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Drinfeld Twist
In mathematics, a Hopf algebra, ''H'', is quasitriangularMontgomery & Schneider (2002), p. 72 if there exists an invertible element, ''R'', of H \otimes H such that :*R \ \Delta(x)R^ = (T \circ \Delta)(x) for all x \in H, where \Delta is the coproduct on ''H'', and the linear map T : H \otimes H \to H \otimes H is given by T(x \otimes y) = y \otimes x, :*(\Delta \otimes 1)(R) = R_ \ R_, :*(1 \otimes \Delta)(R) = R_ \ R_, where R_ = \phi_(R), R_ = \phi_(R), and R_ = \phi_(R), where \phi_ : H \otimes H \to H \otimes H \otimes H, \phi_ : H \otimes H \to H \otimes H \otimes H, and \phi_ : H \otimes H \to H \otimes H \otimes H, are algebra morphisms determined by :\phi_(a \otimes b) = a \otimes b \otimes 1, :\phi_(a \otimes b) = a \otimes 1 \otimes b, :\phi_(a \otimes b) = 1 \otimes a \otimes b. ''R'' is called the R-matrix. As a consequence of the properties of quasitriangularity, the R-matrix, ''R'', is a solution of the Yang–Baxter equation (and so a module ''V'' of ''H'' c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Link (knot Theory)
In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a ''trivial'' reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link. For example, a co-dimension 2 link in 3-dimensional space is a subspace of 3-dimensional Euclidean space (or often the 3-sphere) whose connected components are homeomorphic to circles. The simplest nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles (or unknots) linked together once. The circles in the Borromean rings are collectively linked despite the fact that no two of them are directly linked. The Borromean rings thus form a Brunnian link and in fact constitut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-triangular Quasi-Hopf Algebra
A quasi-triangular quasi-Hopf algebra is a specialized form of a quasi-Hopf algebra defined by the Ukraine, Ukrainian mathematician Vladimir Drinfeld in 1989. It is also a generalized form of a quasi-triangular Hopf algebra. A quasi-triangular quasi-Hopf algebra is a set \mathcal = (\mathcal, R, \Delta, \varepsilon, \Phi) where \mathcal = (\mathcal, \Delta, \varepsilon, \Phi) is a quasi-Hopf algebra and R \in \mathcal known as the R-matrix, is an invertible element such that : R \Delta(a) = \sigma \circ \Delta(a) R for all a \in \mathcal, where \sigma\colon \mathcal \rightarrow \mathcal is the switch map given by x \otimes y \rightarrow y \otimes x, and : (\Delta \otimes \operatorname)R = \Phi_R_\Phi_^R_\Phi_ : (\operatorname \otimes \Delta)R = \Phi_^R_\Phi_R_\Phi_^ where \Phi_ = x_a \otimes x_b \otimes x_c and \Phi_= \Phi = x_1 \otimes x_2 \otimes x_3 \in \mathcal. The quasi-Hopf algebra becomes ''triangular'' if in addition, R_R_=1. The twisting of \mathcal by F \in \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-triangular Quasi-Hopf Algebra
A quasi-triangular quasi-Hopf algebra is a specialized form of a quasi-Hopf algebra defined by the Ukraine, Ukrainian mathematician Vladimir Drinfeld in 1989. It is also a generalized form of a quasi-triangular Hopf algebra. A quasi-triangular quasi-Hopf algebra is a set \mathcal = (\mathcal, R, \Delta, \varepsilon, \Phi) where \mathcal = (\mathcal, \Delta, \varepsilon, \Phi) is a quasi-Hopf algebra and R \in \mathcal known as the R-matrix, is an invertible element such that : R \Delta(a) = \sigma \circ \Delta(a) R for all a \in \mathcal, where \sigma\colon \mathcal \rightarrow \mathcal is the switch map given by x \otimes y \rightarrow y \otimes x, and : (\Delta \otimes \operatorname)R = \Phi_R_\Phi_^R_\Phi_ : (\operatorname \otimes \Delta)R = \Phi_^R_\Phi_R_\Phi_^ where \Phi_ = x_a \otimes x_b \otimes x_c and \Phi_= \Phi = x_1 \otimes x_2 \otimes x_3 \in \mathcal. The quasi-Hopf algebra becomes ''triangular'' if in addition, R_R_=1. The twisting of \mathcal by F \in \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-bialgebra
In mathematics, quasi-bialgebras are a generalization of bialgebras: they were first defined by the Ukrainian mathematician Vladimir Drinfeld in 1990. A quasi-bialgebra differs from a bialgebra by having coassociativity replaced by an invertible element \Phi which controls the non- coassociativity. One of their key properties is that the corresponding category of modules forms a tensor category. Definition A quasi-bialgebra \mathcal = (\mathcal, \Delta, \varepsilon, \Phi,l,r) is an algebra \mathcal over a field \mathbb equipped with morphisms of algebras :\Delta : \mathcal \rightarrow \mathcal :\varepsilon : \mathcal \rightarrow \mathbb along with invertible elements \Phi \in \mathcal, and r,l \in A such that the following identities hold: :(id \otimes \Delta) \circ \Delta(a) = \Phi \lbrack (\Delta \otimes id) \circ \Delta (a) \rbrack \Phi^, \quad \forall a \in \mathcal :\lbrack (id \otimes id \otimes \Delta)(\Phi) \rbrack \ \lbrack (\Delta \otimes id \otimes id)(\Phi) \rbra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-triangular Hopf Algebra
In mathematics, a Hopf algebra, ''H'', is quasitriangularMontgomery & Schneider (2002), p. 72 if there exists an invertible element, ''R'', of H \otimes H such that :*R \ \Delta(x)R^ = (T \circ \Delta)(x) for all x \in H, where \Delta is the coproduct on ''H'', and the linear map T : H \otimes H \to H \otimes H is given by T(x \otimes y) = y \otimes x, :*(\Delta \otimes 1)(R) = R_ \ R_, :*(1 \otimes \Delta)(R) = R_ \ R_, where R_ = \phi_(R), R_ = \phi_(R), and R_ = \phi_(R), where \phi_ : H \otimes H \to H \otimes H \otimes H, \phi_ : H \otimes H \to H \otimes H \otimes H, and \phi_ : H \otimes H \to H \otimes H \otimes H, are algebra morphisms determined by :\phi_(a \otimes b) = a \otimes b \otimes 1, :\phi_(a \otimes b) = a \otimes 1 \otimes b, :\phi_(a \otimes b) = 1 \otimes a \otimes b. ''R'' is called the R-matrix. As a consequence of the properties of quasitriangularity, the R-matrix, ''R'', is a solution of the Yang–Baxter equation (and so a module ''V'' of ''H'' c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Drinfeld
Vladimir Gershonovich Drinfeld ( uk, Володи́мир Ге́ршонович Дрінфельд; russian: Влади́мир Ге́ршонович Дри́нфельд; born February 14, 1954), surname also romanized as Drinfel'd, is a renowned mathematician from the former USSR, who emigrated to the United States and is currently working at the University of Chicago. Drinfeld's work connected algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the geometric Langlands correspondence. Drinfeld introduced the notion of a quantum group (independently discovered by Michio Jimbo at the same time) and made important contributions to mathematical physics, including the ADHM construction of instantons, algebraic formalism of the quantum inverse scattering method, and the Drinfeld–Sokolov reduction in the theory of solitons. He was awarded the Fields Medal in 1990. In 2016, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ribbon Hopf Algebra
A ribbon or riband is a thin band of material, typically cloth but also plastic or sometimes metal, used primarily as decorative binding and tying. Cloth ribbons are made of natural materials such as silk, cotton, and jute and of synthetic materials, such as polyester, nylon, and polypropylene. Ribbon is used for useful, ornamental, and symbolic purposes. Cultures around the world use ribbon in their hair, around the body, and as ornament on non-human animals, buildings, and packaging. Some popular fabrics used to make ribbons are satin, organza, sheer, silk, velvet, and grosgrain. Etymology The word ribbon comes from Middle English ''ribban'' or ''riban'' from Old French ''ruban'', which is probably of Germanic origin. Cloth Along with that of tapes, fringes, and other smallwares, the manufacture of cloth ribbons forms a special department of the textile industries. The essential feature of a ribbon loom is the simultaneous weaving in one loom frame of two or more we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot (mathematics)
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of which takes one knot to the other. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term ''knot'' is also applied to embeddings of in , especially in the case . The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory. Formal definition A knot is an embedding of the circle () into three-dimensional Euclidean space (), or the 3-sphere (), since the 3-sphere is compact. Two knots are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebra
Hopf is a German surname. Notable people with the surname include: *Eberhard Hopf (1902–1983), Austrian mathematician *Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swedish actor *Ludwig Hopf (1884–1939), German physicist *Maria Hopf Maria Hopf (13 September 1913 – 24 August 2008) was a pioneering archaeobotanist, based at the RGZM, Mainz. Career Hopf studied botany from 1941–44, receiving her doctorate in 1947 on the subject of soil microbes. She then worked in phyto ... (1914-2008), German botanist and archaeologist {{surname, Hopf German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid Theory
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] |
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 module conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
