Cohomology in algebra, geometry, physicsand statistics

Information Geometry and Hamiltonian Systems on Lie Groups

The link between Hamiltonian Integrable Systems and Information Geometry was discovered by Amari, Fujiwara and Nakamura (90s). In particular, Nakamura succeeded to define the tau-function for the open Toda Lattice by using Information Geometry .

We propose a more general study of Hamiltonian Systems related with the Information Geometry on Lie groups.

Fisher-Rao semi-definite metric is naturally induced as a left-invariant semi-definite metric on the Lie group, which is regarded as the parameter space of the family of probability density functions. For a specific choice of family of probability density functions on compact semi-simple Lie group, the equation for the geodesic flow is derived through the Euler-Poincaré reduction. Certain perspectives are mentioned about the geodesics equation on the basis of its similarity with the Bloch-Brockett –Ratiu double bracket equation and with the Euler-Arnol'd equation for a generalized free rigid body dynamics.

This is a joint work with Daisuke Tarama (Ritsumeikan University).
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