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Skeletonization Of Fusion Categories
In mathematics, the skeletonization of fusion categories is a process whereby one extracts the core data of a fusion category or related categorical object in terms of minimal Set theory, set-theoretic information. This set-theoretic information is referred to as the skeletal data of the fusion category. This process is related to the general technique of Skeleton (category theory), skeletonization in category theory. Skeletonization is often used for working with examples, doing computations, and classifying fusion categories. The relevant feature of Fusion category, fusion categories which makes the technique of skeletonization effective is the strong finiteness conditions placed on fusion categories, such as the requirements that they have finitely many isomorphism classes of Schur's lemma, simple objects and that all of their Hom space, hom-spaces are finite dimensional. This allows the entire categorical structure of a fusion category to be encoded in a finite amount of complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spacetime Diagram
A spacetime diagram is a graphical illustration of locations in space at various times, especially in the special theory of relativity. Spacetime diagrams can show the geometry underlying phenomena like time dilation and length contraction without mathematical equations. The history of an object's location through time traces out a line or curve on a spacetime diagram, referred to as the object's world line. Each point in a spacetime diagram represents a unique position in space and time and is referred to as an event (relativity), event. The most well-known class of spacetime diagrams are known as Minkowski diagrams, developed by Hermann Minkowski in 1908. Minkowski diagrams are two-dimensional graphs that depict events as happening in a universe consisting of one space dimension and one time dimension. Unlike a regular distance-time graph, the distance is displayed on the horizontal axis and time on the vertical axis. Additionally, the time and space units of measurement are c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Category
In category theory, a branch of mathematics, a spherical category is a pivotal category (a monoidal category with traces) in which left and right traces coincide. Spherical fusion categories give rise to a family of three-dimensional topological state sum models (a particular formulation of a topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. While TQFTs were invented by physicists, they are also of mathemati ...), the Turaev-Viro model, or rather Turaev-Viro-Barrett-Westbury model. References Category theory {{categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Diagrams For Braiding
String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian animated short * ''Strings'' (2004 film), a film directed by Anders Rønnow Klarlund * ''Strings'' (2011 film), an American dramatic thriller film * ''Strings'' (2012 film), a British film by Rob Savage * ''Bravetown'' (2015 film), an American drama film originally titled ''Strings'' * '' The String'' (2009), a French film Music Instruments * String (music), the flexible element that produces vibrations and sound in string instruments * String instrument, a musical instrument that produces sound through vibrating strings ** List of string instruments * String piano, a pianistic extended technique in which sound is produced by direct manipulation of the strings, rather than striking the piano's keys Types of groups * String band, musical e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Transformations
In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a certain transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom. Although the unphysical axes in the space of detailed configurations are a fundamental property of the physical model, there is no special set of directions "perpendicular" to them. Hence there is an enormous amount of freedom involved in taking a "cross section" representing each physical configuration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-j Symbol
Wigner's 6-''j'' symbols were introduced by Eugene Paul Wigner in 1940 and published in 1965. They are defined as a sum over products of four Wigner 3-j symbols, Wigner 3-''j'' symbols, : \begin \begin j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end &= \sum_ (-1)^ \begin j_1 & j_2 & j_3\\ -m_1 & -m_2 & -m_3 \end\times\\ &\times \begin j_1 & j_5 & j_6\\ m_1 & -m_5 & m_6 \end \begin j_4 & j_2 & j_6\\ m_4 & m_2 & -m_6 \end \begin j_4 & j_5 & j_3\\ -m_4 & m_5 & m_3 \end . \end The summation is over all six allowed by the selection rules of the 3-''j'' symbols. They are closely related to the Racah W-coefficients, which are used for recoupling 3 angular momenta, although Wigner 6-''j'' symbols have higher symmetry and therefore provide a more efficient means of storing the recoupling coefficients. Their relationship is given by: : \begin j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end = (-1)^ W(j_1 j_2 j_5 j_4; j_3 j_6) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
