|
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] |
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] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles. 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 microscopic parameters that fluctuate about average values and are characterized by probability distributions. This established the fields of statistical thermodynamics and statistical physics. The founding of the field of statistical mechanics is generally credited to three physicists: *Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates *James Clerk Maxwell, who developed models of probability distr ... [...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] |
Quasitriangular 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] |
Quantum Inverse Scattering Method
In quantum physics, the quantum inverse scattering method is a method for solving integrable models in 1+1 dimensions, introduced by L. D. Faddeev in 1979. The quantum inverse scattering method relates two different approaches: #the Bethe ansatz, a method of solving integrable quantum models in one space and one time dimension; #the Inverse scattering transform, a method of solving classical integrable differential equations of the evolutionary type. This method led to the formulation of quantum groups. Especially interesting is the Yangian, and the center of the Yangian is given by the quantum determinant. An important concept in the Inverse scattering transform is the Lax representation; the quantum inverse scattering method starts by the quantization of the Lax representation and reproduces the results of the Bethe ansatz. In fact, it allows the Bethe ansatz to be written in a new form: the ''algebraic Bethe ansatz''.cf. e.g. the lectures by N.A. Slavnov, This led to fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable Model
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from more ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisenberg XXZ Model
Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series of papers with Max Born and Pascual Jordan, during the same year, his matrix formulation of quantum mechanics was substantially elaborated. He is known for the uncertainty principle, which he published in 1927. Heisenberg was awarded the 1932 Nobel Prize in Physics "for the creation of quantum mechanics". Heisenberg also made contributions to the theories of the hydrodynamics of turbulent flows, the atomic nucleus, ferromagnetism, cosmic rays, and subatomic particles. He was a principal scientist in the German nuclear weapons program during World War II. He was also instrumental in planning the first West German nuclear reactor at Karlsruhe, together with a research reactor in Munich, in 1957. Following World War II, he was appointed di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
