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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-Hopf Algebra
A quasi-Hopf algebra is a generalization of a Hopf algebra, which was defined by the Russian mathematician Vladimir Drinfeld in 1989. A ''quasi-Hopf algebra'' is a quasi-bialgebra \mathcal = (\mathcal, \Delta, \varepsilon, \Phi) for which there exist \alpha, \beta \in \mathcal and a bijective antihomomorphism ''S'' ( antipode) of \mathcal such that : \sum_i S(b_i) \alpha c_i = \varepsilon(a) \alpha : \sum_i b_i \beta S(c_i) = \varepsilon(a) \beta for all a \in \mathcal and where :\Delta(a) = \sum_i b_i \otimes c_i and :\sum_i X_i \beta S(Y_i) \alpha Z_i = \mathbb, :\sum_j S(P_j) \alpha Q_j \beta S(R_j) = \mathbb. where the expansions for the quantities \Phiand \Phi^ are given by :\Phi = \sum_i X_i \otimes Y_i \otimes Z_i and :\Phi^{-1}= \sum_j P_j \otimes Q_j \otimes R_j. As for a quasi-bialgebra, the property of being quasi-Hopf is preserved under twisting. Usage Quasi-Hopf algebras form the basis of the study of Drinfeld twists and the representations in terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ukraine
Ukraine ( uk, Україна, Ukraïna, ) is a country in Eastern Europe. It is the second-largest European country after Russia, which it borders to the east and northeast. Ukraine covers approximately . Prior to the ongoing Russian invasion, it was the eighth-most populous country in Europe, with a population of around 41 million people. It is also bordered by Belarus to the north; by Poland, Slovakia, and Hungary to the west; and by Romania and Moldova to the southwest; with a coastline along the Black Sea and the Sea of Azov to the south and southeast. Kyiv is the nation's capital and largest city. Ukraine's state language is Ukrainian; Russian is also widely spoken, especially in the east and south. During the Middle Ages, Ukraine was the site of early Slavic expansion and the area later became a key centre of East Slavic culture under the state of Kievan Rus', which emerged in the 9th century. The state eventually disintegrated into rival regional po ... [...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] |
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] |
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] |
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] |
