Differential calculus on quantum homogeneous spaces
type: Item
instance of: scholarly article
title: Differential calculus on quantum homogeneous spaces
zbMATH Open document ID: 1028.58004
DOI: 10.1023/A:1024496304742
author: I. Heckenberger
author: S. Kolb
published in: Letters in Mathematical Physics
publication date: 2003-08-20T00:00:00Z
review text: The quantum tangent space of a covariant first-order differential calculus (FODC) over a quantum homogeneous space is established, and the generators of FODC over the Podleś' quantum sphere \(C_q [\mathbb{S}_c^2]\) are determined as an application. An FODC over \({\mathcal B}\), an algebra over \(\mathbb{C}\) is a \({\mathcal B}\)-bimodule \(\Gamma\) together with a \(\mathbb{C}\)-linear map d\(: {\mathcal B}\to\Gamma\), where \[ \Gamma=\text{Lin}_\mathbb{C}\{adb c\mid a.b.c\in {\mathcal B}\},\quad \text{d}ab=\text{d}a b+ a\text{d}b, \] [cd. \textit{A. Klimyk} and \textit{K. Schmüdgen}, ``Quantum Groups and their Representations'', Springer, Heidelberg (1997; Zbl 0891.17010)]. Let \({\mathcal A}\) be a Hopf algebra and \(\Delta_{\mathcal B}:{\mathcal B} \to {\mathcal A}\otimes {\mathcal B}\) a left comodule structure on \({\mathcal B}\). \(\Gamma\) is called left-covariant if \(\Gamma\) possesses a left \({\mathcal A}\)-comodule structure \(\Delta_\Gamma:\Gamma\to{\mathcal A}\otimes\Gamma\) such that \[ \Delta_\Gamma (a\text{d}b c)= (\Delta_{\mathcal B}a)(\text{Id} \otimes\text{d})\Delta_{\mathcal B} b)(\Delta_{{\mathcal B}}{\mathcal C}). \] Let \({\mathcal A}=U^0_{\mathcal C}\) be the dual Hopf algebra generated by the matrix coefficients of all \(U\)-modules in \({\mathcal C}\), where \(U\) is a Hopf algebra over \(\mathbb{C}\) with bijective antipode \({\mathcal S}\) and a right coideal subalgebra \(K\), and \({\mathcal C}\) is a tensor category of finite-dimensional left \(U\)-modules (Sect.2). If \({\mathcal A}\) separates the elements of \(U\) and the antipode is bijective, then the quantum homogeneous space is the left coideal subalgebra \[ {\mathcal B}=\bigl\{b\in {\mathcal A}\mid b_{(1)}b_{(2)}(k)= 0,\;k\in K^+\bigr\} K^+= \bigl\{k-\varepsilon (k)\mid k\in K\bigr\} \] (Sect. 3), cf. \textit{E. F. Müller} and \textit{H. Schneider} [Isr. J. Math. 111, 157-190 (1999; Zbl 1001.17015)]. Let \({\mathcal B}^+= \{b\in{\mathcal B}\mid \varepsilon (b)=0\}\) and \[ {\…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 58B32
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 16T05
zbMATH DE Number: 1967857
zbMATH Keywords: quantum homogeneous spaces
zbMATH Keywords: first-order differential calculus
zbMATH Keywords: Hopf algebra
zbMATH Keywords: quantum tangent space
zbMATH Keywords: quantum sphere
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1967857