J ) [13,52,55].This algebra is the direct sum Der 0 (J ) ⊕ J of the inner derivations (those generated by left multiplications l x with algebra elements x) and J itself.The Lie bracket on Der 0 (J ) is the commutator, while, for x, y ∈ J , it is [x, y] := [l x , l y ] ∈ Der 0 (J ), and, for d ∈ Der 0 (J ) and , we have a transvective symmetric pair (see Definition 2.5) which provides us with further structure to work with.
The generalized distribution H J on J ⋆ is still not integrable in general, but the bilinear form G ξ induced by equation (1.1) on H J ξ is symmetric, and therefore it defines a pseudo-Riemannian metric on the m J -regular part of each G(J )-orbit O reg m J ⊂ J ⋆ (see Definition 2.3).On J there is the symmetric, bilinear form τ (x, y) := tr l {x,y} , which is also associative with respect to the Jordan product, and we can use it to decompose J [52, p. 59].Specifically, in the case of positive Jordan algebras, τ is positive definite so that there is a canonical identification J ∼ = J ⋆ , and we can also derive additional properties from the above-mentioned decomposition.Each ξ ∈ J ⋆ has a spectral decomposition associating with ξ its spectral coefficients (λ i ) r i=1 ∈ R r , where r denotes the rank of J .The pair (n + , n − ) counting the number of positive and negative spectral coefficients is called the spectral signature of ξ.We then show the following: Theorem A (cf. Theorem 4.7).If J is a positive simple real Jordan algebra, then the orbits of the structure group G(J ) consist of all elements with the same spectral signature.
We also characterize the regular points (i.e., those where the generalized distribution H J is integrable) in such a G(J )-orbit in Theorem 4.8 and describe the pseudo-Riemannian metric G ξ at each regular point ξ ∈ J ⋆ in Proposition 4.9.Specifically, let Ω J denote the cone of squares of J , i.e., the interior of the set x 2 | x ∈ J .Then Ω J is the G(J )-orbit of the identity 1 J .The characterizations in Theorem 4.8, Proposition 3.3, and Proposition 4.9 lead to the following remarkable description: Theorem B. Let J be a positive simple real Jordan algebra.Then all points of a G(J )-orbit O ⊂ J ⋆ are m J -regular iff O ⊂ Ω J or O ⊂ −Ω J .The form G on O is positive definite in the first and negative definite in the second case, thus defining a Riemannian metric on O which is invariant with respect to the action of the automorphism group of J .
For all regular ξ / ∈ ±Ω J the form G ξ is indefinite, so the definiteness of G ξ gives a new characterization of Ω J .We provide descriptions of the orbits O and the metric 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 finitedimensional algebra 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.

Preliminary material 2.1 Notational conventions
For a finite-dimensional (real or complex) vector space V , we denote by Gl(V ) the Lie group of linear automorphisms, whose Lie algebra gl(V ) consists of all linear endomorphisms of V .
For finite-dimensional vector spaces V , W and U , we define the contraction map where V ⋆ 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 Finally, the dual of a linear map ϕ : V → W is denoted as

Generalized distributions
Let M be a finite-dimensional, real smooth manifold.A generalized distribution on M is a family D = (D p ) p∈M of subspaces D p ⊂ T p M .We let Γ(D) be the set of (smooth) vector fields X on M with X p ∈ D p for all p, and we call D smooth if for each v ∈ D p there is a vector field X ∈ Γ(D) with X p = v.Given a smooth generalized distribution D, we define the Frobenius tensor F p at p ∈ M as It is straightforward to verify that F p (X, Y ) depends on X p and Y p only, i.e., F = (F p ) p∈M is a well-defined tensor field.We also define the generalized distribution Definition 2.1.We call a smooth generalized distribution D involutive at p ∈ M , if F p = 0 or, equivalently, if [D, D] p = D p , and we call it involutive, if this holds for every p.Furthermore, an (immersed) submanifold N ⊂ M with T p N = D p for all p ∈ N is called an integral leaf of D. If there is an integral leaf of D containing p, then we call D integrable at p ∈ M and call p an integral point of D; if this is the case for each p ∈ M , then we call D integrable.
Clearly, if D is integrable (at p), then it is also involutive (at p); according to Frobenius' theorem, the converse of this statement holds if D has constant rank.However, if the rank of D is non-constant, then the converse may fail to hold [72].

G-manifolds
Let G be a finite-dimensional, real Lie group with identity element e, Lie algebra g ∼ = T e G, and let M be a G-manifold, i.e., a finite-dimensional, real smooth manifold with a smooth left action For p ∈ M we define the stabilizer of p to be the subgroup Evidently, H g•p = gHg −1 , and h g•p = Ad g (h p ), so that the stabilizer on each G-orbit is unique up to conjugation.
We define the orbit distribution on M by where X * denotes the action field on M .Evidently, D g is integrable, as the G-orbits are integral leaves of D g .Moreover, Also recall that the map X → X * is a anti-homomorphism of Lie algebras, i.e., For any linear subspace m ⊂ g, we define the smooth generalized distribution D m by and evidently, Let m ⊂ g be a linear subspace.Then the following are equivalent: p , and, by equation (2.4), this is the case iff [X, Y ] ∈ m + h p , showing the equivalence of the first two conditions.
The second condition is equivalent to saying that for each As the rank of D m p is a lower semicontinuous function in p, O reg ⊂ O is open (but possibly empty).As we shall see in later sections, O reg m may be a proper subset of O and is not necessarily connected.
Corollary 2.4.Suppose that m ⊂ g is a linear subspace such that (2.5) Then for each p ∈ M the following are equivalent: (1) p is involutive, p is an m-regular point.
In this case, the maximal integral leaf through p is the connected component of p in We call this pair transvective, if g is generated by m as a Lie algebra, i.e., if [m, m] = k.
Clearly, equation (2.6) is equivalent to saying that the involution σ : g → g with k and m as the (+1)-and (−1)-eigenspace, respectively, is a Lie algebra automorphism.

Structure groups and canonical distributions on duals of algebras
In this section, we shall consider a finite-dimensional algebra A, by which we simply mean a finite-dimensional (real or complex) vector space with a bilinear product • : A × A → 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 • is a map A ⋆ → A ⋆ ⊗ A ⋆ , and as T ξ A ⋆ ∼ = A ⋆ , we may regard this as a linear bivector field on A ⋆ : for all a, b ∈ A. Therefore, there is an induced multiplication on the space C ∞ (A ⋆ ) of real-valued, smooth functions on A ⋆ which by abuse of notation we also denote by •, given by with the canonical identification justifying the ambiguous use of the symbol •.Contraction in the first entry yields a linear map whose image we call the A-dual distribution H A on A ⋆ by For a finite-dimensional algebra (A, •) we define the following: (1) For a ∈ A we let l a ∈ gl(A) be the map (b → a • b) ∈ gl(A).
(2) The structure Lie algebra of A is the Lie subalgebra g(A) ⊂ gl(A) generated by m A := {l a | a ∈ A}.
(3) The structure Lie group of A is the connected Lie subgroup G(A) ⊂ Gl(A) with Lie algebra g(A).
(5) An automorphism of A is a linear isomorphism g : It is straightforward to verify that the automorphisms and derivations form a regular Lie subgroup and a Lie subalgebra Aut(A) ⊂ Gl(A) and Der(A) ⊂ gl(A), respectively, called the automorphism group and derivation algebra of A, respectively.In fact, Der(A) is the Lie algebra of Aut(A).Moreover, g ∈ Aut(A) and d ∈ Der(A) iff for all a ∈ A we have That is, the adjoint action of Aut(A) and Der(A) on gl(A) preserves the subspace m A and hence the structure Lie algebra g(A) and structure Lie group G(A). 1  For ξ ∈ A ⋆ and a, b ∈ A, we have = ⟨♯a, b⟩ ξ , so that l * a = ♯a ∈ Γ(A ⋆ , T * A ⋆ ), regarded as a linear 1-form on A ⋆ .It follows that More generally, for a smooth function f ∈ C ∞ (A ⋆ ), we define the A-dual vector field of f as Since ♯a = ♯f a = ∇ A (f a ), it follows that the A-dual distribution may also be characterized by and unwinding the definitions, it easily follows that for f, g ∈ C ∞ (A ⋆ ) that 1 Actually, it would be more accurate to call g(A) and G(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.
Remark 3.2.If the multiplication • is symmetric (e.g., if A is a Jordan algebra), the dual vector field ∇ A (f ) is usually referred to as the gradient vector field of f , while in the case of an antisymmetric multiplication • (e.g., if A is a Lie algebra), it is called the Hamiltonian vector field of f .That is, the term A-dual vector field subsumes both cases.
We wish to caution the reader that in case of a skew-symmetric multiplication •, the notation ∇ A (f ) for the Hamiltonian vector does not match the standard convention.We use it nevertheless to unify our notation.
If g ∈ Aut(A) is an automorphism, then by equation (3.5) the action of g * on A ⋆ preserves the distribution H A and hence permutes integral leaves of equal dimensions, preserving m A -regular points.
As it turns out, if the product • is symmetric or skew-symmetric, then there is a canonical bilinear pairing on H A .Proposition 3.3.Let (A, •) be a finite-dimensional real algebra such that • is symmetric (skewsymmetric, respectively).Then on there is a canonical non-degenerate symmetric (skew-symmetric, respectively) bilinear form, given by Furthermore, G is preserved by the action of the automorphism group Aut(A).
Finally, if g ∈ Aut(A) is an automorphism, then equation (3.5) implies that and from here, the invariance of G under the action of the automorphism group follows.
We shall now give classes of examples of these notions.
Comparing our notions with those established for Poisson manifolds, we observe that for a function f ∈ C ∞ (g ⋆ ), the g-gradient vector field ∇ A (f ) corresponds to the Hamiltonian vector field X f for Poisson manifolds, so that the dual distribution H A ξ from equation (3.4) is the Hamiltonian distribution of the Poisson manifold.It is integrable, as the Hamiltonian vector fields satisfy the identity The Jacobi identity implies that l a = ad a satisfies [l a , l b ] = l [a,b] , so that m A is closed under the commutator bracket and therefore, g(A) = m A ∼ = g/z(g).That is, the action of the structure group is induced by the coadjoint action of G on g ⋆ , and the skew-symmetric non-degenerate bilinear form G on H A from equation (3.8) coincides with the symplectic form on each coadjoint orbit in g ⋆ .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 m A are the coadjoint orbits of g ⋆ , 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 • is equivalent to saying that l a l b = l a•b , so that {l a | a ∈ A} ⊂ gl(A) is a subalgebra, which means that the structure algebra g(A) equals m A with the Lie bracket being the commutator.
Thus, if we regard A as a Lie algebra with the Lie bracket [a, b] := a • b − b • a, then the G(A)-orbits are the coadjoint orbits on A ⋆ , regarded as the dual of a Lie algebra and thus described in the preceding paragraph.
Note that by Proposition 3.3 the bilinear form G on these orbits only exists if • is symmetric or antisymmetric.
If A is a commutative and associative algebra, then G(A) and g(A) are abelian Lie groups, respectively.In this case, the G(A)-orbits of A ⋆ are diffeomorphic to the direct product of a torus and Euclidean space.
In the two preceding cases, m A is closed under Lie brackets, so that it coincides with the structure algebra g(A).This implies that, by the very definition, m A is integrable having the G(A)-orbits in A ⋆ as leaves.In particular, all orbits are m A -regular.
In contrast, for a Jordan algebra J , it is no longer true that m J is a Lie algebra, so that not all G(J )-orbits on the dual J ⋆ are m J -regular in our sense.Since the Jordan product is symmetric, the non-degenerate form G from (3.8) defines a pseudo-Riemannian metric on the regular part O reg m J of each orbit.We shall describe these structures on the G(J )-orbits on J ⋆ and the pseudo-Riemannian metric G in more details, and we will see how, for some specific type of positive Jordan algebras, and suitable orbits, 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].

Jordan algebras and Jordan distributions
Let J be a real, finite-dimensional Jordan algebra, that is, a real vector space endowed with a bilinear symmetric product {•, •}, satisfying for x, y ∈ 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 R J ∈ Γ J ⋆ , S 2 (T J ⋆ ) from equation (3.1), the musical operator # J : T * J ⋆ → T J ⋆ from equation (3.3), the J -dual vector field ∇ J f = #df ∈ X(J ⋆ ) from (3.7), and the induced J -dual distribution H J ⊂ T J ⋆ from equation (3.4).
In particular, we have H J = D m J by equation (3.6), where is the space of (left-)multiplications with elements x ∈ J , acting on the dual space J ⋆ .For every m J -regular point ξ, the vector space carries the non-degenerate symmetric bilinear form G ξ defined in equation (3.8).
As we pointed out before, the space m J of (left-)multiplication in J is not closed under Lie brackets in general.However, the following is known.We denote by Der 0 (J ) the span of all elements of the form [l x , l y ] for x, y ∈ J .By equation (4.1), Der 0 (J ) ⊂ Der(J ) is an ideal whose elements are called inner derivations of J .This fact can be used to describe the structure Lie algebra of J .Definition 4.2.We define the extended structure Lie algebra ĝ(J ) of J as follows.As a vector space, ĝ(J ) is defined by ĝ(J ) = Der 0 (J ) ⊕ J .
The Lie bracket on ĝ(J ) is defined as follows: on Der 0 (J ) ⊂ Der(J ) ⊂ gl(J ), the Lie bracket is just the commutator between linear maps; 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 (ĝ(J ), Der 0 (J )) is a transvective symmetric pair in the sense of Definition 2.5.
There is a canonical Lie algebra representation of ĝ(J ) on J , called the standard representation, defined by Indeed, this defines a Lie algebra homomorphism by the definition of the Lie bracket on ĝ(J ) and by equation (3.5).
Observe that the image ϕ(ĝ(J )) ⊂ gl(J ) is generated by all l x , x ∈ J , whence equals the structure Lie algebra g(J ) from Definition 3.1.Thus, there is a surjective Lie group homomorphism Ĝ(J ) → G(J ) with differential ϕ, where G(J ) ⊂ Gl(J ) is the structure group from Definition 3.1. 2n general, ϕ may fail to be injective (the kernel of ϕ contains the center of z(J ) ⊂ J ⊂ g(J )), so that the structure algebra g(J ) and the extended structure algebra ĝ(J ) may not be isomorphic.
Then we obtain the following integrability criterion.
Proposition 4.3.Let J be a Jordan algebra.Then, for the distribution H J from equation (3.4), the following assertions are equivalent. (1) ) Der 0 (J )•ξ ⊂ J •ξ, where the multiplication refers to the dual action of Der 0 (J ), J ⊂ ĝ(J ) on J ⋆ .
If this is the case, then the maximal integral leaf through ξ is the connected component of ξ Proof .The map ϕ from equation (4.2) defines an action of Ĝ(J ) on J ⋆ such that, by equation (3.6), it is H J ξ = D J ξ .Evidently, Ĝ(J )•ξ = G(J )•ξ.Since (ĝ(J ), Der 0 (J )) is a transvective symmetric pair, equation (2.5) is satisfied for m := J ⊂ ĝ(J ), and the assertion now follows from Corollary 2.4, as by equation (2.3) it is
A symmetric bilinear form β on J is called associative, if for all x, y, z ∈ J i.e., if all l x are self-adjoint w.r.t.β.Then the following is known.An element c ∈ 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 ̸ = 0.For an idempotent c ∈ J , l c is diagonalizable with eigenvalues in 0, 1  2 , 1 [30, Proposition III.1.2].Therefore, it follows that τ (c, c) = tr l {c,c} = tr l c ≥ 1, as the trace is the sum of the eigenvalues, and c is in the 1-eigenspace of l c .
If J has an identity element 1 J , then a Jordan frame of J is a set (c i ) r i=1 ⊂ J of primitive idempotents such that = τ (c i , {c i , c j }) = 0 for i ̸ = j, so that (c i ) r i=1 is an τ -orthogonal system.From this, one can show that the maps l c i commute pairwise [30,Lemma IV.1.3].
Let c := span{c i }.Then, as all l c i are diagonalizable, there is a τ -orthogonal decomposition of J into the common eigenspace of l c i , i.e., into spaces of the form for some ρ ∈ c ⋆ .Since c i has only eigenvalues 0, 1  2 , 1 and ρ(1 J ) = 1, it follows that ρ = 1 2 (θ i +θ j ), i ≤ j, where (θ i ) r i=1 ∈ c ⋆ is the dual basis to (c i ) r i=1 .That is, we have the τ -orthogonal eigenspace decomposition where J ij := J 1 2 (θ i +θ j ) .This is called the Peirce decomposition of J with respect to the Jordan frame (c i ) r i=1 .For convenience, we let J ji := J ij for i < j.

Semi-simple and positive Jordan algebras
For a Jordan algebra J , we define the radical of J as the null space of τ , i.e., r(J ) := {a ∈ J | τ (a, x) = 0 for all x ∈ J }.
Evidently, r(J ) ⊂ J is an ideal by Proposition 4.4.We call J semi-simple if r(J ) = 0, i.e., if τ is non-degenerate.Moreover, we call J positive or formally real, if τ is positive definite.
We shall now collect some known results on semi-simple and positive Jordan algebras.
Proposition 4.5.Let J be a semi-simple real Jordan algebra.Then the following hold.
(1) J has a decomposition J = J 1 ⊕ • • • ⊕ J k into simple Jordan algebras J i , i.e., such that J i does not contain a non-trivial ideal [52,Theorem III.11].
(4) If J is positive, then for every x ∈ J there is a Jordan frame {c i } r i=1 with x ∈ span({c i } r i=1 ) [30,Theorem III.1.2].In particular, J has Jordan frames.
(5) If J is simple and positive and (c i ) r i=1 and (c ′ i ) r i=1 are Jordan frames, then there is an automorphism h ∈ Aut 0 (J ) with h(c i ) = c ′ i for all i [30, Theorem IV.2.5], 3 where Aut 0 (J ) ⊂ Aut(J ) is the identity component.In particular, all Jordan frames have the same number r of elements, and r is called the rank of J .Proof .We only need to show point (7), as it appears not to be explicitly stated in the literature.Note that J ii is a subalgebra, as {J ii , J ii } ⊂ J ii by the product relations in point ( 6), and by definition c i = 1 J ii .Since τ | J ii is a positive definite associative bilinear form, it follows from point (3) that J ii is a positive Jordan algebra as well.However, since c i = 1 J ii is primitive, it follows that each Jordan frame of J ii consists of c i only, so that by ( 4) each x ∈ J ii must be a multiple of The fourth of these results is called the spectral theorem of positive Jordan algebras.It shows that each x ∈ J admits a decomposition for a Jordan frame (c i ) r i=1 , and the decomposition in equation (4.7) is referred to as the spectral decomposition of x.The λ i 's are called the spectral coefficients of x.Evidently, the tuple (λ i ) r i=1 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 J be a semi-simple, positive Jordan algebra, (c i ) r i=1 a Jordan frame of J and x = i λ i c i .Then, it holds where we use the index convention that i, j run over 1, . . ., r, while a, b run over those indices with λ a ̸ = 0, and ν, µ over those indices with λ ν = 0.
Proof .By point (7) in Proposition 4.5, the Peirce decomposition in equation (4.6) reads As l x (x ij ) = 1 2 (λ i + λ j )x ij for x ij ∈ J ij , the first equality in equation (4.8) is immediate.For the second equality, recall that Der 0 (J ) is spanned by [l x ij , l y kl ] for x ij , y ij ∈ J ij , and we compute Therefore, the relation "⊂" 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 The third equation then follows as g(J ) For a positive simple Jordan algebra J , the following hold: (1) The orbits of Aut 0 (J ) are the sets of elements with equal spectral coefficients.
(2) The orbits of the structure group G(J ) consist of all elements with equal spectral signature.
Proof .Any automorphism maps (primitive) idempotents to (primitive) idempotents and fixes 1 J , whence it maps Jordan frames to Jordan frames.Thus, if x = r i=1 λ i c i for a Jordan frame (c i ) r i=1 , it follows that for h ∈ Aut 0 (J ) and (h(c i )) r i=1 is again a Jordan frame, so that x, h(x) have the same spectral coefficients (λ i ) r i=1 .
Conversely, if x, y have the same spectral coefficients (λ i ) r i=1 , then . Thus, by point (5) of Proposition 4.5, there is a h ∈ Aut 0 (J ) with h(c i ) = c ′ i and hence, h(x) = y.This shows the first statement.Concerning the second statement, we define the following subsets of J : Evidently, Moreover, by the first assertion, all these sets are Aut 0 (J )-invariant.
There are continuous (in fact, polynomial) functions a k : J → R such that is the minimal polynomial of all generic x ∈ J , i.e., elements with pairwise distinct spectral coefficients x) = 0 by equation (4.5), and as the roots λ i (x) are pairwise distinct, it follows that and as generic x's are dense in J [30, Proposition II.2.1], it follows that equation (4.11) holds for any x ∈ J .Then Σ ≤m are those elements where λ = 0 is a root of f (x, •) of multiplicity ≥ r −m; that is, As the spectral coefficients of x are unchanged under the automorphism group, equation (4.10) and equation (4.11) imply We assert that Σ ≤m is invariant under G(J ).For this, fix a Jordan frame (c i ) r i=1 and let x = λ a c a ∈ c ∩ Σ m , using the index summation convention from Lemma 4.6.Define the map and as e ta λ a c a ∈ Σ m , the Aut 0 (J )-invariance of Σ m implies that Im(Φ x ) ⊂ Σ m .Moreover, it follows that the image of the differential d (e,0) In fact, equation (4.8) implies that Im d (e,0) This implies that for x ∈ c ∩ Σ m generic, there is an open neighborhood U ⊂ G(J ) of the identity such that Thus, for X ∈ g(J ) and x ∈ c ∩ Σ m generic, equation (4.12) implies for |t| small enough such that exp(tX) ∈ 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 ∈ R, in particular for t = 1.That is, we conclude that for x ∈ c ∩ Σ m generic, and taking the closure, it follows that equation (4.16) holds for all x ∈ c ∩ Σ ≤m .Moreover, by the first part, each x ∈ Σ ≤m can be written as x = h • x for x ∈ c ∩ Σ ≤m and h ∈ Aut 0 (J ).Thus, it holds = 0, so that equation (4.16) holds for all x ∈ Σ ≤m and X ∈ g(J ).Thus, by equation (4.12) it follows that exp(g(J )) • Σ ≤m ⊂ Σ ≤m , and as the connected group G(J ) is generated by exp(g(J )), the asserted G(J )-invariance of Σ ≤m follows.
Since Σ m = Σ ≤m \Σ ≤m−1 is the difference of two G(J )-invariant sets, it follows that Σ m is G(J )-invariant as well.
Next, we assert that Σ n + ,n − ⊂ Σ m is relatively closed.For if (x k ) k∈N ∈ Σ n + ,n − converges to x 0 ∈ Σ m , then, fixing a Jordan frame (c i ) r i=1 , we find h k ∈ Aut 0 (J ) such that Since y k ∈ Σ n + ,n − as well, it follows that the signs of 0 ̸ = λ a,k are equal for all k.As Aut 0 (J ) is compact, we may pass to a subsequence to assume that h k → h 0 , whence y k → h 0 x 0 , i.e., Since x 0 and hence h 0 x 0 ∈ Σ m , it follows that λ a,0 ̸ = 0 for all a, whence λ a,0 has the same sign as all λ a,k , so that h 0 x 0 ∈ Σ n + ,n − , i.e., x 0 ∈ Σ n + ,n − .Thus, equation (4.9) is the disjoint decomposition of Σ m into finitely many relatively closed subsets, and since G(J ) and hence all orbits are connected, it follows that each G(J )-orbit must be contained in some Σ n + ,n − .
On the other hand, as elements with equal spectral coefficients lie in the same Aut 0 (J )-orbit, equation (4.14) immediately implies that G(J ) acts transitively on Σ n + ,n − , which completes the proof. ■ For a positive Jordan algebra J we identify J and J ⋆ by the isomorphism By the spectral theorem (cf.point (4) of Proposition 4.5), for each ξ ∈ J ⋆ there is a Jordan frame (c i ) r i=1 on J such that and we define the spectral coefficients (λ i ) i and the spectral signature (n + , n − ) of ξ to be the spectral coefficients and signature of ξ # .We let be the set of elements of spectral signature (n + , n − ).Furthermore, we define the dual of τ to be the scalar product on J ⋆ given by For x, y ∈ J and ξ ∈ J ⋆ , we have (l Therefore, by the definition of the dual action, it follows that so that, by Theorem 4.7, we obtain that the orbits of the action of G(J ) on J ⋆ are the sets O n + ,n − .The open cone of squares in J is Looking at the spectral decomposition in equation (4.7), it follows that x ∈ Ω J iff all its spectral coefficients are positive iff l x is positive definite, and the latter description shows that Ω J is indeed a convex cone; in fact, it easily follows from this characterization that Theorem 4.8.Let J be a positive, simple Jordan algebra with structure group G(J ) ⊂ Gl(J ).
Then ξ ∈ J ⋆ is m J -regular iff the spectral coefficients (λ i ) of ξ satisfy: In particular, the G(J )-orbit O n + ,n − is m J -regular iff n + = 0 or n − = 0, i.e., iff it is contained in Ω J or −Ω J .
On the other hand, on O n + ,0 (O 0,n − , respectively) λ a , λ b > 0 (< 0, respectively) so that equation (4.20) holds; whence O n + ,0 and O 0,n − are the only m J -regular orbits, and by equation (4.19) these are the orbits contained in ΩJ or − ΩJ , respectively.■ Let us now describe the pseudo-Riemannian metric G on O reg m J .Take ξ ∈ J ⋆ with spectral decomposition for some Jordan frame (c i ) r i=1 , and assume it satisfies equation (4.20).Then, it holds and we have the following Proposition.
Proposition 4.9.Let ξ = λ a c ♭ a ∈ J ⋆ be as above.Then, it holds which is equivalent to Therefore, evaluating both sides of equation (3.8) gives us Since both equations must be equal, (4.22) follows as the Peirce decomposition J = ij J ij is τ -orthogonal.■ Remark 4.10.
(1) Comparing the description of the regular points in O in Theorem 4.8 and equation (4.22), it follows that G ξ has a pole of order 1 on O\O reg m J .(2) As G ξ is positive or negative definite on J ♭ aa , depending on the sign of λ a ̸ = 0, it follows that G ξ is indefinite at any regular point of spectral signature (n + , n − ) with n + , n − > 0.
That is, G on O is definite (and hence defines a Riemannian metric (3) It is also evident from the description of regular points in Theorem 4.8 that for a non- is not path connected.
(4) Note that the Riemannian metric G on O n + ,0 (and, similarly, −G on O 0,n − ) is not complete.Namely, for t > 0, the curve for a Jordan frame (c i ) r i=1 has constant speed with respect to G because However, α cannot be extended in O n + ,0 at t = 0.

Examples
We shall now describe the metric G for the standard examples of positive Jordan algebras.

The Fisher-Rao metric for finite sample spaces
We regard J := 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 Ω J can be identified with the first orthant R n + ⊂ R n ∼ = J ⋆ .The metric G ξ at ξ = (ξ 1 , . . ., ξ n ) ∈ Ω J is given by When interpreting Ω J as the set of positive finite measures on X n = {1, . . ., n}, it is clear that G is such its pullback to the submanifold of strictly positive probability distributions on X n (i.e., open interior of the unit simplex inside 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 Ω J = R n + as the analogue of the homogeneous symplectic form on co-adjoint orbits in the case of Lie algebras.

The Jordan algebras
Let K denote either the real, complex or quaternionic numbers, and we define the Jordan algebra of self-adjoint matrices For convenience, we replace τ from equation (4.3) by the associative inner product τ (A, B) := Tr(AB), so that τ and τ only differ by the multiplicative constant 1 n dim R M sa n (K).Let E ij ∈ K n×n denote the matrix with a 1 in the (i, j)-entry.Then {E 11 , . . ., E nn } is a Jordan frame of M sa n (K), and the remaining Pierce spaces with respect to this frame are given as For K = R, C and H, the automorphism group of M sa n (K) is SO(n), U(n) and Sp(n), respectively, acting on M sa n (K) by conjugation.Thus, in particular, each A ∈ M sa n (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 ξ Moreover, because of Proposition 3.3, it follows that G is preserved by the automorphism group of M sa n (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 K = C.In this case, we may identify M sa n (C) with the Jordan algebra of selfadjoint observables of a finite-level quantum system with Hilbert space H ∼ = C n .Then, if we focus on the m J -regular orbit Ω J of faithful, non-normalized quantum states, which can be identified with the dual of the orbit of invertible positive matrices in M sa n (C), the metric tensor G is such that its pullback to the submanifold of faithful quantum states, determined by the condition 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 m J -regular orbit through non-normalized pure states, which are identified with rank-one matrices in M sa n (C), the metric tensor G is such that its pullback to the submanifold of pure states, determined by the condition 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 sa n (C).

4.6
The spin-factor Jordan algebra J Spin(n) Denoting the standard inner product of R n by ⟨•, •⟩, we let and where 1 is the identity element of J Spin(n).An associative inner product is given by The automorphism group is SO(n), acting on R n and fixing 1.Every Jordan frame is given by for a fixed unit vector e 0 ∈ R n , and the Peirce space complementary to the Jordan frame is The two spectral coefficients of an element X = t1 + x are is regular, where e 0 ∈ R n is a unit vector, then the tangent vectors X 1 , X 2 ∈ T ξ O are of the form where x i ∈ e ⊥ 0 , and where t 0 = ±s 0 ⇒ t i = ±s i .The spectral coefficients of ξ are λ 1 = 1 2 (t 0 + s 0 ) and λ 2 = 1 2 (t 0 − s 0 ), and Therefore, While it would be possible but elaborate to calculate the regular points and the inner product 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.

Non-simple, semisimple positive Jordan algebras
By point (1) of Proposition 4.5, each positive, semisimple Jordan algebra admits a decomposition J = J 1 ⊕ • • • ⊕ J k into positive simple Jordan algebras, so that both the automorphism and the structure group of J are the direct sum of the automorphism and structure group of the simple factors J i , respectively.Then, applying Theorems 4.7 and 4.8, and Proposition 4.9, it follows that the G(J )-orbits are of the form where In particular, such an orbit is regular iff n i + n i − = 0 for all i, and the metric G on the regular part of this orbit is given by (4.22).

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 G(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 G(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 J , while we consider positive linear functionals on 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 J (e.g., classical probability distributions, quantum states).It is true that it is possible to identify 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 Ω J of invertible positive elements of a suitable Jordan algebra J , are given from the onset.From our point of view, the open cone Ω 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 H J .Therefore, Ω 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 G(J ) of the cone itself, while the Riemannian structure we obtain is invariant only under the subgroup of G(J ) composed by automorphisms of 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 R n , the G(J )-invariant Riemannian metric tensor g on Ω J = R n + can be written as [18,Theorem 2.3.19] where {p j } j=1,...,n is a Cartesian coordinate system adapted to Ω J in the sense that elements in Ω 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 G(J ) invariance of g implies it is invariant with respect to dilations also when J = B sa (C n ) ∼ = M sa n (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 J = B sa (C n ) ∼ = M sa n (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 socalled 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 G(J )-invariant Riemannian metric tensor discussed above is defined in terms of the Jordan triple product of 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 methodlike theory, our constructions and results are different from previous ones and, taken together, they yield a fuller picture of the structure of Jordan algebras.

Definition 2 . 3 .
and this is evidently equivalent to the third condition.■ Let M be a G-manifold and m ⊂ g a linear subspace.We call p ∈ M an m-regular point if D m p = T p O, where O ⊂ M is the G-orbit of p.The subset of m-regular points in O is denoted by O reg m ⊂ O.
[30,rk 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 .
This metric is positive definite if t 0 ≥ |s 0 | and negative definite if t 0 ≤ −|s 0 |, as predicted by Proposition 4.9.