What Lie algebras can tell us about Jordan algebras

Florio Maria Ciaglia, Jürgen Jost, and Lorenz J. Schwachhöfer

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Submission date: 20. Dec. 2021
Pages: 29
Bibtex
MSC-Numbers: 17C20, 17C27, 17B60, 17B60
Keywords and phrases: Jordan algebras, Lie algebras, Kirillov orbit method, generalized distributions, Peirce decomposition, Fisher-Rao metric, Bures-Helstrom metric
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Abstract:
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_ξ 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.