duality in mathematics

0.1 Duality in mathematics

The following is a mathematical topic entry on different types of duality encountered in different areas of mathematics; accordingly there is a string of distinct definitions associated with this topic rather than a single, general definition, although some of the linked definitions, that is, categorical duality, are more general than others.

0.1.1 Duality definitions in mathematics:

  1. 1.

    Categorical duality and Dual category (http://planetmath.org/IndexOfCategoryTheory): reversing arrows

  2. 2.

    Duality principleMathworldPlanetmath (http://planetmath.org/DualityPrinciple)

  3. 3.

    Double duality

  4. 4.


  5. 5.


  6. 6.

    Duality functorsMathworldPlanetmath, (for example the duality functor Homk(--,k) )

  7. 7.

    Poincaré duality/Poincaré isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/PoincareDuality)

  8. 8.

    Poincaré-Lefschetz duality, and Alexander-Lefschetz duality

  9. 9.

    Alexander duality: J. W. Alexander’s duality theory (cca. 1915)

  10. 10.

    Serre duality : example- in the proof of the Riemann-Roch theorem for curves (http://planetmath.org/ProofOfRiemannRochTheorem).

  11. 11.

    Dualities in logic, example: De Morgan dual (http://planetmath.org/IdealInvertingInPruferRing), Boolean algebraMathworldPlanetmath

  12. 12.

    Stone duality: Boolean algebras and Stone spaces

  13. 13.

    Dual numbers- as in an associative algebra; (almost synonymous with double)

  14. 14.

    Geometric dualities: dual polyhedron, dual of a planar graph, duality in order theory, the Legendre transformation -an application of the duality between points and lines; generalized Legendre, that is, the Legendre-Fenchel transformation.

  15. 15.

    Hamilton–Lagrange duality in theoretical mechanics and optics

  16. 16.

    Dual spacePlanetmathPlanetmath (http://planetmath.org/DualSpace)

  17. 17.

    Dual space example (http://planetmath.org/DoubleDualEmbedding)

  18. 18.

    Dual homomorphisms (http://planetmath.org/DualHomomorphism)

  19. 19.

    Duality of Projective Geometry (http://planetmath.org/Polarity2)

  20. 20.

    Analytic dualities

  21. 21.

    Duals of an algebraPlanetmathPlanetmath/algebraic duality (http://planetmath.org/DualOfACoalgebraIsAnAlgebra), for example, dual pairs of Hopf *-algebras and duality of cross productsMathworldPlanetmath of C*-algebras

  22. 22.

    Tangled, or Mirror, duality (http://planetmath.org/GrassmanHopfAlgebrasAndTheirDualCoAlgebras): interchanging morphismsMathworldPlanetmath and objects

  23. 23.

    Duality as a homological mirror symmetry

  24. 24.
  25. 25.

    Hodge dual

  26. 26.

    Duality of locally compact groups (http://planetmath.org/CompactQuantumGroup)

  27. 27.

    Pontryagin dualityMathworldPlanetmath (http://planetmath.org/PontryaginDuality), for locally compact commutativePlanetmathPlanetmathPlanetmath topological groupsMathworldPlanetmath and their linear representations

  28. 28.

    Tannaka-Krein duality (http://planetmath.org/CompactQuantumGroup): for compact matrix pseudogroups and non-commutative topological groups; its generalizationPlanetmathPlanetmath leads to quantum groupsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath in Quantum theoriesPlanetmathPlanetmath; Tannaka’s theorem provides the means to reconstruct a compact group G from its category of representations Π(G); Krein’s theorem shows which categories arise as a dual object to a compact group; the finite-dimensional representations of Drinfel’d ’s quantum groups form a braided monoidal category, whereas Π(G) is a symmetric monoidal category.

  29. 29.

    Tannaka duality: an extensionPlanetmathPlanetmathPlanetmath of Tannakian duality by Alexander Grothendieck (http://planetmath.org/AlexanderGrothendieckABiographyOf) to algebraic groups and Tannakian categories.

  30. 30.

    Contravariant dualities

  31. 31.

    Weak duality, example : weak duality theorem in linear programming (http://planetmath.org/LinearProgrammingProblem); dual problems in optimization theory

  32. 32.

    Dual codes

  33. 33.

    Duality in Electrical Engineering

0.1.2 Examples of duals:

  1. 1.

    a category 𝒞 and its dual 𝒞op

  2. 2.

    the category of Hopf algebrasPlanetmathPlanetmathPlanetmath over a field is (equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to) the opposite category of affine group schemes over speck

  3. 3.

    Dual Abelian variety

  4. 4.

    Example of a dual space theorem (http://planetmath.org/DualSpaceSeparatesPoints)

  5. 5.

    Example of Pontryagin duality (http://planetmath.org/DualGroupOfGIsHomeomorphicToTheCharacterSpaceOfL1G)

  6. 6.

    initial and final object

  7. 7.

    kernel and cokernelPlanetmathPlanetmath

  8. 8.

    limit and colimitMathworldPlanetmath

  9. 9.


  • 1 S. Doplicher and J. Roberts. A new duality theory for compact groups. Inventiones Mathematicae, 98:157–218, 1989.
  • 2 André Joyal and Ross Street, An introduction to Tannaka duality and quantum groups, in Part II of Category TheoryMathworldPlanetmathPlanetmathPlanetmathPlanetmath, Proceedings, Como 1990, eds. A. Carboni, M. C. Pedicchio and G. Rosolini, Lectures Notes in Mathematics No.1488, Springer, Berlin, 1991, 411-492.
Title duality in mathematics
Canonical name DualityInMathematics
Date of creation 2013-03-22 18:24:50
Last modified on 2013-03-22 18:24:50
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 51
Author bci1 (20947)
Entry type Topic
Classification msc 51A10
Classification msc 14F25
Classification msc 55M05
Classification msc 18-00
Synonym categorical duality
Synonym Poincaré duality
Synonym polarity
Related topic IndexOfCategoryTheory
Related topic SerreDuality
Related topic StoneSpace
Related topic CompactQuantumGroup
Related topic PoincareDuality
Related topic Polarity2
Related topic DualOfACoalgebraIsAnAlgebra
Related topic GrassmanHopfAlgebrasAndTheirDualCoAlgebras
Related topic PontryaginDuality
Related topic LinearProgrammingProblem
Related topic IdealInvertingInPruferRing
Related topic IndexOfCategories