In mathematics, a source for the representation theory of the group of diffeomorphisms of a smooth manifold M is the initial observation that (for M connected) that group acts transitively on M. A survey paper from 1975 of the subject by Anatoly Vershik, Israel Gelfand and M. I. Graev[1] attributes the original interest in the topic to research in theoretical physics of the local current algebra, in the preceding years.
be the ideal of smooth functions which vanish up to the n-1th partial derivative at x.
For n > 0 the group Diffxn(M) is defined as the subgroup of Diffx1(M) which acts as the identity on
So, we have a descending chain Here Diffxn(M) is a normal subgroup of Diffx1(M), which means we can look at the quotient group Using harmonic analysis, a real- or complex-valued function (with some sufficiently nice topological properties) on the diffeomorphism group can be decomposed into Diffx1(M) representation-valued functions over M. So what are the representations of Diffx1(M)?
Let's use the fact that if we have a group homomorphism φ:G → H, then if we have a H-representation, we can obtain a restricted G-representation.
This is isomorphic to the general linear group GL+(n, R) (and because we're only considering orientation preserving diffeomorphisms and so the determinant is positive).
That corresponds to the density, or in other words, how the tensor rescales under the determinant of the Jacobian of the diffeomorphism at x.
So, we have just discovered the tensor reps (with density) of the diffeomorphism group.
We have the chain Here, the "⊂" signs should really be read to mean an injective homomorphism, but since it is canonical, we can pretend these quotient groups are embedded one within the other.
In general, the space of sections of the tensor and jet bundles would be an irreducible representation and we often look at a subrepresentation of them.
Also, the exterior derivative is an intertwiner from the space of differential forms to another of higher order.
(Other derivatives are not, because connections aren't invariant under diffeomorphisms, though they are covariant.)