What are the properties of this Poisson algebra?

Question

I have the following (real) quantities (which are from a Classical Mechanics problem):

$$A_1=\frac 1 4(x^2 +p_x^2-y^2-p_y^2 ) \quad A_2=\frac 1 2(x y +p_x p_y)$$

$$A_3=\frac 1 2(x p_y - y p_x )$$

This should be a Poisson algebra with the following Lie bracket:

$$\{f , g\} =\frac{df}{dx}\frac{dg}{dp_x}+\frac{df}{dy}\frac{dg}{dp_y}-\frac{dg}{dx}\frac{df}{dp_x}-\frac{dg}{dy}\frac{df}{dp_y}$$

And if we define: $H=x^2 + y^2 + p_x^2+p^2_y $

I calculated the following relations: $$\{A_i,A_j\}=\sum_k\epsilon_{ijk}A_k$$

$$\{A_i,H\}=0$$

I see that $\epsilon_{ijk}$ are the structure constants, so this Poisson algebra is isomorphic to $\mathfrak{su}(2)$ and $\mathfrak{so}(3)$.

Shouldn't $H$ be a Casimir operator? But what is the universal enveloping algebra where it belongs to? I tried to construct a Casimir operator using the Killing form but it was not $H$.

And does this Lie algebra have an associated group? If $A$ where matrices, I could do $e^{A\theta}$, to obtain the group elements.

I think that my confusion arises because this is not a matrix algebra, where the Lie bracket is just the commutator.

Answer 1

The universal enveloping algebra is the right object to consider in the case of representations in a noncommutive Poisson algebra. But you have a representation in a commutive Poisson algebra. Here the analogue of the universal enveloping algebra is the Lie-Poisson algebra generated by the Lie algebra generators.

Your computations reveal that your $H$ (which is an element of this Lie-Poisson algebra) is a central element, hence a (classical) Casimir element.

The exponentials $e^{u\cdot A}$ are also elements of the Lie-Poisson algebra, and you can compute their Poisson brackets using the chain rule. You'll find that you get a nice and well-known Lie group.

If you want to learn more about Lie-Poisson algebras, you might want to look at my online book

Arnold Neumaier and Dennis Westra, Classical and Quantum Mechanics via Lie algebras, 2011. http://lanl.arxiv.org/abs/0810.1019