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MiS Preprint

What Lie algebras can tell us about Jordan algebras

Florio Maria Ciaglia, Jürgen Jost and Lorenz J. Schwachhöfer


Inspired by Kirillov’s theory of coadjoint orbits, we develop a structure theory for finite dimensional Jordan algebras. Given a Jordan algebra $J$ , we define a generalized distribution $H^J$ on its dual space $J^*$ which is canonically determined by the Jordan product in $J$ , is invariant under the action of what we call the structure group of $J$ , and carries a naturally-defined pseudo-Riemannian bilinear form $G_{\xi}$ at each point. We then turn to the case of positive Jordan algebras and classify the orbits of $J^*$ under the structure group action. We show that the only orbits which are also leaves of $H^J$ are those in the closure of the cone of squares or its negative, and these are the only orbits where this pseudo-Riemannian bilinear form determines a Riemannian metric tensor $G$.

We discuss applications of our construction to both classical and quantum information geometry by showing that, for appropriate choices of $J$ , the Riemannian metric tensor $G$ 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.

MSC Codes:
17C20, 17C27, 17B60, 17B60
Jordan algebras, Lie algebras, Kirillov orbit method, generalized distributions, Peirce decomposition, Fisher-Rao metric, Bures-Helstrom metric

Related publications

2023 Journal Open Access
Florio M. Ciaglia, Jürgen Jost and Lorenz J. Schwachhöfer

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

In: Symmetry, integrability and geometry : methods and applications (SIGMA), 19 (2023), p. 078