# 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
9. Hermitian duality
10. Duality functors, (for example the duality functor ${\displaystyle Hom_{k}(--,k)}$ )
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 ${\displaystyle \leftarrow \rightarrow }$ 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 ${\displaystyle G}$ from its category of representations ${\displaystyle \Pi (G)}$; 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 ${\displaystyle \Pi (G)}$ 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 ${\displaystyle {\mathcal {C}}}$ and its dual ${\displaystyle {\mathcal {C}}^{op}}$
2. the category of Hopf algebras over a field ${\displaystyle k}$ 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

## References

1. S. Doplicher and J. Roberts. A new duality theory for compact groups. Inventiones Mathematicae , 98:157--218, 1989.
2. Andr\'e 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.