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Hypergeometric solutions of trigonometric KZ equations satisfy dynamical difference equations
- Adv. Math
"... {yavmar, anv} @ email.unc.edu Abstract. The trigonometric KZ equations associated to a Lie algebra g depend on a parameter λ ∈ h where h ⊂ g is a Cartan subalgebra. A system of dynamical difference equations with respect to λ compatible with the KZ equations is introduced in [TV]. We prove that the ..."
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Cited by 16 (8 self)
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{yavmar, anv} @ email.unc.edu Abstract. The trigonometric KZ equations associated to a Lie algebra g depend on a parameter λ ∈ h where h ⊂ g is a Cartan subalgebra. A system of dynamical difference equations with respect to λ compatible with the KZ equations is introduced in [TV]. We prove that the standard hypergeometric solutions of the trigonometric KZ equations associated to slN also satisfy the dynamical difference equations.
Tarasov: Dynamical Yang-Baxter equations, quasi-Poisson homogeneous spaces, and quantization
"... This paper is a continuation of [10]. Let us recall the main result of [10]. Let G be a Lie group, g = Lie G, U ⊂ G a connected closed Lie subgroup such that the corresponding subalgebra u ⊂ g is reductive in g (i.e., there exists an ..."
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Cited by 7 (2 self)
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This paper is a continuation of [10]. Let us recall the main result of [10]. Let G be a Lie group, g = Lie G, U ⊂ G a connected closed Lie subgroup such that the corresponding subalgebra u ⊂ g is reductive in g (i.e., there exists an
Quantum Dynamical coboundary Equation for finite dimensional simple Lie algebras
, 2006
"... For a finite dimensional simple Lie algebra g, the standard universal solution R(x) ∈ Uq(g) ⊗2 of the Quantum Dynamical Yang–Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang–Baxter Equation. It can be built from the standard R–matrix and from the solution ..."
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Cited by 4 (0 self)
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For a finite dimensional simple Lie algebra g, the standard universal solution R(x) ∈ Uq(g) ⊗2 of the Quantum Dynamical Yang–Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang–Baxter Equation. It can be built from the standard R–matrix and from the solution F(x) ∈ Uq(g) ⊗2 of the Quantum Dynamical co-(x)R F12(x). F(x) can be computed explicitely as an infinite product through the use of an auxiliary linear equation, the ABRR equation. Inspired by explicit results in the fundamental representation, it has been conjectured that, in the case where g = sl(n + 1) (n ≥ 1) only, there could exist an element M(x) ∈ Uq(sl(n + 1)) such that the dynamical gauge transform R J of R(x) by M(x), Cycle Equation as R(x) = F −1
Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review
, 2009
"... A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromod ..."
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Cited by 3 (3 self)
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A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non–Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier–Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasitriangular, quasiHopf algebras, bialgebroids, Grassmann-Hopf algebras and Higher Dimensional Algebra. On the one hand, this quantum
SUPER SOLUTIONS OF THE DYNAMICAL YANG-BAXTER EQUATION
, 2005
"... Abstract. A super dynamical r-matrix r satisfies the zero weight condition if: [h ⊗ 1 + 1 ⊗ h, r(λ)] = 0 for all h ∈ h, λ ∈ h ∗. In this paper we classify super dynamical r−matrices with zero weight, thus extending the results of [6] to the graded case. 1. ..."
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Abstract. A super dynamical r-matrix r satisfies the zero weight condition if: [h ⊗ 1 + 1 ⊗ h, r(λ)] = 0 for all h ∈ h, λ ∈ h ∗. In this paper we classify super dynamical r−matrices with zero weight, thus extending the results of [6] to the graded case. 1.
Universal Vertex-IRF Transformation for Quantum Affine Algebras
, 2008
"... We construct a universal Vertex-IRF transformation between Vertex type universal solution and Face type universal solution of the quantum dynamical Yang-Baxter equation. This universal Vertex-IRF transformation satisfies the generalized coBoundary equation) case. This solution has a simple Gauss dec ..."
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Cited by 3 (0 self)
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We construct a universal Vertex-IRF transformation between Vertex type universal solution and Face type universal solution of the quantum dynamical Yang-Baxter equation. This universal Vertex-IRF transformation satisfies the generalized coBoundary equation) case. This solution has a simple Gauss decomposition which is constructed using Sevostyanov’s characters of twisted quantum Borel algebras. We show that the evaluation of this universal solution in the evaluation representation of Uq(A (1) 1) gives the standard Baxter’s transformation between the 8-Vertex model and the IRF height model. and is an extension of our previous work to the quantum affine Uq(A (1) r 1
DYNAMICAL TWISTS IN HOPF ALGEBRAS
, 2007
"... Abstract. We establish a bijective correspondence between gauge equivalence classes of dynamical twists in a finite-dimensional Hopf algebra H based on a finite abelian group A and equivalence classes of pairs (K, {Vλ} λ ∈ b A), where K is an H-simple left H-comodule semisimple algebra and {Vλ} λ ∈ ..."
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Abstract. We establish a bijective correspondence between gauge equivalence classes of dynamical twists in a finite-dimensional Hopf algebra H based on a finite abelian group A and equivalence classes of pairs (K, {Vλ} λ ∈ b A), where K is an H-simple left H-comodule semisimple algebra and {Vλ} λ ∈ b A is a family of irreducible representations satisfying certain conditions. Our results generalize the results obtained by Etingof-Nikshych on the classification of dynamical twists in group algebras. 1.
Chern-Simons Theory with Sources and Dynamical Quantum Groups I: Canonical Analysis and Algebraic Structures.
, 2005
"... We study the quantization of Chern-Simons theory with group G coupled to dynamical sources. We first study the dynamics of Chern-Simons sources in the Hamiltonian framework. The gauge group of this system is reduced to the Cartan subgroup of G. We show that the Dirac bracket between the basic dynami ..."
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We study the quantization of Chern-Simons theory with group G coupled to dynamical sources. We first study the dynamics of Chern-Simons sources in the Hamiltonian framework. The gauge group of this system is reduced to the Cartan subgroup of G. We show that the Dirac bracket between the basic dynamical variables can be expressed in term of dynamical r−matrix of rational type. We then couple minimally these sources to Chern-Simons theory with the use of a regularisation at the location of the sources. In this case, the gauge symmetries of this theory split in two classes, the bulk gauge transformation associated to the group G and world lines gauge transformations associated to the Cartan subgroup of G. We give a complete hamiltonian analysis of this system and analyze in detail the Poisson algebras of functions invariant under the action of bulk gauge transformations. This algebra is larger than the algebra of Dirac observables because it contains in particular functions which are not invariant under reparametrization of the world line of the sources. We show that the elements of this Poisson algebra have Poisson brackets expressed in term of dynamical r−matrix of trigonometric type. This algebra is a dynamical generalization of Fock-Rosly structure. We analyze the quantization of these structures and describe different star structures on these algebras, with a special care to the case where G = SL(2, R) and G = SL(2, C)R, having in mind to apply these results to the study of the quantization of massive spinning point particles coupled to gravity with a cosmological constant in 2+1 dimensions. 1
