Lecture 2 Groups and Homogeneous Spaces

Group theory and representation theory are themselves two enormous and interesting subjects. However, they are auxiliary in our presentation and we are forced to restrict our consideration to a brief overview.

Besides introduction to that areas presented in [254, 333] we recommend additionally the books [159, 321]. The representation theory intensively uses tools of functional analysis and on the other hand inspires its future development. We use the book [164] for references on functional analysis here and recommend it as a nice reading too.

2.1 Groups and Transformations

We start from the definition of the central object, which formalises the universal notion of symmetries [159]*§ 2.1.

It is worth (and often done) to push abstraction one level higher and to keep the group alone without the underlying space:

If we forget the nature of the elements of a transformation group G as transformations of a set X then we need to supply a separate “multiplication table” for elements of G. By the previous Exercise both concepts are mathematically equivalent. However, an advantage of a transition to abstract groups is that the same abstract group can act by transformations of apparently different sets.

It is much simpler to study groups with the following additional property.

However, most of the interesting and important groups are non-commutative.

Groups may have some additional analytical structures, e.g. they can be a topological space with a corresponding notion of limit and respective continuity. We also assume that our topological groups are always locally compact [159]*§ 2.4, that is there exists a compact neighbourhood of every point. It is common to assume that the topological and group structures are in agreement:

An even better structure can be found among Lie groups [159]*§ 6, e.g. groups with a differentiable law of multiplication. In the investigation of such groups, we could employ the whole arsenal of analytical tools. Hereafter, most of the groups studied will be Lie groups.

The above three groups are behind many important results of real and complex analysis [138, 140, 185, 240] and we meet them many times later.

2.2 Subgroups and Homogeneous Spaces

A study of any mathematical object is facilitated by a decomposition into smaller or simpler blocks. In the case of groups, we need the following:

While abstract groups are a suitable language for investigation of their general properties, we meet groups in applications as transformation groups acting on a set X. We will describe the connections between those two viewpoints. It can be approached either by having a homogeneous space build the class of isotropy subgroups or by having a subgroup define respective homogeneous spaces. The next two subsections explore both directions in detail.

2.2.1 From a Homogeneous Space to the Isotropy Subgroup

Let X be a set and let us define, for a group G, an operation G: X→ X of G on X. We say that a subset S⊂ X is G-invariant if g· s∈ S for all g∈ G and s∈ S.

Thus, if X has a non-trivial invariant subset, we can split X into disjoint parts. The finest such decomposition is obtained from the following equivalence relation on X, say, x₁∼ x₂, if and only if there exists g∈ G such that gx₁=x₂, with respect to which X is a disjoint union of distinct orbits [239]*§ I.5, that is subsets of all gx₀ with a fixed x₀∈ X and arbitrary g∈ G.

Thus, from now on, without loss of generality, we assume that the action of G on X is transitive, i.e. for every x∈ X we have

In this case, X is G-homogeneous space.

This provides a transition from a G-action on a homogeneous space X to a subgroup of G, or even to an equivalence class of such subgroups under conjugation.

2.2.2 From a Subgroup to the Homogeneous Space

We can also go in the opposite direction—given a subgroup of G, find the corresponding homogeneous space. Let G be a group and H be its subgroup. Let us define the space of cosets X=G/H by the equivalence relation: g₁∼ g₂ if there exists h ∈ H such that g₁=g₂h.

There is an important type of subgroups:

The special role of normal subgroups is explained by the following property:

In our studies normal subgroup will not appear and the set X=G / H will not be a group. However, for any subgroup H⊂ G the set X=G / H is a homogeneous space under the left G-action g: g₁H↦ (gg₁)H. For practical purposes it is more convenient to have a parametrisation of X and express the above G-action through those parameters, as shown below.

We define a function (section) [159]*§ 13.2 s: X→ G such that it is a right inverse to the natural projection p: G→ G/H, i.e. p(s(x))=x for all x∈ X. Depending on situation some additional properties of s may be required, e.g. continuity. In our work we will usually need only that the section s is a measurable function.

Then, any g∈ G has a unique decomposition of the form g=s(x)h, where x=p(g)∈ X and h∈ H. We define a map r associated to s through the identities:

Thus, starting from a subgroup H of a group G, we can define a G-homogeneous space X=G/H.

2.3 Differentiation on Lie Groups and Lie Algebras

To do some analysis on groups, we need suitably-defined basic operations: differentiation and integration.

Differentiation is naturally defined for Lie groups. If G is a Lie group and G_x is its closed subgroup, then the homogeneous space G/G_x considered above is a smooth manifold (and a loop as an algebraic object) for every x∈ X [159]*Thm. 2 in § 6.1. Therefore, the one-to-one mapping G/G_x → X from § 2.2.2 induces a structure of C^∞-manifold on X. Thus, the class C₀^∞(X) of smooth functions with compact supports on X has the natural definition.

For every Lie group G there is an associated Lie algebra g. This algebra can be realised in many different ways. We will use the following two out of four listed in [159]*§ 6.3.

2.3.1 One-parameter Subgroups and Lie Algebras

For the first realisation, we consider a one-dimensional continuous subgroup x(t) of G as a group homomorphism of x: (ℝ,+)→ G. For such a homomorphism x, we have x(s+t)=x(s)x(t) and x(0)=e.

The one-parameter subgroup x(t) defines a tangent vector X=x′(0) belonging to the tangent space T_e of G at e=x(0). The Lie algebra g can be identified with this tangent space. The important exponential map exp: g → G works in the opposite direction and is defined by expX=x(1) in the previous notations. For the case of a matrix group, the exponent map can be explicitly realised through the exponentiation of the matrix representing a tangent vector:

2.3.2 Invariant Vector Fields and Lie Algebras

In the second realisation of the Lie algebra, g is identified with the left (right) invariant vector fields on the group G, that is, first-order differential operators X defined at every point of G and invariant under the left (right) shifts: XΛ = Λ X (XR=RX). This realisation is particularly usable for a Lie group with an appropriate parametrisation. The following examples describe different techniques for finding such invariant fields.

2.3.3 Commutator in Lie Algebras

The important operation in a Lie algebra is a commutator. If the Lie algebra of a matrix group is realised by matrices, e.g. Exercise 24, then the commutator is defined by the expression [A,B]=AB−BA in terms of the respective matrix operations. If the Lie algebra is realised through left (right) invariant first-order differential operators, then the commutator [A,B]=AB−BA again defines a left (right) invariant first-order operator—an element of the same Lie algebra.

Among the important properties of the commutator are its anti-com­mu­ta­ti­vi­ty ([A,B]=−[B,A]) and the Jacobi identity

The procedure from Example 27 can also be used to calculate the derived action of a G-action on a homogeneous space.

2.4 Integration on Groups

In order to perform an integration we need a suitable measure. A measure dµ on X is called (left) invariant measure with respect to an operation of G on X if

Left invariant measures on X=G is called the (left) Haar measure. It always exists and is uniquely defined up to a scalar multiplier [321]*§ 0.2. An equivalent formulation of (14) is: G operates on L₂(X,dµ) by unitary operators. We will transfer the Haar measure dµ from G to g via the exponential map exp: g→ G and will call it as the invariant measure on a Lie algebra g.

In this notes we assume all integrations on groups performed over the Haar measures.

The above simple result has surprisingly important consequences for representation theory of compact groups.

The following Lemma characterizes linear subspaces of L₁(G,dµ) invariant under shifts in the term of ideals of convolution algebra L₁(G,dµ) and is of the separate interest.

Proof. Of course we consider only the “right-invariance and right-convolution” case. Then the other three cases are analogous. Let H be a closed linear subspace of L₁(G,dµ) invariant under right shifts and k(g)∈ H. We will show the inclusion

for any f∈L₁(G,dµ). Indeed, we can treat integral (15) as a limit of sums

But the last sum is simply a linear combination of vectors k(hg_j)∈ H (by the invariance of H) with coefficients f(g_j). Therefore sum (16) belongs to H and this is true for integral (15) by the closeness of H.

Otherwise, let H be a right ideal in the group convolution algebra L₁(G,dµ) and let φ_j(g)∈L₁(G,dµ) be an approximate unit of the algebra [84]*§ 13.2, i.e. for any f∈L₁(G,dµ) we have

Then for k(g)∈ H and for any h′∈ G the right convolution

from the first expression is tensing to k(hh′) and from the second one belongs to H (as a right ideal). Again the closeness of H implies k(hh′)∈ H that proves the assertion.