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

2.
Duality principle^{} (http://planetmath.org/DualityPrinciple)

3.
Double duality

4.
Triality

5.
Selfduality

6.
Duality functors^{}, (for example the duality functor $Ho{m}_{k}(,k)$ )

7.
Poincaré duality/Poincaré isomorphism^{} (http://planetmath.org/PoincareDuality)

8.
PoincaréLefschetz duality, and AlexanderLefschetz duality

9.
Alexander duality: J. W. Alexander’s duality theory (cca. 1915)

10.
Serre duality : example in the proof of the RiemannRoch 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 LegendreFenchel 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 \to $ AlexanderSpanier 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.
TannakaKrein duality (http://planetmath.org/CompactQuantumGroup): for compact matrix pseudogroups and noncommutative 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 $\mathrm{\Pi}(G)$; Krein’s theorem shows which categories arise as a dual object to a compact group; the finitedimensional representations of Drinfel’d ’s quantum groups form a braided monoidal category, whereas $\mathrm{\Pi}(G)$ is a symmetric monoidal category.

29.
Tannaka duality: an extension^{} of Tannakian duality by Alexander Grothendieck (http://planetmath.org/AlexanderGrothendieckABiographyOf) to algebraic groups and Tannakian 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 $\mathrm{spec}k$

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, 411492.
Title  duality in mathematics 
Canonical name  DualityInMathematics 
Date of creation  20130322 18:24:50 
Last modified on  20130322 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 1800 
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 