space of a point particle. If, however, we wished to describe a “gas” composed of freely spinning (fixed) rigid bodies instead of point particles, we expect that the inner bracket for this kinetic theory would be the so(3) bracket mentioned in Section \[S-FINITE\]. Another, more exotic, possibility would be a kinetic theory that describes a gas of Kida vortices. The dynamics of a Kida vortex takes place on the Lie algebra so(2,1) (see [@mmof]), and is governed by the corresponding Lie-Poisson bracket.

$$\label{ALGBRAC}
[f, g] (z) = c_i^{jk} z^i {\frac{\partial f }{\partial z^j}} {\frac{\partial
g }{\partial z^k}} \, ,$$

where $c_i^{jk}$ are the structure constants of the Lie coalgebra.

Integration by parts (assuming $f$ is constant on the boundary) shows that the bracket (\[INFBRAC\]) with inner bracket of form (\[ALGBRAC\]) satisfies equation (\[INFCOSYM\]) if and only if the structure constants obey the condition $c_i^{ik} = 0$ for all $k.$ This happens to be satisfied by the structure constants of any semi-simple Lie algebra. (Incidentally, any Hamiltonian flow generated by a Lie-Poisson bracket satisfying this condition obeys the naive Liouville’s theorem in noncanonical coordinates [@mrmp].)

Now, to calculate the Casimir invariants for these examples, we again must solve equation (\[INFNULL\]) for 