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
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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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 HJ 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 Gξ 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 HJ 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.