PlanetPhysics/Duality and Triality

Duality in Mathematics and Categorical Physics

The following is a contributed mathematical, physical mathematics and engineering topic entry (rather than a philosophical one), concerning different types of duality encountered in different areas of mathematics and categorical/algebraic theoretical physics; 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.

Duality definitions in mathematics:

1. Categorical duality/Dual category: reversing arrows
2. Duality principle
3. Double duality
4. Triality
5. Self-duality: concept whose dual is itself
6. Harmonic duality
7. topological duality: Conjugation
8. Adjointness
9. Hermitian duality
10. Duality functors, (for example the duality functor $Hom_{k}(--,k)$ )
11. Poincar\'e duality/Poincar\'e isomorphism
12. Poincar\'e-Lefschetz duality, and Alexander-Lefschetz duality
13. Alexander duality: J. W. Alexander's duality theory (cca. 1915)
14. Serre duality :

example- in the proof of the Riemann-Roch theorem for curves.

1. Logic duality: Dualities in logic, example: De Morgan dual, Boolean algebra
2. Stone duality: Boolean algebras and Stone spaces
3. Dual numbers- as in an associative algebra; (almost synonymous with double)
4. 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.

1. Hamilton--Lagrange duality in theoretical mechanics and optics
2. Dual space
3. Dual space example
4. Dual homomorphisms
5. Duality of Projective Geometry
6. Analytic dualities
7. Duals of an algebra/algebraic duality,

for example, dual pairs of Hopf *-algebras and duality of cross products of C*-algebras #Tangled, or Mirror, duality: interchanging morphisms and objects #Duality as a homological mirror symmetry

1. cohomology theory duals: de Rham cohomology $\leftarrow \rightarrow$ Alexander-Spanier cohomology
2. Hodge dual
3. Duality of locally compact groups
4. Pontryagin duality, for locally compact commutative topological groups and their linear representations #Tannaka-Krein duality: 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.

1. Tannaka duality: an extension of Tannakian duality by

Alexander Grothendieck to algebraic groups and Tannakian categories.

1. Contravariant dualities
2. Weak duality, example : weak duality theorem in linear programming;

dual problems in optimization theory

1. Dual codes
2. Duality in Electrical Engineering

Examples of duals:

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