Representation theory of diffeomorphism groups
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a source for the
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
s of a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
''M'' is the initial observation that (for ''M'' connected) that group acts transitively on ''M''.


History

A survey paper from 1975 of the subject by
Anatoly Vershik Anatoly Moiseevich Vershik (russian: Анато́лий Моисе́евич Ве́ршик; born on 28 December 1933 in Leningrad) is a Soviet and Russian mathematician. He is most famous for his joint work with Sergei V. Kerov on representati ...
,
Israel Gelfand Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гел ...
and M. I. Graev attributes the original interest in the topic to research in
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
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 In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
''C''(''M'')·Diff(''M'').


Constructions

Let therefore ''M'' be a ''n''-dimensional
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, and ''x'' be any point on it. Let Diff(''M'') be the orientation-preserving
diffeomorphism group In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
of ''M'' (only the
identity component In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity componen ...
of mappings
homotopic In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to the identity diffeomorphism if you wish) and Diff''x''1(''M'') the stabilizer of ''x''. Then, ''M'' is identified as a
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
:Diff(''M'')/Diff''x''1(''M''). From the algebraic point of view instead, C^\infty(M) is the
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
of
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s over ''M'' and I_x(M) is the
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
of smooth functions vanishing at ''x''. Let I_x^n(M) be the ideal of smooth functions which vanish up to the n-1th
partial derivative In 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 (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
at ''x''. I_x^n(M) is invariant under the group Diff''x''1(''M'') of diffeomorphisms fixing x. For ''n'' > 0 the group Diff''x''''n''(''M'') is defined as the
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of Diff''x''1(''M'') which acts as the identity on I_x(M)/I_x^n(M). So, we have a descending chain :Diff(''M'') ⊃ Diff''x''1(M) ⊃ ... ⊃ Diff''x''''n''(''M'') ⊃ ... Here Diff''x''''n''(''M'') is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of Diff''x''1(''M''), which means we can look at the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
:Diff''x''1(''M'')/Diff''x''''n''(''M''). Using
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
, a real- or complex-valued function (with some sufficiently nice topological properties) on the diffeomorphism group can be
decompose Decomposition or rot is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is ...
d into Diff''x''1(''M'') representation-valued functions over ''M''.


The supply of representations

So what are the representations of Diff''x''1(''M'')? Let's use the fact that if we have a
group homomorphism In mathematics, given 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) \cdot h(v) wh ...
φ:''G'' → ''H'', then if we have a ''H''-representation, we can obtain a restricted ''G''-representation. So, if we have a rep of :Diff''x''1(''M'')/Diff''x''''n''(''M''), we can obtain a rep of Diff''x''1(''M''). Let's look at :Diff''x''1(''M'')/Diff''x''2(''M'') first. This is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
GL+(''n'', R) (and because we're only considering orientation preserving diffeomorphisms and so the determinant is positive). What are the reps of GL+(''n'', R)? :GL^+(n,\mathbb)\cong \mathbb^+\times SL(n,\mathbb{R}). We know the reps of SL(''n'', R) are simply
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
s over ''n'' dimensions. How about the R+ part? That corresponds to the ''density'', or in other words, how the tensor rescales under the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the Jacobian of the diffeomorphism at ''x''. (Think of it as the
conformal weight A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...
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''x''1(''M'')/Diff''x''''n''(''M''). This is a finite-dimensional group. We have the chain :Diff''x''1(''M'')/Diff''x''1(''M'') ⊂ ... ⊂ Diff''x''1(''M'')/Diff''x''''n''(''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''x''1(''M'')/Diff''x''''m''(''M'') can automatically be turned into a rep of :Diff''x''1/Diff''x''''n''(''M'') if ''n'' > ''m''. Let's say we have a rep of :Diff''x''1/Diff''x''''p'' + 2 which doesn't arise from a rep of :Diff''x''1/Diff''x''''p'' + 1. Then, we call the
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
with that rep as the
fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
(i.e. Diff''x''1/Diff''x''''p'' + 2 is the structure group) a
jet bundle In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Je ...
of order ''p''. Side remark: This is really the method of
induced representation In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" represent ...
s with the smaller group being Diffx1(M) and the larger group being Diff(''M'').


Intertwining structure

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. We can study the structure of these reps through the study of the
intertwiner In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
s between them. If the fiber is not an irreducible representation of Diff''x''1(''M''), then we can have a nonzero intertwiner mapping each fiber pointwise into a smaller quotient representation. Also, the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
is an intertwiner from the space of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s to another of higher order. (Other derivatives are not, because connections aren't invariant under diffeomorphisms, though they are covariant.) The
partial derivative In 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 (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
isn't diffeomorphism invariant. There is a derivative intertwiner taking sections of a jet bundle of order ''p'' into sections of a jet bundle of order ''p'' + 1. Diffeomorphisms Representation theory of groups