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Pontrjagin dual

Context

Group Theory

Duality

Contents

Definition

Definition

Let AA be a commutative (Hausdorff) topological group. A (continuous) group character of AA is any continuous homomorphism χ:AS 1\chi: A\to S^1 to the circle group. The Pontrjagin dual group

A^TopGrps(A,S 1) \widehat{A} \;\coloneqq\; TopGrps \big( A ,\, S^1 \big)

is the abelian group of all these group characters of AA, equipped with pointwise multiplication (that is multiplication induced by multiplication in the circle group, the multiplication of norm-11 complex numbers in S 1S^1\subset\mathbb{C}) and with the topology of uniform convergence on each compact KAK\subset A (this is equivalent to the compact-open topology).

Examples

Example

The Pontrjagin dual of the additive group of integers \mathbb{Z} is the circle group S 1S^1, and conversely, \mathbb{Z} is the Pontrjagin dual of S 1S^1. This pairing of dual topological groups, given by (n,z)z n(n,z) \mapsto z^n, is related to the subject of Fourier transforms.

In general, the dual of a discrete group is a compact group and conversely. In particular, therefore, the dual of a finite group is again finite.

Example

The finite cyclic groups are Pontrjagin self-dual: /n^/n\widehat{\mathbb{Z}/n} \,\simeq\, \mathbb{Z}/n.

Proposition

(Pontrjagin dual of compact group as second group cohomology group)
If GG is a finite group (more generally: a compact Lie group) then its Pontrjagin dual is equivalently its cohomology group in degree-2 group cohomology (more generally: refined Lie group cohomology) with integer coefficients:

G^H grp 2(G;)=H 2(BG;). \widehat{G} \;\simeq\; H^2_{grp} \big( G ;\, \mathbb{Z} \big) \;=\; H^2 \big( B G ;\, \mathbb{Z} \big) \,.

Proof

The key point is that, by assumption on GG, we have

(1)H grp 1(G;)=0. H^{\bullet \geq 1}_{grp} \big( G;\, \mathbb{R} \big) \;=\; 0 \,.

Using this, the conclusion is obtained as follows: The defining short exact sequence of groups

S 1 \mathbb{Z} \xhookrightarrow{\;} \mathbb{R} \twoheadrightarrow S^1

extends to a homotopy fiber sequence of 2-groups (and further of n-groups)

S 1BB. \mathbb{Z} \xrightarrow{\;\;} \mathbb{R} \xrightarrow{\;\;} S^1 \xrightarrow{\;\;} B \mathbb{Z} \xrightarrow{\;\;} B \mathbb{R} \xrightarrow{\;\;} \cdots \,.

This induces a long exact sequence of cohomology groups induced from the long exact sequence of homotopy groups of the image of this fiber sequence under the derived hom-space H(BG,)Maps(BG;)\mathbf{H}(B G,-) \coloneqq Maps(B G;\, -) (of H=\mathbf{H} = ∞Grpd):

π 1(H(BG,B 2)) π 1(H(BG,B 2S 1)) π 0(H(BG,B 2)) π 0(H(BG,B 2)) = = = = H 1(BG,)=0 H 1(BG,S 1) H 2(BG,) H 2(BG,)=0 . \array{ \cdots &\to& \pi_1 \left( \mathbf{H}(B G, \, B^2 \mathbb{R}) \right) &\xrightarrow{\;\;}& \pi_1 \left( \mathbf{H}(B G, \, B^2 S^1) \right) &\xrightarrow{\;\;}& \pi_0 \left( \mathbf{H}(B G, \, B^2 \mathbb{Z}) \right) &\xrightarrow{\;\;}& \pi_0 \left( \mathbf{H}(B G, \, B^2 \mathbb{R}) \right) &\to& \cdots \\ && = && = && = && = \\ \cdots &\to& \underset{ = 0 }{ \underbrace{ H^1 \big( B G \;, \mathbb{R} \big) } } &\xrightarrow{\;\;}& H^1 \big( B G \;, S^1 \big) &\xrightarrow{\;\simeq\;}& H^2 \big( B G \;, \mathbb{Z} \big) &\xrightarrow{\;\;}& \underset{ = 0 }{ \underbrace{ H^2 \big( B G \;, \mathbb{R} \big) } } &\to& \cdots \,. }

Using the assumption (1) under the braces, this implies the middle isomorphism, as shown.

Now the claim follows by re-expressing H 1(BG;S 1)H^1(B G;\, S^1) as follows:

H 2(BG;)H 1(BG;S 1) π 0H(BG,BS 1) π 0Groupoids(G*,S 1*) π 0(Groups(G,S 1)S 1) π 0(Groups(G,S 1)×BS 1) Groups(G,S 1)G^, \begin{aligned} H^2(B G;\, \mathbb{Z}) \;\simeq\; H^1(B G;\, S^1) & \;\simeq\; \pi_0 \mathbf{H}\big( B G, \, B S^1 \big) \\ & \;\simeq\; \pi_0 Groupoids\big( G \rightrightarrows \ast, \, S^1 \rightrightarrows \ast \big) \\ & \;\simeq\; \pi_0 \Big( Groups\big(G,S^1\big) \sslash S^1 \Big) \\ & \;\simeq\; \pi_0 \Big( Groups\big(G,\,S^1\big) \times B S^1 \Big) \\ & \;\simeq\; Groups(G,S^1) \;\simeq\; \widehat G \,, \end{aligned}

where the third line expresses the functor groupoid of functors and natural transformations between delooping groupoids, while the last step uses that the circle group, being abelian, has trivial conjugation action on the hom-set of group homomorphisms. (For GG a compact Lie group the analogous argument applies to the delooping/quotient stacks BG\mathbf{B}G in H=\mathbf{H} = Smooth∞Grpd.)

Example

(equivariant fundamental group of 3-twists of equivariant K-theory)
For GG a finite group, the fundamental group π 1()\pi_1(-) of the GG-fixed locus () G(-)^G of the base space PU()\mathcal{B} PU(\mathcal{H}) of the universal equivariant PU ( ) PU(\mathbb{H}) -bundle (classifying 3-twists in twisted equivariant K-theory) is

π 1((PU()) G)Grps(G,S 1)=G^ \pi_1 \Big( \big( \mathcal{B} PU(\mathcal{H}) \big)^G \Big) \;\simeq\; Grps(G, S^1) \,=\, \widehat G

(in any connected component of a “stable map” GPU()G \to PU(\mathcal{H}), that is) and hence is the Pontrjagin dual group (Def. ) when GG is abelian.

By BEJU 2014, Thm. 1.10, see this Prop..

Example

The Pontrjagin dual ^\hat{\mathbb{R}} of the additive group of real numbers is isomorphic again to \mathbb{R} itself, with the pairing given by (x,p)e ixp(x,p) \mapsto \mathrm{e}^{\mathrm{i} x p}. More generally, n^= n\widehat{\mathbb{R}^n} \,=\, \mathbb{R}^n.

Properties

Pontrjagin duality theorem

Theorem

For every abelian Hausdorff locally compact topological group AA, the natural function AA^^A \mapsto \widehat{\widehat{A}} from AA into the Pontrjagin dual of the Pontrjagin dual of AA, assigning to every gAg\in A the continuous character f gf_g given by f g(χ)=χ(g)f_g(\chi)=\chi(g), is an isomorphism of topological groups (that is, a group isomorphism that is also a homeomorphism).

Thus, the functor

LocCompAb opLocCompAb:GG^LocCompAb^{op} \to LocCompAb: G \to \widehat{G}

is an equivalence of categories, in fact an adjoint equivalence whose unit is

AA^^:gf gA \to \widehat{\widehat{A}}: g \mapsto f_g

and whose counit (the same arrow read in the opposite category) are isomorphisms. This contravariant self-equivalence restricts to equivalences

Ab opCompAbAb^{op} \to CompAb
CompAb opAbCompAb^{op} \to Ab

where Ab is the category of (discrete topological) groups and CompAbCompAb is the category of abelian Hausdorff compact topological groups, each embedded in LocCompAbLocCompAb in the evident way.

The Fourier transform on abelian locally compact groups is formulated in terms of Pontrjagin duals (see below).

Basic properties of dual groups

There are many properties of Hausdorff abelian locally compact groups that implies properties of their Pontrjagin duals. For example:

For a discussion of these facts see Morris 77, Armacost 81, exposition in:

Another statement of this type is presented in Mackey 1948:

Applications

Pontrjagin duality underlies the abstract framework of Fourier analysis on locally compact Hausdorff abelian groups AA: by Fourier duality on AA, there is a Hilbert space isomorphism (Fourier transform)

A:L 2(A,dμ)L 2(A^,dμ^),\mathcal{F}_A: L^2(A, d\mu) \to L^2(\hat{A}, d\hat{\mu})\,,

where dμd\mu is a suitable choice of Haar measure on AA, and dμ^d\hat{\mu} is a suitable choice of Haar measure on the dual group. Fourier duality is compatible with Pontrjagin duality in the sense that if A^^\hat{\hat{A}} is identified with AA, then A^\mathcal{F}_{\hat{A}} is the inverse of A\mathcal{F}_A.

References

General

The original article:

Gentle exposition:

Textbook accounts:

See also

Also:

For higher groups

Discussion of categorified Pontrjagin duality for 2-groups and application to topological T-duality: