Operator representations of \(\mathcal{U}_ q(sl_ 2(\mathbb{R}))\)
type: Item
instance of: scholarly article
title: Operator representations of \(\mathcal{U}_ q(sl_ 2(\mathbb{R}))\)
zbMATH Open document ID: 0862.17012
DOI: 10.1007/BF00416024
author: Konrad Schmüdgen
published in: Letters in Mathematical Physics
publication date: 1996-07-16T00:00:00Z
review text: Let \(q\) be a complex number, different from 0, \(-1\), and 1. The quantized universal enveloping algebra \(U_q (sl(2))\) can be described as the algebra over \(\mathbb{C}\) with unit generated by an invertible element \(K\) and two elements \(X_+\), \(X_-\), satisfying the commutation rules \[ KK_+ =qX_+ K, \quad KX_- =q^{-1} X_-K, \quad X_+ X_-- X_-X_+ = (q- q^{-1})^{-1} (K^2- K^{-2}). \] This algebra was first introduced by \textit{P. Kulish} and \textit{N. Reshetikhin} [J. Sov. Math. 23, 2435-2441 (1983); translation from Zap. Nauchn. LOMI 101, 101-110 (1981)]. It carries a natural comultiplication making it into a Hopf algebra. Then, it is the simplest example of the quantum groups introduced by \textit{V. Drinfel'd} [Proc. Int. Congr. Math., Berkeley 1986, Vol 1, 798-820 (1987; Zbl 0476.16008)] and \textit{M. Jimbo} [Lett. Math. Phys. 10, 63-69 (1985; Zbl 0602.17005)]. There are several natural \(*\)-structures on this algebra (for associated special values of \(q)\) making this Hopf algebra into a Hopf \(*\)-algebra. The best known one is given by \(K^* =K\) and \(X_+^* = X_-\). Here one must have \(q\in \mathbb{R}\). This gives the quantization of \(su(2)\). For \(|q |= 1\), we can take \(K^* =K\) and \(X_+^*= -X_+\), \(X^*_- = -X_-\). This is considered as the quantization of \(sl(2,\mathbb{R})\). The first one gives a compact quantum group while the second one yields a non-compact quantum group. In the first case, it is not so hard to construct operator representations on Hilbert spaces (respecting the \(*\)-structure) because there are finite-dimensional representations (hence with bounded operators). In the second case, it is much more subtle to get representations because the operators are necessarily unbounded (in the nontrivial case) and in particular, one has to say precisely what is meant by an operator representation of this algebra. In this paper, the author defines and studies certain types of operator representations of the \(*\)-algebra \(U_q …
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 46L89
Mathematics Subject Classification ID: 46K10
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 888968
zbMATH Keywords: Hopf \(*\)-algebra
zbMATH Keywords: quantized universal enveloping algebra
zbMATH Keywords: compact quantum group
zbMATH Keywords: non-compact quantum group
zbMATH Keywords: operator representations
zbMATH Keywords: integrable representations
zbMATH Keywords: quantum plane
zbMATH Keywords: self-adjoint operator
zbMATH Keywords: non-self-adjoint operators
reviewed by: Alfons Van Daele
MaRDI profile type: Publication
cites work: \(q\)-deformed orthogonal and pseudo-orthogonal algebras and their representations
cites work: A q-difference analogue of \(U({\mathfrak g})\) and the Yang-Baxter equation
cites work: Q4317887
cites work: Quantum group \(\text{SU}(1,1)\rtimes\mathbb Z_ 2\) and ``super-tensor'' products
cites work: Unbounded operator algebras and representation theory.
cites work: Integrable operator representations of \(\mathbb{R}_ q^ 2\), \(X_{q,\gamma}\) and \(SL_ q(2,\mathbb{R})\)
cites work: An Operator-Theoretic Approach to a Cocycle Problem in the Complex Plane
cites work: Q4026484
cites work: Q3876492
cites work: Q4026485
cites work: Unbounded elements affiliated with \(C^*\)-algebras and non-compact quantum groups
full work available at URL: https://doi.org/10.1007/bf00416024
OpenAlex ID: W24569106
description: scientific article; zbMATH DE number 888968