nLab Pontrjagin dual

Definition

Definition

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

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

is the abelian group of all these group characters of $A$, equipped with pointwise multiplication (that is multiplication induced by multiplication in the circle group, the multiplication of norm-$1$ complex numbers in $S^1\subset\mathbb{C}$) and with the topology of uniform convergence on each compact $K\subset A$ (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 $S^1$, and conversely, $\mathbb{Z}$ is the Pontrjagin dual of $S^1$. This pairing of dual topological groups, given by $(n,z) \mapsto z^n$, is related to the subject of Fourier transforms.

Example

More generally, the Pontrjagin dual of $\mathbb{Z}^n$ is the n-torus $(S^1)^n$. In solid state physics this example appears in the guise of the Brillouin torus.

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: $\widehat{\mathbb{Z}/n} \,\simeq\, \mathbb{Z}/n$.

Proposition

(Pontrjagin dual of compact group as second group cohomology group)
If $G$ 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:

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

Proof

The key point is that, by assumption on $G$, we have

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

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

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

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

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

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 $\mathbf{H}(B G,-) \coloneqq Maps(B G;\, -)$ (of $\mathbf{H} =$ ∞Grpd):

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

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

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

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

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 $G$ a compact Lie group the analogous argument applies to the delooping/quotient stacks $\mathbf{B}G$ in $\mathbf{H} =$Smooth∞Grpd.)

Example

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

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

(in any connected component of a “stable map” $G \to PU(\mathcal{H})$, that is) and hence is the Pontrjagin dual group (Def. ) when $G$ 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 $(x,p) \mapsto \mathrm{e}^{\mathrm{i} x p}$. More generally, $\widehat{\mathbb{R}^n} \,=\, \mathbb{R}^n$.

Properties

Pontrjagin duality theorem

Theorem

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

Thus, the functor

$$LocCompAb^{op} \to LocCompAb: G \to \widehat{G}$$

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

$$A \to \widehat{\widehat{A}}: g \mapsto f_g$$

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

$$Ab^{op} \to CompAb$$

$$CompAb^{op} \to Ab$$

where Ab is the category of (discrete topological) groups and $CompAb$ is the category of abelian Hausdorff compact topological groups, each embedded in $LocCompAb$ 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:

• If $A$ is finite, then $\widehat{A}$ is finite.

• If $A$ is compact, then $\widehat{A}$ is discrete.

  (see also at nearby homomorphisms from compact Lie groups are conjugate)

• If $A$ is discrete, then $\widehat{A}$ is compact.

• If $A$ is torsion-free and discrete, then $\widehat{A}$ is connected and compact.

• If $A$ is an abelian torsion group, then $\widehat A$ is an abelian profinite group (for more see at Pontryagin duality for torsion abelian groups)

• If $A$ is connected and compact, then $\widehat{A}$ is torsion-free and discrete.

• If $A$ is a Lie group, then $\widehat{A}$ is compactly generated in that there is a compact neighborhood of the neutral element that generates $\widehat{A}$ as a group.

• If $A$ is compactly generated, then $\widehat{A}$ is a Lie group.

• If $A$ is second countable, then $\widehat{A}$ is second countable.

• If $A$ is separable, then $\widehat{A}$ is metrizable.

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

• Variations on Pontryagin duality, (nCafe)

Another statement of this type is presented in Mackey 1948:

• $A$ is connected if and only if $\widehat{A}$ is a product of a discrete torsion-free group with a finite dimensional real vector space.

Applications

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

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

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

Related concepts

• Pontryagin duality for torsion abelian groups

• locally compact topological group

• Fourier transform

• character group

• dual object

• Cartier duality, Poincaré line bundle

References

General

The original article:

• Lev Pontrjagin, Theory of topological commutative groups, Annals of Mathematics Second Series, Vol. 35, No. 2 (Apr., 1934), pp. 361-388 (doi:10.2307/1968438)

  Russian translation: Uspekhi Mat. Nauk, 1936, no. 2, 177–195 (mathnet:umn8882)

Gentle exposition:

• Partha Sarathi Chakraborty, Pontrjagin duality for finite groups (pdf)

Textbook accounts:

• Sidney A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Soc. Lecture Notes 29, Cambridge U. Press, 1977 (doi:10.1017/CBO9780511600722)

• David L. Armacost, The Structure of Locally Compact Abelian Groups, Dekker, New York, 1981.

See also

• Wikipedia, Pontryagin duality

Also:

• George Mackey, The Laplace Transform For Locally Compact Abelian Groups, Proceedings of the National Academy of Sciences of the United States of America Vol. 34, No. 4 (Apr. 15, 1948), pp. 156-162 (jstor:87968)

• Michael Barr, On duality of topological abelian groups (pdf, pdf)

  which provides a perhaps better context for Pontryagin duality than the category of locally compact Hausdorff abelian groups (also known as ‘LCA groups’). Barr explains:

  Did you know that there is a *-autonomous category of topological abelian groups that includes all the LCA groups and whose duality extends that of Pontrjagin? The groups are characterized by the property that among all topological groups on the same underlying abelian group and with the same set of continuous homomorphisms to the circle, these have the finest topology. It is not obvious that such a finest exists, but it does and that is the key.

For higher groups

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

• Ulrich Bunke, Thomas Schick, Markus Spitzweck, Andreas Thom, Duality for topological abelian group stacks and T-duality, Proceedings of the ICM 2006 satellite conference, Valladolid, Spain, 2006. Zürich: EMS. Series of Congress Reports, 227-347 (2008) (arXiv:math/0701428)

• Calder Daenzer, A groupoid approach to noncommutative T-duality, PhD, 2007 (pdf)