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Bratteli Diagram
In mathematics, a Bratteli diagram is a combinatorial structure: a Graph (discrete mathematics), graph composed of vertices labelled by positive integers ("level") and unoriented edges between vertices having levels differing by one. The notion was introduced by Ola Bratteli in 1972 in the theory of operator algebras to describe directed sequences of finite-dimensional algebras: it played an important role in Elliott's classification of approximately finite-dimensional C*-algebra, AF-algebras and the theory of subfactors. Subsequently Anatoly Vershik associated dynamical systems with infinite paths in such graphs. Definition A Bratteli diagram is given by the following objects: * A sequence of sets ''V''''n'' ('the vertices at level ''n'' ') labeled by positive integer set N. In some literature each element v of ''V''''n'' is accompanied by a positive integer ''b''''v'' > 0. * A sequence of sets ''E''''n'' ('the edges from level ''n'' to ''n'' + 1 ') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the modu ... [...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 King Henry VIII in 1534, it is the oldest university press 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 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 university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Bratteli–Vershik Diagram
In mathematics, a Bratteli–Veršik diagram is an ordered, essentially simple Bratteli diagram (''V'', ''E'') with a homeomorphism on the set of all infinite paths called the Veršhik transformation. It is named after Ola Bratteli and Anatoly Vershik. Definition Let ''X'' = be the set of all paths in the essentially simple Bratteli diagram (''V'', ''E''). Let ''E''min be the set of all minimal edges in ''E'', similarly let ''E''max be the set of all maximal edges. Let ''y'' be the unique infinite path in ''E''max. (Diagrams which possess a unique infinite path are called "essentially simple".) The Veršhik transformation is a homeomorphism φ : ''X'' → ''X'' defined such that φ(''x'') is the unique minimal path if ''x'' = ''y''. Otherwise ''x'' = (''e''1, ''e''2,...) , ''e''''i'' ∈ ''E''''i'' where at least one ''e''''i'' ∉ ''E''max. Let ''k'' be the smallest such integer. Then φ(''x ... [...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 matrix, transfer matrices, invented by Harold Neville Vazeille Temperley, Neville Temperley and Elliott H. Lieb, 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 algebra (ring theory), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birman–Wenzl Algebra
In mathematics, the Birman–Murakami–Wenzl (BMW) algebra, introduced by and , is a two-parameter family of algebras \mathrm_n(\ell,m) of dimension 1\cdot 3\cdot 5\cdots (2n-1) having the Hecke algebra of the symmetric group as a quotient. It is related to the Kauffman polynomial of a link. It is a deformation of the Brauer algebra in much the same way that Hecke algebras are deformations of the group algebra of the symmetric group. Definition For each natural number ''n'', the BMW algebra \mathrm_n(\ell,m) is generated by G_1^,G_2^,\dots , G_^,E_1,E_2,\dots ,E_ and relations: : G_iG_j=G_jG_i, \mathrm \left\vert i-j \right\vert \geqslant 2, :G_i G_ G_i=G_ G_i G_, E_i E_ E_i=E_i, : G_i + ^=m(1+E_i), : G_ G_i E_ = E_i G_ G_i = E_i E_, G_ E_i G_ =^ E_ ^, : G_ E_i E_=^ E_, E_ E_i G_ =E_ ^, : G_i E_i= E_i G_i = l^ E_i, E_i G_ E_i =l E_i. These relations impl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brauer Algebra
In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer in the context of the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the general linear group in Schur–Weyl duality. Structure The Brauer algebra \mathfrak_n(\delta) is a \mathbbdelta/math>-algebra depending on the choice of a positive integer n. Here \delta is an indeterminate, but in practice \delta is often specialised to the dimension of the fundamental representation of an orthogonal group O(\delta). The Brauer algebra has the dimension :\dim\mathfrak_n(\delta) = \frac = (2n-1)!! = (2n-1)(2n-3)\cdots 5\cdot 3\cdot 1 Diagrammatic definition A basis of \mathfrak_n(\delta) consists of all pairings on a set of 2n elements X_1, ..., X_n, Y_1, ..., Y_n (that is, all perfect matchings of a complete graph K_n: any two of the 2n elements may be matched to each other, regardless of their symbols). The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young's Lattice
In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers ''On quantitative substitutional analysis,'' developed the representation theory of the symmetric group. In Young's theory, the objects now called Young diagrams and the partial order on them played a key, even decisive, role. Young's lattice prominently figures in algebraic combinatorics, forming the simplest example of a differential poset in the sense of . It is also closely connected with the crystal bases for affine Lie algebras. Definition Young's lattice is a lattice (and hence also a partially ordered set) ''Y'' formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers diagrams). Significance The traditional application of Young's lattice is to the description of the irreducible representations of symmetric groups S''n'' for all ''n'', together with their branching properties, in chara ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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