text: 171,167,1474089465408888872,1304363050791772160,Florio M. Ciaglia,"What can Lie algebras tell us about Jordan algebras? Among other things which I surely do not know, they can tell us how to find the Fisher-Rao and the Bures-Helstrom metric tensor as if we were looking for the canonical symplectic form on coadjoint orbits <LINK>",https://arxiv.org/abs/2112.09781,"Inspired by Kirillov's theory of coadjoint orbits, we develop a structure theory for finite dimensional Jordan algebras. Given a Jordan algebra ${\mathcal{J}}$, we define a generalized distribution $\mathcal{H}^{{\mathcal{J}}}$ on its dual space ${\mathcal{J}}^\star$ which is canonically determined by the Jordan product in ${\mathcal{J}}$, is invariant under the action of what we call the structure group of ${\mathcal{J}}$, and carries a naturally-defined pseudo-Riemannian bilinear form ${\mathcal{G}}_{\xi}$ at each point. We then turn to the case of positive Jordan algebras and classify the orbits of ${\mathcal{J}}^\star$ under the structure group action. We show that the only orbits which are also leaves of $\mathcal{H}^{{\mathcal{J}}}$ 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 ${\mathcal{G}}$. We discuss applications of our construction to both classical and quantum information geometry by showing that, for appropriate choices of ${\mathcal{J}}$, the Riemannian metric tensor ${\mathcal{G}}$ 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. ",What Lie algebras can tell us about Jordan algebras,1,"['What can Lie algebras tell us about Jordan algebras? Among other things which I surely do not know, they can tell us how to find the Fisher-Rao and the Bures-Helstrom metric tensor as if we were looking for the canonical symplectic form on coadjoint orbits\n<LINK>']",21,12,263