Representations of infinite dimensional Lie algebras as vector fields on manifolds

2012-05-22T13:15:55Z

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 by Robert Bryant for Representations of infinite dimensional Lie algebras as vector fields on manifolds

2012-05-22T15:05:18Z

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

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

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($M$).

Representations of infinite dimensional Lie algebras as vector fields on manifolds - MathOverflow

Representations of infinite dimensional Lie algebras as vector fields on manifolds

Answer by Robert Bryant for Representations of infinite dimensional Lie algebras as vector fields on manifolds