Solutions of the qKZB equation in tensor products of finite dimensional modules over the elliptic quantum group 𝐸_{𝜏,𝜂}𝑠𝑙₂

1999 ◽  
pp. 385-396
Author(s):  
E. Mukhin ◽  
A. Varchenko
Keyword(s):  
Quantum Group ◽  
Tensor Products ◽  
Finite Dimensional ◽  
Elliptic Quantum

A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yan-an Cai ◽  
Hongjia Chen ◽  
Xiangqian Guo ◽  
Yao Ma ◽  
Mianmian Zhu
Keyword(s):  
Quantum Group ◽  
Tensor Products ◽  
Simple Modules ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Direct Sum Decomposition ◽  
Free Modules ◽  
Weight Modules

Abstract In this paper, we construct a class of new modules for the quantum group U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] . The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) -module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free modules with finite-dimensional simple modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) , and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) .

Invariants of 3-manifolds derived from universal invariants of framed links

1995 ◽  
Vol 117 (2) ◽  
pp. 259-273 ◽  
Author(s):  
Tomotada Ohtsuki
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Independent Method ◽  
Topological Invariant ◽  
Finite Dimensional

Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

Structure Induced on Subsets of a Lattice Via Information Gathered from Tensor Products

1994 ◽  
Vol 37 (1) ◽  
pp. 100-111
Author(s):  
Brent Smith
Keyword(s):  
Tensor Products ◽  
Cartesian Products ◽  
Dimensional Lattice ◽  
Finite Dimensional

AbstractSuppose that a subset of a finite dimensional lattice has the property that there are many orthogonal tensor products that are almost 1 on the set, then the set is forced to have unusual concentrations of points in small cartesian products.

scholarly journals Tensor Products and Bimorphisms

1976 ◽  
Vol 19 (4) ◽  
pp. 385-402 ◽  
Author(s):  
Bernhard Banaschewski ◽  
Evelyn Nelson
Keyword(s):  
Tensor Product ◽  
Commutative Ring ◽  
Tensor Products ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bilinear Maps ◽  
Special Situation ◽  
Dual Spaces ◽  
Finite Dimensional

The binary tensor product, for modules over a commutative ring, has two different aspects: its connection with universal bilinear maps and its adjointness to the internal hom-functor. Furthermore, in the special situation of finite-dimensional vector spaces, the tensor product can also be described in terms of dual spaces and the internal hom-functor. The aim of this paper is to investigate these relationships in the setting of arbitrary concrete categories.

Simple Cn modules with multiplicities 1 and applications

1994 ◽  
Vol 72 (7-8) ◽  
pp. 326-335 ◽  
Author(s):  
D. J. Britten ◽  
J. Hooper ◽  
F. W. Lemire
Keyword(s):  
Weyl Group ◽  
Tensor Products ◽  
Weight Space ◽  
One Dimensional ◽  
Infinite Dimensional ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Recursion Formulas ◽  
Completely Reducible ◽  
Index 2

In this paper we show that there exist exactly two nonequivalent simple infinite dimensional highest weight Cn modules having the property that every weight space is one dimensional. The tensor products of these modules with any finite-dimensional simple Cn module are proven to be completely reducible and we provide an explicit decomposition for such tensor products. As an application of these decompositions, we obtain two recursion formulas for computing the multiplicities of simple finite dimensional Cn modules. These formulas involve a sum over subgroups of index 2 in the Weyl group of Cn.

scholarly journals WAKIMOTO REALIZATION OF THE ELLIPTIC QUANTUM GROUP $U_{q,p}(\widehat{{\rm sl}_N})$

2009 ◽  
Vol 24 (30) ◽  
pp. 5561-5578
Author(s):  
TAKEO KOJIMA
Keyword(s):  
Quantum Group ◽  
Free Field ◽  
Quantum Algebra ◽  
Free Field Realization ◽  
Wakimoto Realization ◽  
Arbitrary Level ◽  
Elliptic Quantum

We construct a free field realization of the elliptic quantum algebra [Formula: see text] for arbitrary level k ≠ 0, -N. We study Drinfeld current and the screening current associated with [Formula: see text] for arbitrary level k. In the limit p → 0 this realization becomes q-Wakimoto realization for [Formula: see text].

scholarly journals Algebraic Bethe ansatz for the elliptic quantum group Eτ,η(sl2)

1996 ◽  
Vol 480 (1-2) ◽  
pp. 485-503 ◽  
Author(s):  
Giovanni Felder ◽  
Alexander Varchenko
Keyword(s):  
Quantum Group ◽  
Bethe Ansatz ◽  
Algebraic Bethe Ansatz ◽  
Elliptic Quantum

scholarly journals A categorical reconstruction of crystals and quantum groups at q = 0

2019 ◽  
Vol 70 (3) ◽  
pp. 895-925
Author(s):  
Craig Smith
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Analogous Result ◽  
Crystal Bases ◽  
Direct Sums ◽  
Finite Dimensional ◽  
Monoidal Structure

Abstract The quantum co-ordinate algebra Aq(g) associated to a Kac–Moody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite-dimensional irreducible Uq(g) modules. In this paper, we investigate whether an analogous result is true when q=0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this, we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over Z whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig’s quantum group at v=∞.

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