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Homeduality in mathematics

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# 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: reversing arrows

2. 3. Double duality

4. Triality

5. Self-duality

6. Duality functors, (for example the duality functor $Hom_{k}(--,k)$ )

7. 8. Poincaré-Lefschetz duality, and Alexander-Lefschetz duality

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

10. 11. Dualities in logic, example: De Morgan dual, 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. 17. 18. 19. 20. Analytic dualities

21. Duals of an algebra/algebraic duality, for example, dual pairs of Hopf *-algebras and duality of cross products of C*-algebras

22. Tangled, or Mirror, duality: interchanging morphisms and objects

23. Duality as a homological mirror symmetry

24. Cohomology theory duals: de Rham cohomology $\leftarrow\rightarrow$ Alexander-Spanier cohomology

25. Hodge dual

26. 27. Pontryagin duality, for locally compact commutative topological groups and their linear representations

28. 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.

29. Tannaka duality: an extension of Tannakian duality by Alexander Grothendieck to algebraic groups and Tannakian categories.

30. Contravariant dualities

31. Weak duality, example : weak duality theorem in linear programming; dual problems in optimization theory

32. 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 $\operatorname{spec}k$

3. Dual Abelian variety

4. Example of a dual space theorem

5. 6. initial and final object

7. 8. 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.

## Mathematics Subject Classification

51A10*no label found*14F25

*no label found*55M05

*no label found*18-00

*no label found*

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