Quantum anchor: \(U_q (sl(2))\) case
type: Item
instance of: scholarly article
title: Quantum anchor: \(U_q (sl(2))\) case
zbMATH Open document ID: 1094.81509
DOI: 10.1007/BF03042373
author: P. Akueson
published in: Advances in Applied Clifford Algebras
publication date: 1999-01-01T00:00:00Z
review text: This is an overview article on a noncommutative analog of a hyperboloid together with the action of the Lie algebra \(\mathfrak s\mathfrak l(2, \mathbb C)\) on its complexified tangent bundle. The basic idea is that the fundamental vector fields for the latter action generate the space of vector fields on the hyperboloid as a module over functions. Starting from the quantum group \(U_{q}(\mathfrak s\mathfrak l(2))\) (for generic \(q\)), the noncommutative hyperboloid is defined as a quotient of the tensor algebra of the fundamental representation of \(U_{q}(\mathfrak s\mathfrak l(2))\). Next, vector fields on the quantum hyperboloid are defined as a quotient of functions with values in the \(q\)-analog of the three-dimensional representation of \(\mathfrak s\mathfrak l(2, \mathbb C)\). This turns out to be a flat deformation of its classical counterpart. This left module admits a ``quantum anchor'', i.e. a right module structure such that three specific elements realize an action of the braided Lie algebra \(\mathfrak s\mathfrak l(2)_{q}\).
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 46L87
zbMATH DE Number: 1502064
reviewed by: Andreas Cap
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03042373
OpenAlex ID: W2067826178
description: scientific article; zbMATH DE number 1502064