Talk:PlanetPhysics/Duality and Triality

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\subsection{Duality in Mathematics and Categorical Physics} The following is a contributed mathematical, \htmladdnormallink{physical mathematics}{http://planetphysics.us/encyclopedia/NonNewtonian2.html} and engineering topic entry (rather than a philosophical one), concerning different \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html} of \emph{duality} encountered in different areas of mathematics and categorical/algebraic \htmladdnormallink{theoretical physics}{http://planetphysics.us/encyclopedia/NonNewtonian2.html}; accordingly there is a string of distinct definitions associated with this topic rather than a single, \htmladdnormallink{general definition}{http://planetphysics.us/encyclopedia/PreciseIdea.html}, although some of the linked definitions, that is, categorical duality, are more general than others.

\subsubsection{Duality definitions in mathematics:} \begin{enumerate}

\item \htmladdnormallink{Categorical duality/Dual category}{http://planetphysics.us/encyclopedia/IndexOfCategoryTheory.html}: reversing arrows \item \htmladdnormallink{Duality principle}{http://planetphysics.us/encyclopedia/DualityPrinciple.html} \item Double duality \item Triality \item Self-duality: concept whose dual is itself \item Harmonic duality \item \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} duality: Conjugation \item Adjointness \item Hermitian duality \item Duality \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, (for example the duality functor $Hom_k(--,k)$ ) \item Poincar\'e duality/Poincar\'e isomorphism \item Poincar\'e-Lefschetz duality, and Alexander-Lefschetz duality \item Alexander duality: J. W. Alexander's duality theory (cca. 1915) \item Serre duality : example- in the proof of the Riemann-Roch theorem for curves. \item Logic duality: Dualities in logic, example: De Morgan dual, Boolean algebra \item Stone duality: Boolean algebras and Stone spaces \item Dual numbers- as in an associative algebra; (almost synonymous with double) \item Geometric dualities: dual polyhedron, dual of a planar \htmladdnormallink{graph}{http://planetphysics.us/encyclopedia/Cod.html}, duality in order theory, the Legendre transformation -an application of the duality between points and lines; generalized Legendre, that is, the Legendre-Fenchel transformation. \item Hamilton--Lagrange duality in theoretical \htmladdnormallink{mechanics}{http://planetphysics.us/encyclopedia/Mechanics.html} and optics \item \htmladdnormallink{Dual space}{http://planetphysics.us/encyclopedia/DualSpace.html} \item Dual space example \item Dual homomorphisms \item Duality of Projective Geometry \item Analytic dualities \item Duals of an algebra/algebraic duality, for example, dual pairs of Hopf *-algebras and duality of \htmladdnormallink{cross products}{http://planetphysics.us/encyclopedia/VectorProduct.html} of \htmladdnormallink{C*-algebras}{http://planetphysics.us/encyclopedia/VonNeumannAlgebra2.html} \item Tangled, or Mirror, duality: interchanging \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item Duality as a homological mirror symmetry \item \htmladdnormallink{cohomology theory}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} duals: de Rham cohomology $\leftarrow \rightarrow$ Alexander-Spanier cohomology \item Hodge dual \item Duality of locally compact groups \item Pontryagin duality, for locally compact commutative \htmladdnormallink{topological groups}{http://planetphysics.us/encyclopedia/PolishGroup.html} and their linear \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} \item Tannaka-Krein duality: for compact \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html} pseudogroups and \htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} topological groups; its generalization leads to \htmladdnormallink{quantum groups}{http://planetphysics.us/encyclopedia/QuantumGroup4.html} in \htmladdnormallink{quantum theories}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}; Tannaka's \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} provides the means to reconstruct a compact \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $G$ from its \htmladdnormallink{category of representations}{http://planetphysics.us/encyclopedia/CategoryOfRepresentations.html} $\Pi(G)$; Krein's theorem shows which \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} arise as a dual object to a compact group; the finite-dimensional representations of Drinfel'd 's quantum groups form a \htmladdnormallink{braided monoidal category}{http://planetphysics.us/encyclopedia/QuantumCategories.html}, whereas $\Pi(G)$ is a symmetric monoidal category. \item Tannaka duality: an extension of Tannakian duality by Alexander Grothendieck to \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} groups and Tannakian categories. \item Contravariant dualities \item Weak duality, example : weak duality theorem in linear programming; dual problems in optimization theory \item Dual codes \item Duality in Electrical Engineering \end{enumerate}

\subsubsection{Examples of duals:}

\begin{enumerate} \item a category $\mathcal{C}$ and its dual $\mathcal{C}^{op}$ \item the category of \htmladdnormallink{Hopf algebras}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} over a \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} $k$ is (equivalent to) the opposite category of affine group schemes over the \htmladdnormallink{spectrum}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html}. \item Dual Abelian variety \item Example of a dual space theorem \item Example of Pontryagin duality \item initial and final object \item kernel and cokernel \item limit and colimit \item direct sum and product \end{enumerate}

\begin{thebibliography}{99} \bibitem{SD-JR1989} S. Doplicher and J. Roberts. A new duality theory for compact groups. {\em Inventiones Mathematicae}, 98:157--218, 1989.

\bibitem{AJ-RS1991} 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.

\end{thebibliography}

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