# 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 principle (http://planetmath.org/DualityPrinciple)

3. 3.

Double duality

4. 4.

Triality

5. 5.

Self-duality

6. 6.

Duality functors, (for example the duality functor $Hom_{k}(--,k)$ )

7. 7.

Poincaré duality/Poincaré isomorphism (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 algebra

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 space (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 algebra/algebraic duality (http://planetmath.org/DualOfACoalgebraIsAnAlgebra), for example, dual pairs of Hopf *-algebras and duality of cross products of C*-algebras

22. 22.

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

23. 23.

Duality as a homological mirror symmetry

24. 24.

Cohomology theory duals: de Rham cohomology $\leftarrow\rightarrow$ Alexander-Spanier cohomology

25. 25.

Hodge dual

26. 26.

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

27. 27.

Pontryagin duality (http://planetmath.org/PontryaginDuality), for locally compact commutative topological groups and their linear representations

28. 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 quantum groups form a braided monoidal category, whereas $\Pi(G)$ is a symmetric monoidal category.

29. 29.

Tannaka duality: an extension 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 $\mathcal{C}$ and its dual $\mathcal{C}^{op}$

2. 2.

the category of Hopf algebras over a field is (equivalent to) the opposite category of affine group schemes over $\operatorname{spec}k$

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 cokernel

8. 8.

limit and colimit

9. 9.

## 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.
 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