page_title: Pontryagin duality
section_title: Dualities for non-commutative topological groups
hierarchical_section_title: Pontryagin duality / Generalizations / Dualities for non-commutative topological groups
caption_attribution_description: English: Duality for finite groups.
context_page_description: In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as, the circle, or finite cyclic groups. The Pontryagin duality theorem itself states that locally compact abelian groups identify naturally with their bidual.
  The subject is named after Lev Semenovich Pontryagin who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the group being second-countable and either compact or discrete. This was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940.
context_section_description: For non-commutative locally compact groups  the classical Pontryagin construction stops working for various reasons, in particular, because the characters don't always separate the points of , and the irreducible representations of  are not always one-dimensional. At the same time it is not clear how to introduce multiplication on the set of irreducible unitary representations of , and it is even not clear whether this set is a good choice for the role of the dual object for . So the problem of constructing duality in this situation requires complete rethinking.
  Theories built to date are divided into two main groups: the theories where the dual object has the same nature as the source one (like in the Pontryagin duality itself), and the theories where the source object and its dual differ from each other so radically that it is impossible to count them as objects of one class.
  The second type theories were historically the first: soon after Pontryagin's work Tadao Tannaka (1938) and Mark Krein (1949) constructed a duality theory for arbitrary compact groups known now as the Tannaka–Krein duality. In this theory the dual object for a group  is not a group but a category of its representations .
  The theories of first type appeared later and the key example for them was the duality theory for finite groups. In this theory the category of finite groups is embedded by the operation  of taking group algebra  (over ) into the category of finite dimensional Hopf algebras, so that the Pontryagin duality functor  turns into the operation  of taking the dual vector space (which is a duality functor in the category of finite dimensional Hopf algebras).
  In 1973 Leonid I. Vainerman, George I. Kac, Michel Enock, and Jean-Marie Schwartz built a general theory of this type for all locally compact groups. From the 1980s the research in this area was resumed after the discovery of quantum groups, to which the constructed theories began to be actively transferred. These theories are formulated in the language of C*-algebras, or Von Neumann algebras, and one of its variants is the recent theory of locally compact quantum groups.
  One of the drawbacks of these general theories, however, is that in them the objects generalizing the concept of group are not Hopf algebras in the usual algebraic sense. This deficiency can be corrected (for some classes of groups) within the framework of duality theories constructed on the basis of the notion of envelope of topological algebra.
caption: Duality for finite groups.
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