|
Planar Algebra
In mathematics, planar algebras first appeared in the work of Vaughan Jones on the Subfactor#Standard invariant, standard invariant of a II-1 subfactor, II1 subfactor. They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial), and have been used in describing the properties of Khovanov homology with respect to tangle (mathematics), tangle composition. Any subfactor planar algebra provides a family of unitary representations of Thompson groups. Any finite group (and quantum generalization) can be encoded as a planar algebra. Definition The idea of the planar algebra is to be a diagrammatic axiomatization of the Subfactor#Standard invariant, standard invariant. Planar tangle A (shaded) planar tangle is the data of finitely many ''input'' disks, one ''output'' disk, non-intersecting strings giving an even number, say 2n , intervals per disk and one \star-marked interval per disk. image:Tangle.png, 200px Here, the mark ... [...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] |
