INFORMATION GEOMETRY FOR SOME LIE ALGEBRAS

For certain unitary representations of a Lie algebra [Formula: see text] we define the statistical manifold ℳ of states as the convex cone of [Formula: see text] for which the partition function Z= Tr exp {-X} is finite. The Hessian of Ψ= log Z defines a Riemannian metric g on [Formula: see text], (the Bogoliubov–Kubo–Mori metric); [Formula: see text] foliates into the union of coadjoint orbits, each of which can be given a complex structure (that of Kostant). The program is carried out for so(3), and for sl(2,R) in the discrete series. We show that ℳ=R+× CP 1 and R+×H respectively. We show that for the metaplectic representation of the quadratic canonical algebra, ℳ=R+× CP 2/Z2. Exactly solvable model dynamics is constructed in each case.