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.
Categorical duality and Dual category (http://planetmath.org/IndexOfCategoryTheory): reversing arrows
2.
Duality principle (http://planetmath.org/DualityPrinciple)
3.
Double duality
4.
Triality
5.
Self-duality
6.
Duality functors, (for example the duality functor $Hom_{k}(--,k)$ )
7.
Poincaré duality/Poincaré isomorphism (http://planetmath.org/PoincareDuality)
8.
Poincaré-Lefschetz duality, and Alexander-Lefschetz duality
9.
Alexander duality: J. W. Alexander’s duality theory (cca. 1915)
10.
Serre duality :example- in the proof of the Riemann-Roch theorem for curves (http://planetmath.org/ProofOfRiemannRochTheorem).
11.
Dualities in logic, example: De Morgan dual (http://planetmath.org/IdealInvertingInPruferRing), Boolean algebra
12.
Stone duality: Boolean algebras and Stone spaces
13.
Dual numbers- as in an associative algebra; (almost synonymous with double)
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.
Hamilton–Lagrange duality in theoretical mechanics and optics
16.
Dual space (http://planetmath.org/DualSpace)
17.
Dual space example (http://planetmath.org/DoubleDualEmbedding)
18.
Dual homomorphisms (http://planetmath.org/DualHomomorphism)
19.
Duality of Projective Geometry (http://planetmath.org/Polarity2)
20.
Analytic dualities
21.
Duals of an algebra/algebraic duality (http://planetmath.org/DualOfACoalgebraIsAnAlgebra),for example, dual pairs of Hopf *-algebras and duality of cross products of C*-algebras
22.
Tangled, or Mirror, duality (http://planetmath.org/GrassmanHopfAlgebrasAndTheirDualCoAlgebras):interchanging morphisms and objects
23.
Duality as a homological mirror symmetry
24.
Cohomology theory duals: de Rham cohomology $\\leftarrow\\rightarrow$ Alexander-Spanier cohomology
25.
Hodge dual
26.
Duality of locally compact groups (http://planetmath.org/CompactQuantumGroup)
27.
Pontryagin duality (http://planetmath.org/PontryaginDuality), for locally compact commutative topological groups and their linear representations
28.
Tannaka-Krein duality (http://planetmath.org/CompactQuantumGroup): for compact matrix pseudogroups and non-commutative topological groups; its generalization leads to quantum groups in Quantum theories; Tannaka’s theorem provides the means to reconstruct a compact group $G$ from its category of representations $\\Pi(G)$ ; 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 $\\Pi(G)$ is a symmetric monoidal category.
29.
Tannaka duality: an extension of Tannakian duality byAlexander Grothendieck (http://planetmath.org/AlexanderGrothendieckABiographyOf) to algebraic groups andTannakian categories.
30.
Contravariant dualities
31.
Weak duality, example : weak duality theorem in linear programming (http://planetmath.org/LinearProgrammingProblem);dual problems in optimization theory
32.
Dual codes
33.
Duality in Electrical Engineering

0.1.2 Examples of duals:

1.
a category $\\mathcal{C}$ and its dual $\\mathcal{C}^{op}$
2.
the category of Hopf algebras over a field is (equivalent to) the opposite category of affine group schemes over $\\operatorname{spec}k$
3.
Dual Abelian variety
4.
Example of a dual space theorem (http://planetmath.org/DualSpaceSeparatesPoints)
5.
Example of Pontryagin duality (http://planetmath.org/DualGroupOfGIsHomeomorphicToTheCharacterSpaceOfL1G)
6.
initial and final object
7.
kernel and cokernel
8.
limit and colimit
9.
direct sum and product

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 Theory, Proceedings, Como 1990, eds. A. Carboni, M. C. Pedicchio and G. Rosolini, Lectures Notes in Mathematics No.1488, Springer, Berlin, 1991, 411-492.

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. Categorical duality and Dual category (http://planetmath.org/IndexOfCategoryTheory): reversing arrows 2. Duality principle (http://planetmath.org/DualityPrinciple) 3. Double duality 4. Triality 5. Self-duality 6. Duality functors, (for example the duality functor $Hom_{k}(--,k)$ ) 7. Poincaré duality/Poincaré isomorphism (http://planetmath.org/PoincareDuality) 8. Poincaré-Lefschetz duality, and Alexander-Lefschetz duality 9. Alexander duality: J. W. Alexander’s duality theory (cca. 1915) 10. Serre duality :example- in the proof of the Riemann-Roch theorem for curves (http://planetmath.org/ProofOfRiemannRochTheorem). 11. Dualities in logic, example: De Morgan dual (http://planetmath.org/IdealInvertingInPruferRing), Boolean algebra 12. Stone duality: Boolean algebras and Stone spaces 13. Dual numbers- as in an associative algebra; (almost synonymous with double) 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. Hamilton–Lagrange duality in theoretical mechanics and optics 16. Dual space (http://planetmath.org/DualSpace) 17. Dual space example (http://planetmath.org/DoubleDualEmbedding) 18. Dual homomorphisms (http://planetmath.org/DualHomomorphism) 19. Duality of Projective Geometry (http://planetmath.org/Polarity2) 20. Analytic dualities 21. Duals of an algebra/algebraic duality (http://planetmath.org/DualOfACoalgebraIsAnAlgebra),for example, dual pairs of Hopf -algebras and duality of cross products of C-algebras 22. Tangled, or Mirror, duality (http://planetmath.org/GrassmanHopfAlgebrasAndTheirDualCoAlgebras):interchanging morphisms and objects 23. Duality as a homological mirror symmetry 24. Cohomology theory duals: de Rham cohomology $\\leftarrow\\rightarrow$ Alexander-Spanier cohomology 25. Hodge dual 26. Duality of locally compact groups (http://planetmath.org/CompactQuantumGroup) 27. Pontryagin duality (http://planetmath.org/PontryaginDuality), for locally compact commutative topological groups and their linear representations 28. Tannaka-Krein duality (http://planetmath.org/CompactQuantumGroup): for compact matrix pseudogroups and non-commutative topological groups; its generalization leads to quantum groups in Quantum theories; Tannaka’s theorem provides the means to reconstruct a compact group $G$ from its category of representations $\\Pi(G)$ ; 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 $\\Pi(G)$ is a symmetric monoidal category. 29. Tannaka duality: an extension of Tannakian duality byAlexander Grothendieck (http://planetmath.org/AlexanderGrothendieckABiographyOf) to algebraic groups andTannakian categories. 30. Contravariant dualities 31. Weak duality, example : weak duality theorem in linear programming (http://planetmath.org/LinearProgrammingProblem);dual problems in optimization theory 32. Dual codes 33. Duality in Electrical Engineering 0.1.2 Examples of duals: 1. a category $\\mathcal{C}$ and its dual $\\mathcal{C}^{op}$ 2. the category of Hopf algebras over a field is (equivalent to) the opposite category of affine group schemes over $\\operatorname{spec}k$ 3. Dual Abelian variety 4. Example of a dual space theorem (http://planetmath.org/DualSpaceSeparatesPoints) 5. Example of Pontryagin duality (http://planetmath.org/DualGroupOfGIsHomeomorphicToTheCharacterSpaceOfL1G) 6. initial and final object 7. kernel and cokernel 8. limit and colimit 9. direct sum and product 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 Theory, 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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