Differential Realizations of certain algebras

Question

I'm a first year graduate student in Mathematical Physics, and I am trying to generalise a certain method involving the so-called "Differential realizations" of certain algebras. The problem I'm having is that in papers here (Page 8) and here (Page 5-6), they express a certain type of algebra in terms of differential operators and I'm not sure how that is done. My goal is to see if such methods can be generalised to other algebras to test for a certain property called "Shape Invariance" which is mentioned in the latter paper. I'm wondering if anyone here has ever heard of differential realizations and has any idea how they can be derived for other algebras?

Answer 1

Differential realizations of algebras are common in physics. Here's the general idea for Lie algebras.

A little background.

Recall that an algebra is a pair $(V,[\cdot,\cdot])$ where $V$ is a vector space over a field $\mathbb F$ and $[\cdot,\cdot]$ is an $\mathbb F$-bilinear mapping from $V\times V$ to itself. We call the mapping $[\cdot, \cdot]$ the bracket. The algebra is called a lie algebra provided the bracket satisfies the following two properties for all $x,y,z\in V$: \begin{align} 0&=[x,x] \tag{L1}\\ 0&= [x,[y,z]] + [z,[x,y]]+[y,[z,x]]. \tag{L2} \end{align} Property $(\mathrm{L2})$ is called the Jacobi Identity. A linear mapping $\phi$ between two Lie algebras $\mathfrak g$ and $\mathfrak h$ is called a homomorphism provided is preserves the bracket, namely \begin{align} \phi([x,y]_\mathfrak g) = [\phi(x), \phi(y)]_\mathfrak h. \end{align} A representation of a Lie algebra $\mathfrak g$ over a field $\mathbb F$ is a homomorphism $\phi:\mathfrak g\to \mathfrak {gl}(V)$ where $V$ is a vector space over $\mathbb F$. Here $\mathfrak{gl}(V)$ denotes the set of all linear maps from $V$ to $V$ which forms a Lie algebra when the bracket is taken to be the commutator induced by composition of functions; $[f,g] = f\circ g - g\circ f$.

Representations by differential operators.

Often in physics, one considers a representation of a Lie algebra $\mathfrak g$ that maps each element of the lie algebra to a linear differential operator on some vector space of functions. This is what the papers are doing when they're finding differential relations of Lie algebras.

An example.

The three-dimensional Heisenberg algebra, also known to physicists as the harmonic oscillator algebra, is a three-dimensional complex vector space with basis $\{a, a^\dagger, I\}$ and a bracket defined by the following structure relations: \begin{align} [a,a^\dagger] = I, \qquad [a,I]=0, \qquad [a^\dagger, I]=0. \end{align} In general, structure relations are simply relations that specify the action of the Lie bracket on all distinct pairs of basis elements. Now, consider the linear mapping $\phi:\mathfrak g\to \mathfrak {gl}(L^2(\mathbb R))$ given by its action on $a,a^\dagger, I$ as follows: \begin{align} \phi(a) &= \frac{1}{\sqrt{2}}(P - i X), \qquad \phi(a^\dagger) = \frac{1}{\sqrt{2}}(P + i X), \qquad \phi(I) = I \end{align} where $P,X,I$ are defined by \begin{align} (Pf)(x) &= -i\frac{df}{dx}(x), \qquad (Xf)(x) = xf(x), \qquad (If)(x) = f(x) \end{align} One can show (try it!) that this mapping is a Lie algebra homomorphism, so it is a representation of the harmonic oscillator algebra by way of differential operators on the vector space of square integrable, complex-valued functions on the real line.

How to find such representations.

I'm not certain how one could have discovered the representation $\phi$ above for the harmonic oscillator algebra, but a common way of generating such representations of Lie algebras is to determine a Riemannian or semi-Riemannin manifold $(M,g)$ whose algebra of Killing vectors is the algebra you're looking for. The killing vectors then give the desired representation in terms of differential operators acting on the vector space of scalar functions on the manifold provided you can solve Killing's equation.

For example suppose we want to determine a representation of $\mathfrak{so}(3)$ in terms of differential operators. We consider take the Riemannian manifold $S^2$. It's algebra of killing vectors $\mathfrak{so}(3)$, the Lie algebra of the three-dimensional rotation group. What are the killing vectors? Well, in spherical coordinates, the metric is \begin{align} ds^2 = d\theta^2 + \sin^2\theta d\phi^2 \end{align} and in these coordinates, one obtains the following killing vectors: \begin{align} R &= \partial_\phi \\ S &= \cos\phi\partial_\theta - \cot\theta \sin\phi\partial_\phi \\ T &= -\sin\phi\partial_\theta - \cot\theta \cos\phi\partial_\phi \end{align} which, as you can check by taking commutators, gives the desired representation of $\mathfrak {so}(3)$ as differential operators acting on the vector space of scalar functions on $S^2$.