Pontryagin duality

In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. It places in a unified context a number of observations about functions on the real line or on finite abelian groups:

Categorical considerations

It is useful to regard the dual group functorially. In what follows, LCA is the category of locally compact abelian groups and continuous group homomorphisms.

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The dual group construction of G^ is a contravariant functor LCA → LCA. In particular, the iterated functor

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G → (G^)^ is covariant.

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Theorem. The dual group is a category isomorphism from LCA to LCAop.

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Theorem. The iterated dual functor is naturally isomorphic to the identity functor on LCA.

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This isomorphism is comparable to the double dual of finite-dimensional vector spaces (a special case, for real and complex vector spaces).

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The duality interchanges the subcategories of discrete groups and compact groups. If R is a ring and G is a left R-module, the dual group G^ will become a right R-module; in this way we can also see that discrete left R-modules will be Pontryagin dual to compact right R-modules. The ring End(G) of endomorphisms in LCA is changed by duality into its opposite ring (change the multiplication to the other order). For example if G is an infinite cyclic discrete group, G^ is a circle group: the former has End(G) = Z so this is true also of the latter.

Related Topics:
Compact - Ring - Module - Endomorphism - Opposite

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