Home > Representation theory of diffeomorphism groups

A survey paper from 1975 of the subject by A. M. Vershik , I. M. Gel'fand and M. I. Graev attributes the original interest in the topic to research in theoretical physics of the local current algebra , in the preceding years. Research on the finite configuration representations was in papers of R. S. Ismagilov (1971), and A. A. Kirillov (1974). The representations of interest in physics are described as a cross product C^∞(M)·Diff(M).

Let therefore M be a n-dimensional connected differentiable manifold, and x be any point on it. Let Diff(M) be the (orientation preserving) diffeomorphism group of M (only the connected part homotopic to the identity diffeomorphism if you wish) and Diff_x¹(M) the stabilizer of x. Then, M is identified as a homogeneous space

Diff(M)/Diff_x¹(M).

From the algebraic point of view instead, is the algebra of smooth functions over M and is the ideal of smooth functions vanishing at x. Let be the ideal of smooth functions which vanish up to the n-1^th partial derivativeIn mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. They are useful in n-dimensional calculus and differential geometry. The partial derivative of at x. This is a diffeomorphic invariant definition. Diff_xⁿ(M) would then be the subgroupGroup theory In mathematics, given a group G under a binary operation , we say that some subset H of G is a subgroup of G if H also forms a group under the operation . More precisely, H is a subgroup of G if the restriction of to H is a group operation on of Diff_x¹(M) which maps to itself. So, we have a chain

Diff(M)⊃Diff_x¹(M)⊃...⊃Diff_xn(M)⊃...

Here Diff_xⁿ(M) is a normal subgroupIn mathematics, a normal subgroup ''N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G the element g-1ng is still in N''. The statement N is a normal subgroup of G is written: :. Another way to put t of Diff_x¹(M) which means we can look at the quotient groupIn mathematics, given a group G and a normal subgroup N of G the quotient group or factor group of G over N is a group that intuitively "collapses" the normal subgroup N to the identity element. The quotient group is written G ''N and is usually spoken in

Diff_x¹(M)/Diff_xⁿ(M).

Using harmonic analysisHarmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. The basic waves are called ", a real- or complex-valued function (with some sufficiently nice topological properties) on the diffeomorphism group can be decomposed into Diff_x¹(M) representation valued functions over M.

So what are the reps of Diff_x¹(M)? Let's use the fact that if we have a group homomorphismGiven two groups G ) and H ·), a group homomorphism from G ) to H ·) is a function h : G H such that for all u and v in G it holds that : h ''u v h ''u · h ''v From this property, one can deduce that h maps the identity element e of G to the identity elem φ:G→H, then if we have a H-representation, we can obtain a restricted G-representatio;. So, if we have a rep of

Diff_x¹(M)/Diff_xⁿ(M),

we can obtain a rep of Diff_x¹(M).

Let's look at

Diff_x¹(M)/Diff_x²(M)

first. This is isomorphic to the general linear group GL⁺(n,R) (plus because we're only considering orientation preserving diffeomorphisms and so the determinant is positive). What are the reps of GL⁺(n,R)? Well,

.

We know the reps of SL(n,R) are simply tensors over n dimensions. How about the R⁺ part? That corresponds to the density, or in other words, how the tensor rescales under the determinant of the Jacobian of the diffeomorphism at x. (Think of it as the conformal weight if you will, except that there is no conformal structure here). (Incidentally, there is nothing preventing us from having a complex density).

So, we have just discovered the tensor reps (with density) of the diffeomorphism group.

Let's look at

Diff_x¹(M)/Diff_xⁿ(M)

now. This is a finite dimensional group. We have the chain

Diff_x¹(M)/Diff_x¹(M)⊂...⊂Diff_x¹(M)/Diff_xⁿ(M)⊂...

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.

Any rep of

Diff_x¹(M)/Diff_x^m(M)

can automatically be turned into a rep of

Diff_x¹/Diff_xⁿ(M)

if n > m. Let's say we have a rep of

Diff_x¹/Diff_x^p+2

which doesn't arise from a rep of

Diff_x¹/Diff_x^p+1.

Then, we call the fiber bundle with that rep as the fiber (i.e. Diff_x¹/Diff_x^p+2 is the structure group) a jet bundle of order p.

Side remark: This is really the method of induced representations with the smaller group being Diff_x¹(M) and the larger group being Diff(M).

In general, the space of sections of the tensor/jet bundles would be an irreducible rep and we often look at a subrep of them. We can study the structure of these reps through the study of the intertwiners between them. If the fiber is not an irreducible rep of Diff_x¹(M), then we can have a nonzero intertwiner mapping each fiber pointwise into a smaller quotient representation. Also, the exterior derivative (but not other derivatives because connections aren't invariant under diffeomorphisms ( covariant, yes, but invariant, no)) is an intertwiner from the space of differential forms to another of higher order. The partial derivative isn't diffeomorphism invariant. However, there is a derivative intertwiner taking sections of a jet bundle of order p into sections of a jet bundle of order p+1.