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 )
- 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
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 from its category of representations ; 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 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 and its dual
- 2.
the category of Hopf algebras
over a field is (equivalent
to) the opposite category of affine group schemes over
- 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.