Research Briefs

Published Sep 28, 2023

*Florio M. Ciaglia, Jürgen Jost, Lorenz Schwachhöfer*

The theory initiated by Kirillov and extended by Kostant and Souriau relates algebraic structures to differential geometry and mathematical physics in a deep and very productive way. It has led to spectacular results in representation theory, classical and quantum mechanics, and it is closely related to geometric quantization. In that theory, one considers the action of a Lie algebra on its dual, the coadjoint action. Here, the Lie algebra has two roles, it is acted upon, as an algebra, and it acts because it encodes symmetries. In our contribution, we separate these two roles. We take an arbitrary algebra (and later specialize in the case of Jordan algebras - which arise naturally in quantum information theory) and let a Lie algebra act on its dual via infinitesimal symmetries. In the Kirillov-Kostant-Souriau construction, the duality between the Lie algebra and its dual turns the Lie bracket into a symplectic form. Likewise, for a general algebra, the multiplication in the algebra should be turned into a bilinear form. Of course, since the multiplication in a general algebra is not anticommutative, the resulting form will no longer be antisymmetric, that is, symplectic, but rather reflects the structure of the algebra in question.

So, let us see how far we can get with this program, what the technical obstacles are, and what additional assumptions may help us to overcome them.

Thus, we consider an algebra \(\mathcal{A}\), i.e., a vector space with a bilinear product \(\bullet: \mathcal{A} \times \mathcal{A} \to \mathcal{A}\). In order not to have to worry about issues of duality, we assume that \(\mathcal{A}\) is finite-dimensional. \(\mathcal{A}^\star\) is the dual space of \(\mathcal{A}\), and the corresponding pairing is denoted by \(\langle .,.\rangle\). We can identify the bidual \(\mathcal{A}^{\star\star} \) with \(\mathcal{A}\) itself, and for the tangent and cotangent spaces of \(\mathcal{A}^\star\), we then have \(T_\xi \mathcal{A}^\star \cong \mathcal{A}^\star\) and \(T^\ast_\xi \mathcal{A}^\star \cong \mathcal{A}^{\star\star} \cong \mathcal{A}\). Duality allows us to represent the product \(\bullet\) as

\begin{equation} \langle \xi, a \bullet b\rangle \text{ for }\xi \in \mathcal{A}^\star, a, b \in \mathcal{A}. \tag{1} \end{equation}

Thus, from the multiplication \(\bullet\) in the algebra, we have defined a bilinear form on the cotangent spaces (identified with \(\mathcal{A}\)) of \(\mathcal{A}^\star\). For \(C^\infty(\mathcal{A}^\star)\) and \(\xi \in \mathcal{A}^\star\), the differential \(d_\xi f\) then is in \(\mathcal{A}\), and we can define some kind of pseudo-gradient vector field

\[ \Psi^\mathcal{A}(f)_\xi(g) :=\langle \xi, d_\xi f \bullet d_\xi g \rangle. \tag{2} \]

If we now want to carry out a Kirillov-type construction, we encounter the problem that the distribution \(\mathcal{H}^\mathcal{A}\) on \(\mathcal{A}^\star\) defined by

\[ \mathcal{H}^\mathcal{A}_\xi := \{ (\Psi^\mathcal{A}(f))_\xi | f \in C^\infty(\mathcal{A}^\star)\} \subset T_\xi \mathcal{A}^\star. \tag{3} \]

is, in general, not integrable. (In Kirillov's theory, the leaves of this distribution are precisely the coadjoint orbits.)

We, therefore, need a bit more structure on \(\mathcal{A}\), and here, we assume that it is a real, finite-dimensional Jordan algebra \(\mathcal{J}\), i.e., a real vector space with a bilinear symmetric product \(\{\cdot,\cdot\}\), satisfying for \(x,y\in\mathcal{J}\) the *Jordan identity*

\[ \{ \{x, y\}, \{x, x\} \} = \{x, \{y, \{x, x\} \} \}. \ \tag{4} \]

(for instance, obtained by symmetrizing an associative product $\bullet$ via $\{x, y\}:=\frac{x\bullet y +y\bullet x}{2}$, as for self-adjoint matrices). Now, the automorphisms of \(\mathcal{A}\), i.e., the linear isomorphisms \(g: \mathcal{A} \to \mathcal{A}\) with \(g(\{x, y\}) = \{gx, gy\}\), form a Lie group. The Lie algebra of that group consists of the derivations, i.e., the linear maps \(d \in gl(\mathcal{A})\) with \(d(\{x, y\}) = \{dx, y\} + \{x, dy\}\). We also have the structure Lie group \(\mathcal{G}(\mathcal{A})\) whose Lie algebra is generated by left multiplications, i.e., by the maps \(l_x :(y \mapsto \{x, y\}) \in \mathfrak{gl}(\mathcal{A})\). Among those, the brackets \([l_x,l_y]\in \mathfrak{gl}(\mathcal{A}) \) generate the inner derivations \(\mathcal{D}er_0(\mathcal{A})\), and by defining \([x,y]:=[l_y,l_y]\), we get a Lie algebra structure on \(\mathcal{A} \oplus \mathcal{D}er_0(\mathcal{A})\).

Those orbits of the Lie group generated by \(\mathcal{A} \oplus \mathcal{D}er_0(\mathcal{A})\) that satisfy a suitable regularity condition will now play the role of the coadjoint orbits in Kirillov's theory. The regularity condition is essentially connected with the fact that still not all the \(\mathcal{H}^\mathcal{A}_\xi \) need to be integrable in the case of Jordan algebras, but in fact, they are integrable when involutive, i.e., \([ \mathcal{H}^\mathcal{A}_\xi, \mathcal{H}^\mathcal{A}_\xi]=\mathcal{H}^\mathcal{A}_\xi \), and this holds for almost all \(\xi\). Also, since the Jordan product is commutative, now the bilinear form (1) is symmetric. When \(\mathcal{J}\) is semi-simple and positive, we then get Riemannian metrics on the regular orbits, and we can use that for a detailed structure analysis.

We also want to briefly explain what we believe is a relevant connection between this work and information geometry, both classical and quantum.

Specifically, when the Jordan algebra is that of continuous functions on a discrete and finite outcome space (the unique, up to isomorphism, finite-dimensional associative Jordan algebra), the Riemannian metric obtained with our construction turns out to be the *Fisher-Rao* metric tensor ubiquitous in classical information theory and statistics.

Moreover, when the Jordan algebra is that of observables of a finite-level quantum system (i.e., the algebra of self-adjoint linear operators on a finite-dimensional complex Hilbert space), the Riemannian metric tensor obtained with our construction turns out to be the *Fubini-Study* metric tensor on the orbit of pure quantum states, or the *Bures-Helstrom* metric tensor on the orbit of faithful quantum states, both of which represent possible generalizations of the Fisher-Rao metric tensor to the quantum realm.

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