2 edition of **Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups** found in the catalog.

Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups

Vitaly O. Tarasov

- 318 Want to read
- 18 Currently reading

Published
**1997**
by Société mathématique de France in Paris
.

Written in English

- Hypergeometric functions.,
- Affine algebraic groups.,
- Quantum groups.

**Edition Notes**

Statement | V. Tarasov, A. Varchenko. |

Series | Astérisque -- 246. |

Contributions | Varchenko, A. N. |

The Physical Object | |
---|---|

Pagination | vi, 135 p. ; |

Number of Pages | 135 |

ID Numbers | |

Open Library | OL18125693M |

By quantum matrix algebras I mean algebras related to quantum groups and close in a sense to that Mat(m). These algebras have numerous applications. In particular, by using them (more precisely, the so-called reflection equation algebras) we succeeded in defining partial derivatives on the enveloping algebras U(gl(m)). We study the properties of one-dimensional hypergeometric integral solutions of the q-difference (“quantum”) analogue of the Knizhnik–Zamolodchikov–Bernard equations on show that they also obey a difference KZB heat equation in the modular parameter, give formulae for modular transformations, and prove a completeness result, by showing that the associated Fourier transform is Cited by:

1. Introduction. It has long been conjectured that there is a parallelism between the infinite dimensional (quantum) algebras and (equivariant) cohomology, |$\mathrm{K}$|-theory and elliptic cohomology [1, 2].In [3–5], finite-dimensional representations of symmetrizable Kac–Moody algebras |$\mathfrak{g}$| were constructed in terms of homology groups of quiver varieties. Tarasov and A. Varchenko, Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Asterisque, v (), arXiv:q-alg/

2. G-bundles on elliptic curves and quantum groups at roots of 1, 3. Categorications (of cluster algebras and of quantum groups), 4. Langlands duality for quantum groups, 5. Proof of (geometric) character formulas and applications. The resources would be used for the following: (1) Hiring of 2 PhD students (in and ). The book is intended as a supplement to courses in classical or quantum mechanics, electrodynamics, or any other physics course in which one encounters hypergeometric functions. It will serve as a reference for hypergeometric functions, for the relationship of hypergeometric functions to special functions, and for those areas of special Cited by:

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GEOMETRY OF g-HYPERGEOMETRIC FUNCTIONS, QUANTUM AFFINE ALGEBRAS AND ELLIPTIC QUANTUM GROUPS V. Tarasov, A. Varchenko Abstract. The trigonometric quantized Knizhnik-Zamolodchikov (qKZ) equation associated with the quantum grou Up q(sl2) is a system of lin ear difference equations with values in a tensor product of U q(sl2) Verma mod ules.

R-matrix is a matrix acting in the tensor product of two evaluation modules over the elliptic quantum group Eρ,γ(sl2), and the elliptic quantum group is an elliptic analogue of the quantum aﬃne algebra Uq(slf2) associated with sl2 [F], [FV]. The elliptic quantum group depends on two parameters: an elliptic curve C/(Z+ρZ) and Planck’sconstant γ.

Abstract:The trigonometric quantized Knizhnik-Zamolodchikov equation (qKZ equation)associated with the quantum group $U_q(sl_2)$ is a system of linear differenceequations with values in a tensor product of $U_q(sl_2)$ Verma modules. Wesolve the equation in terms of multidimensional $q$-hypergeometric functionsand define a natural isomorphism between the space of solutions and the Author: Vitaly Tarasov, Alexander Varchenko.

BibTeX @MISC{Tarasov97geometryof, author = {V. Tarasov and A. Varchenko}, title = {Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups }, year = {}}.

BibTeX @MISC{Tarasov97geometryof, author = {V. Tarasov and A. Varchenko}, title = {Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups}, year = {}}. In order to establish these results we construct a discrete Gauss-Manin connection, in particular, a suitable discrete local system, Geometry of q-hypergeometric functions homology and cohomology groups with coefficients in this local system, and identify an associated difference equation with the qKZ equationAuthor: V Tarasov and A N Varchenko.

Tarasov, A. Varchenko: Geometry of q-hypergeometric functions as a bridge between quantum affine algebras and elliptic quantum affine algebras.

In preparation Google Scholar [V1]Cited by: Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups By Vitaly Tarasov and Alexander Varchenko Download PDF ( KB)Author: Vitaly Tarasov and Alexander Varchenko.

Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups by V. Tarasov, A. Varchenko, Abstract - Cited by 39 (11 self) - Add to MetaCart.

mensional Lie algebras and groups, algebraic geometry and Hamiltonian mechanics. The physical intuition arises from quantum field theory in two dimensions, integrable models in statistical.

Quantum groups have also appeared in the loop approach at quantum gravity [MS96]. Both indicate an important role for quantum geometry in fundamental physics.

The idea of noncommutativity as a regulator has found a successful application in \fuzzy physics", where function algebras are approximated by nite dimensional algebras [Mad92, Mad95, GKP96].File Size: KB. Two currently active research areas arising from the 8VSOS model are dynamical quantum groups and elliptic hypergeometric functions.

For surveys, see and [16, Chapter 11] or, respectively. The main goal of the present work is to find a new Cited by: 6. Varchenko, Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Adv.

Ser. in Math. Phys. 21, World Sci. Publ. x+ pp. Tarasov, A. Varchenko, Geometry of q q-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque (), vi+ pp.

Add tags for "Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups". Be the first. We consider the q-hypergeometric equation with q N = 1 and α, β, γ ∈ ℤ.

We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic Author: Yoshihiro Takeyama. The p,q-hypergeometric functions are introduced.

Their relation to the basic hypergeometric functions is studied. It is emphasized that investigation of some operators of representations of quantum algebras leads to orthogonal polynomials determining their spectral measures and their by: A universal weight function for a quantum affine algebra is a family of functions with values in a quotient of its Borel subalgebra, satisfying certain coalgebraic properties.

In representations of the quantum affine algebra it gives off-shell Bethe vectors and is used in the construction of solutions of the qKZ by: Similar techniques can be applied to other algebras, e.g. the Double-affine Hecke algebras, Elliptic algebras, quantum toroidal algebras.

These lectures given in Montreal in Summer are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser Cited by: Subjects Primary: 17B Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] Secondary: 32G Moduli and deformations for ordinary differential equations (e.g.

Knizhnik-Zamolodchikov equation) [See also 34Mxx] 33D Basic orthogonal polynomials and functions associated with root Cited by: Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory. Tarasov V and Varchenko A Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups Astérisque Author: Atsushi Mukaihira.In mathematics, the Gaussian or ordinary hypergeometric function 2 F 1 (a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting is a solution of a second-order linear ordinary differential equation (ODE).

Every second-order linear ODE with three regular singular points can be transformed into this.We formulate a two-parameter generalization of the geometric Langlands correspondence, which we prove for all simply-laced Lie algebras.

It identifies the q-conformal blocks of the quantum affine algebra and the deformed W-algebra associated to two Langlands dual Lie algebras. Our proof relies on recent results in quantum K-theory of the Nakajima quiver by: