duality in mathematics

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.

Self-duality

6.

Duality

functors

, (for example the duality functor

H

o

m

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

←

→

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

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 quantum
groups form a braided monoidal category, whereas

Π

⁢

(

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

𝒞

and its dual

𝒞

o

⁢

p

2.

the category of

Hopf algebras

over a field is (

equivalent

to) the opposite category of

affine group

schemes over

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.

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

Generated on Fri Feb 9 21:56:47 2018 by

LaTeXML