By using the moment algebra of the Vlasov kinetic equation, we characterize the integrable Bloch-Iserles system on symmetric matrices [Bloch, A. M., Brînzănescu, V., Iserles, A., Marsden, J. E., and Ratiu, T. S., “A class of integrable flows on the space of symmetric matrices,” Commun. Math. Phys. 290, 399–435 (2009)] as a geodesic flow on the Jacobi group ${\rm Jac}(\mathbb {R}^{2n})={\rm Sp}(\mathbb {R}^{2n})\,\circledS \,{\rm H}(\mathbb {R}^{2n})$. We analyze the corresponding Lie-Poisson structure by presenting a momentum map, which both untangles the bracket structure and produces particle-type solutions that are inherited from the Vlasov-like interpretation. Moreover, we show how the Vlasov moments associated to Bloch-Iserles dynamics correspond to particular subgroup inclusions into a group central extension (first discovered by Ismagilov, Losik, and Michor [“A 2-cocycle on a group of symplectomorphisms,” Mosc. Math. J. 6, 307–315 (2006)]), which in turn underlies Vlasov kinetic theory. In the most general case of Bloch-Iserles dynamics, a generalization of the Jacobi group also emerges naturally.
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