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单词 DualityInMathematics
释义

duality in mathematics


0.1 Duality in mathematics

The following is a mathematical topic entry on differenttypes of duality encountered in different areas of mathematics; accordingly there isa 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.

    Triality

  5. 5.

    Self-duality

  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.

    CohomologyPlanetmathPlanetmath theory duals: de Rham cohomologyMathworldPlanetmath Alexander-Spanier cohomology

  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 quantumgroups form a braided monoidal category, whereas Π(G) is a symmetric monoidal category.

  29. 29.

    Tannaka duality: an extensionPlanetmathPlanetmathPlanetmath of Tannakian duality byAlexander Grothendieck (http://planetmath.org/AlexanderGrothendieckABiographyOf) to algebraic groups andTannakian 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 overspeck

  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.

    direct sumMathworldPlanetmathPlanetmathPlanetmath and productPlanetmathPlanetmathPlanetmath

References

  • 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.
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