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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, braid groups, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ..., information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branching Rule
In group theory, restriction forms a representation of a subgroup using a known representation of the whole group. Restriction is a fundamental construction in representation theory of groups. Often the restricted representation is simpler to understand. Rules for decomposing the restriction of an irreducible representation into irreducible representations of the subgroup are called branching rules, and have important applications in physics. For example, in case of explicit symmetry breaking, the symmetry group of the problem is reduced from the whole group to one of its subgroups. In quantum mechanics, this reduction in symmetry appears as a splitting of degenerate energy levels into multiplets, as in the Stark or Zeeman effect. The induced representation is a related operation that forms a representation of the whole group from a representation of a subgroup. The relation between restriction and induction is described by Frobenius reciprocity and the Mackey theorem. Restrict ... [...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_: 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] |
Root Of Unity
In mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. It is occasionally called a de Moivre number after French mathematician Abraham de Moivre. Roots of unity can be defined in any field (mathematics), field. If the characteristic of a field, characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, converse (logic), conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if it is invertible; that is, a function f:X\to Y is bijective if and only if there is a function g:Y\to X, the ''inverse'' of , such that each of the two ways for composing the two functions produces an identity function: g(f(x)) = x for each x in X and f(g(y)) = y for each y in Y. For example, the ''multiplication by two'' defines a bijection from the integers to the even numbers, which has the ''division by two'' as its inverse function. A function is bijective if and only if it is both injective (or ''one-to-one'')—meaning that each element in the codomain is mappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula : \binom nk = \frac, which using factorial notation can be compactly expressed as : \binom = \frac. For example, the fourth power of is : \begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation : \binom = \binom + \binom . The binomial coefficients occur in many areas of mathematics, and especia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E 4 Temperley
E, or e, is the fifth letter and the second vowel letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''e'' (pronounced ); plural ''es'', ''Es'', or ''E's''. It is the most commonly used letter in many languages, including Czech, Danish, Dutch, English, French, German, Hungarian, Latin, Latvian, Norwegian, Spanish, and Swedish. Name In English, the name of the letter is the "long E" sound, pronounced . In most other languages, its name matches the letter's pronunciation in open syllables. History The Latin letter 'E' differs little from its source, the Greek letter epsilon, 'Ε'. This in turn comes from the Semitic letter '' hê'', which has been suggested to have started as a praying or calling human figure (''hillul'', 'jubilation'), and was most likely based on a similar Egyptian hieroglyph that indicated a different pronunciation. In Semitic, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E 2 Temperley
E, or e, is the fifth Letter (alphabet), letter and the second vowel#Written vowels, vowel letter of the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphabet#Letter names, ''e'' (pronounced ); plural ''es'', ''Es'', or ''E's''. It is the most commonly used letter in many languages, including Czech language, Czech, Danish language, Danish, Dutch language, Dutch, English language, English, French language, French, German language, German, Hungarian language, Hungarian, Latin language, Latin, Latvian language, Latvian, Norwegian language, Norwegian, Spanish language, Spanish, and Swedish language, Swedish. Name In English, the name of the letter is the "long E" sound, pronounced . In most other languages, its name matches the letter's pronunciation in open syllables. History The Latin letter 'E' differs little from its source, the Greek alphabe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
