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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
30/2021

What Lie algebras can tell us about Jordan algebras

Florio Maria Ciaglia, Jürgen Jost and Lorenz J. Schwachhöfer

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 $G_{\xi}$ 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.

Received:
Dec 20, 2021
Published:
Dec 20, 2021
MSC Codes:
17C20, 17C27, 17B60, 17B60
Keywords:
Jordan algebras, Lie algebras, Kirillov orbit method, generalized distributions, Peirce decomposition, Fisher-Rao metric, Bures-Helstrom metric

Related publications

inJournal
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