|
Quasi-triangular Quasi-Hopf Algebra
A quasi-triangular quasi-Hopf algebra is a specialized form of a quasi-Hopf algebra defined by the 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 \mathcal i ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ukraine
Ukraine is a country in Eastern Europe. It is the List of European countries by area, second-largest country in Europe after Russia, which Russia–Ukraine border, borders it to the east and northeast. Ukraine also borders Belarus to the north; Poland and Slovakia to the west; Hungary, Romania and Moldova to the southwest; and the Black Sea and the Sea of Azov to the south and southeast. Kyiv is the nation's capital and List of cities in Ukraine, largest city, followed by Kharkiv, Odesa, and Dnipro. Ukraine's official language is Ukrainian language, Ukrainian. Humans have inhabited Ukraine since 32,000 BC. During the Middle Ages, it was the site of early Slavs, early Slavic expansion and later became a key centre of East Slavs, East Slavic culture under the state of Kievan Rus', which emerged in the 9th century. Kievan Rus' became the largest and most powerful realm in Europe in the 10th and 11th centuries, but gradually disintegrated into rival regional powers before being d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Drinfeld
Vladimir Gershonovich Drinfeld (; born February 14, 1954), surname also romanized as Drinfel'd, is a mathematician from Ukraine, who immigrated to the United States and works 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, he was elected to the National Academy of Sciences. In 2018 he received the Wolf Prize in Mathematics. In 2023 he was awarded the Shaw Prize in Mathematical Sciences. ... [...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' ... [...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) \rb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ribbon Hopf Algebra
A ribbon Hopf algebra (A,\nabla, \eta,\Delta,\varepsilon,S,\mathcal,\nu) is a quasitriangular Hopf algebra which possess an invertible central element \nu more commonly known as the ribbon element, such that the following conditions hold: :\nu^=uS(u), \; S(\nu)=\nu, \; \varepsilon (\nu)=1 :\Delta (\nu)=(\mathcal_\mathcal_)^(\nu \otimes \nu ) where u=\nabla(S\otimes \text)(\mathcal_). Note that the element ''u'' exists for any quasitriangular Hopf algebra, and uS(u) must always be central and satisfies S(uS(u))=uS(u), \varepsilon(uS(u))=1, \Delta(uS(u)) = (\mathcal_\mathcal_)^(uS(u) \otimes uS(u)), so that all that is required is that it have a central square root with the above properties. Here : A is a vector space : \nabla is the multiplication map \nabla:A \otimes A \rightarrow A : \Delta is the co-product map \Delta: A \rightarrow A \otimes A : \eta is the unit operator \eta:\mathbb \rightarrow A : \varepsilon is the co-unit operator \varepsilon: A \rightarrow \mathbb : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |