Duality_(mathematics)

Duality (mathematics)

In mathematics, duality has numerous meanings. Generally speaking, duality is a metamathematical involution. Some duality concepts are closely related and there are explicit theorems governing their relationships. Others are more intuitively related, with no precise correspondence. Nonetheless, "duality is a very pervasive and important concept in (modern) mathematics".^[1] Generally speaking, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have fixed points, the dual of A is sometimes A itself. For example, Desargues' theorem in projective geometry is self-dual in this sense.

1 Geometric dualities
2 Contravariant dualities
3 Analytic dualities
4 Boolean algebra
5 Poincaré-style dualities
6 See also
7 References

Geometric dualities

In one group of dualities, the concepts and theorems of a certain mathematical theory are mechanically translated into other concepts and theorems of the same theory. The prototypical example here is the duality in projective geometry: given any theorem in plane projective geometry, exchanging the terms "point" and "line" everywhere results in a new, equally valid theorem. Other examples include:

dual polyhedron
dual graph of a planar graph
dual problem in optimization theory
De Morgan dual in logic (and its analogues, the dual quantifiers in first-order logic and modal logic)
duality in order theory

Contravariant dualities

In another group of dualities, the objects of one theory are translated into objects of another theory and the morphisms between objects in the first theory are translated into morphisms in the second theory, but with direction reversed. Using a duality of this type, every statement in the first theory can be translated into a "dual" statement in the second theory, where the direction of all arrows has to be reversed. For the general notion in category theory that underlies these dualities, see opposite category. Examples include:

dual spaces in linear algebra
duality between commutative rings and algebraic varieties; compare with noncommutative geometry
Pontryagin duality, relating certain abelian groups to other abelian groups and the background to Fourier analysis
Tannaka-Krein duality, a non-commutative analogue of Pontryagin duality
Stone duality, relating Boolean algebras to certain topological spaces
duality between submodules and factor modules in algebra
categorical duality between projective modules and injective modules in homological algebra

Analytic dualities

In analysis, frequently problems are solved by passing to the dual description of functions and operators.

Fourier transform switches between functions on a vector space and its dual, and interchanges operations of multiplication and convolution on the corresponding function spaces; its dualizing character has many other manifestations, for example, in alternative descriptions of quantum mechanical systems in terms of coordinate and momentum representations.
Laplace transform is similar to Fourier transform and interchanges operators of multiplication by polynomials with constant coefficient linear differential operators.
Legendre transformation is an important analytic duality which switches between velocities in Lagrangian mechanics and momenta in Hamiltonian mechanics.

Boolean algebra

In Boolean algebra a self-dual function is one such that

$\forall (a_1, \ldots, a_n) \in\{0,1\}^n, \quad f(a_1, \ldots, a_n) = \neg f(\neg a_1, \ldots, \neg a_n).$

Negation is self-dual.

Poincaré-style dualities

Theorems showing that certain objects of interest are the dual spaces (in the sense of linear algebra) of other objects of interest are also often called dualities. Examples:

References

^ Springer, Encyclopedia of Mathematics, Duality, [1]

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