Representations of infinite dimensional Lie algebras as vector fields on manifolds

Question

Suppose I have e.g. the Witt algebra,

$\left[l_n,l_m \right] = -(n-m)l_{n+m}$.

I want to realize the $l_n$ as vector fields on some manifold. The classical example is when the $l_n$ span the Lie algebra of diffeomorphisms of the circle, i.e.

$l_n = -i e^{i n \phi} \partial_\phi, \ \ \ 0 \leq \phi < 2\pi.$

Now I'm interested in actions on higher dimensional manifolds, e.g. $S^1 \times$ something.

As a practical approach I could try an ansatz $l_n = -i e^{i n \phi} \partial_\phi + e^{i n \phi} f_n (x) \partial_x$ and demand that the commutation relation is satisfied, which would lead to differential-difference equations for the $f_n$.

I'm sure there's lots of theory about this somewhere... I'd really appreciate some pointers to the right direction...

P.S. Sorry if I seem lazy for not researching this myself, but maybe I'll find the answer faster with some help from The Community?

Answer 1

You could try É. Cartan's papers on infinite pseudogroups (mostly appearing 1904-05). In particular, see Paragraph 57 of Sur la structure des groupes infinis de transformation (suite). There, for example, he proves that the (pseudo-)group of diffeomorphisms of the line has three distinct (i.e., nonequivalent) $2$-dimensional homogeneous spaces and seven distinct $3$-dimensional homogeneous spaces, etc. Obviously, these induce representations of the Lie algebra of vector fields on the line in dimensions $2$ and $3$. A similar statement can be made for the diffeomorphisms of the circle.

For example, if $M$ is a $1$-dimensional manifold, then Diff($M$) acts transitively on $T^\bullet M$ (the punctured tangent bundle of $M$), the space $A(M)$ (the $0$-jets of affine connections on $M$), and the space $P(M)$ (the $0$-jets of projective connections on $M$). (Of course, these are all bundles over $M$.)

Added information: If you take a (possibly periodic) coordinate $x$ on $M$, the vector fields are in one-to-one correspondence with functions, say $V_f = f(x)\partial_x$. Then one has the corresponding homomorphisms $\phi_i$ from the vector fields on $M$ to vector fields in two dimensions of the form

1. $\phi_1(V_f) = f(x)\partial_x - \bigl(f'(x) y\bigr)\partial_y$. (Take $y\not=0$.)
2. $\phi_2(V_f) = f(x)\partial_x - \bigl(f'(x) y + f''(x)\bigr)\partial_y$.
3. $\phi_3(V_f) = f(x)\partial_x - 2\bigl(f'(x) y + f'''(x)\bigr)\partial_y$.

The $3$-dimensional homogeneous spaces are a little harder to describe. Obvious examples are the spaces of $1$-jets of sections of the above bundles, but these are only three of the seven possibilities.

In high enough dimension (I think Cartan says that it starts in dimension $n=5$), it turns out that there are continuous families of inequivalent $n$-dimensional homogeneous spaces of Diff($M$).