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Subfactor
In the theory of von Neumann algebras, a subfactor of a factor M is a subalgebra that is a factor and contains 1 . The theory of subfactors led to the discovery of the Jones polynomial in knot theory. Index of a subfactor Usually M is taken to be a factor of type _1 , so that it has a finite trace. In this case every Hilbert space module H has a dimension \dim_M(H) which is a non-negative real number or + \infty . The index :N of a subfactor N is defined to be \dim_N(L^2(M)) . Here L^2(M) is the representation of N obtained from the GNS construction of the trace of M . Jones index theorem This states that if N is a subfactor of M (both of type _1 ) then the index :N/math> is either of the form 4 cos(\pi /n)^2 for n = 3,4,5,... , or is at least 4 . All these values occur. The first few values of 4 \cos(\pi /n)^2 are 1, 2, (3 + \sqrt)/2 = 2.618..., 3, 3.247..., ... Basic construction Suppose that N is a subfactor of M , and that both a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Algebra
In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a 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 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 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. 200px Here, the mark is shown as a \star-shape. On each input disk it is placed between two adjacent outgoing strings, and on the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sorin Popa
Sorin Teodor Popa (24 March 1953) is a Romanian American mathematician working on operator algebras. He is a professor at the University of California, Los Angeles. Biography Popa earned his PhD from the University of Bucharest in 1983 under the supervision of Dan-Virgil Voiculescu, with thesis ''Studiul unor clase de subalgebre ale C^*-algebrelor''. He has advised 15 doctoral students at UCLA, including Adrian Ioana. Honors and awards In 1990 Popa was an invited speaker at the International Congress of Mathematicians (ICM) in Kyoto, where he gave a talk on "Subfactors and Classifications in von Neumann algebras". He was a Guggenheim Fellow in 1995. In 2006 he gave a plenary lecture at the ICM in Madrid on "Deformation and Rigidity for group actions and Von Neumann Algebras". In 2009 he was awarded the Ostrowski Prize, and in 2010 the E. H. Moore Prize. He is one of the inaugural fellows of the American Mathematical Society The American Mathematical Society (AMS) is an asso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: *The ring L^\infty(\mathbb R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L^2(\mathbb R) of square-integrable functions. *The algebra \mathcal B(\mathcal H) of all bounded operators on a Hilbert space \mathcal H is a von Neumann algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factor (functional Analysis)
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: *The ring L^\infty(\mathbb R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L^2(\mathbb R) of square-integrable functions. *The algebra \mathcal B(\mathcal H) of all bounded operators on a Hilbert space \mathcal H is a von Neumann algebra, non ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Temperley–Lieb Algebra
In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable models, knot theory and the braid group, quantum groups and subfactors of von Neumann algebras. Structure Generators and relations Let R be a commutative ring and fix \delta \in R. The Temperley–Lieb algebra TL_n(\delta) is the R-algebra generated by the elements e_1, e_2, \ldots, e_, subject to the Jones relations: *e_i^2 = \delta e_i for all 1 \leq i \leq n-1 *e_i e_ e_i = e_i for all 1 \leq i \leq n-2 *e_i e_ e_i = e_i for all 2 \leq i \leq n-1 *e_i e_j = e_j e_i for all 1 \leq i,j \leq n-1 such that , i-j, \neq 1 Using these relations, any product of generators e_i can be brought to Jones' normal form: : E= \big(e_e_\cdots e_\big)\big(e_e_\cdots e_\big)\cdots\big(e_e_\cdots e_\big) where (i_1,i_2,\dots,i_r) and (j_1,j_2,\dots,j_r) are two strictly increasing sequences in \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Temperley–Lieb Algebra
In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable models, knot theory and the braid group, quantum groups and subfactors of von Neumann algebras. Structure Generators and relations Let R be a commutative ring and fix \delta \in R. The Temperley–Lieb algebra TL_n(\delta) is the R-algebra generated by the elements e_1, e_2, \ldots, e_, subject to the Jones relations: *e_i^2 = \delta e_i for all 1 \leq i \leq n-1 *e_i e_ e_i = e_i for all 1 \leq i \leq n-2 *e_i e_ e_i = e_i for all 2 \leq i \leq n-1 *e_i e_j = e_j e_i for all 1 \leq i,j \leq n-1 such that , i-j, \neq 1 Using these relations, any product of generators e_i can be brought to Jones' normal form: : E= \big(e_e_\cdots e_\big)\big(e_e_\cdots e_\big)\cdots\big(e_e_\cdots e_\big) where (i_1,i_2,\dots,i_r) and (j_1,j_2,\dots,j_r) are two strictly increasing sequences in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid Group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-stranded structure. More complex patterns can be constructed from an arbitrary number of strands to create a wider range of structures (such as a fishtail braid, a five-stranded braid, rope braid, a French braid and a waterfall braid). The structure is usually long and narrow with each component strand functionally equivalent in zigzagging forward through the overlapping mass of the others. It can be compared with the process of weaving, which usually involves two separate perpendicular groups of strands (warp and weft). Historically, the materials used have depended on the indigenous plants and animals available in the local area. During the Industrial Revolution, mechanized braiding equipment was invented to increase production. The braiding te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutant
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', or equivalently, such that conjugation by g leaves each element of ''S'' fixed. The normalizer of ''S'' in ''G'' is the set of elements \mathrm_G(S) of ''G'' that satisfy the weaker condition of leaving the set S \subseteq G fixed under conjugation. The centralizer and normalizer of ''S'' are subgroups of ''G''. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets ''S''. Suitably formulated, the definitions also apply to semigroups. In ring theory, the centralizer of a subset of a ring is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring ''R'' is a subring of ''R''. This article also deals with centralizers and normalize ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vanderbilt University. He was awarded the Fields Medal in 1982. Career Source: Academic career timeline: (1966–1970) – Bachelor's degree from the École Normale Supérieure (now part of Paris Sciences et Lettres University). (1973) – doctorate from Pierre and Marie Curie University, Paris, France (1970–1974) – appointment at the French National Centre for Scientific Research, Paris (1975) – Queen's University at Kingston, Ontario, Canada (1976–1980) – the University of Paris VI (1979 – present) – the Institute of Advanced Scientific Studies, Bures-sur-Yvette, France (1981–1984) – the French National Centre for Scientific Research, Paris (1984–2017) – the , Paris (2003–2011) – Vanderbilt University, Na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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