On the \(q\)-commutations in \(U_q({\mathfrak n})\) at roots of one
type: Item
instance of: scholarly article
title: On the \(q\)-commutations in \(U_q({\mathfrak n})\) at roots of one
zbMATH Open document ID: 0929.17014
author: Philippe Caldero
published in: Journal of Algebra
publication date: 1999-04-07T00:00:00Z
review text: Let \({\mathfrak g}\) be a semisimple Lie algebra with triangular decomposition \({\mathfrak g}= {\mathfrak n} + {\mathfrak h} + {\mathfrak n} ^{-}\), and let \(\varepsilon\) be a root of unity. Let \(U ^{+} _{\varepsilon}\) (resp. \(U ^{\text{res} +} _{\varepsilon}\)) be the nonrestricted (resp. restricted) quantum enveloping algebra of \({\mathfrak n}\). In the paper under review the author proves that {Fract} \(U ^{+} _{\varepsilon}\) is a quantum Weyl field and gives a description of the \(\varepsilon\)-center of \(U ^{+} _{\varepsilon}\). Let \(U_{\varepsilon} ^{\text{fin} +}\) be the finite part of \(U_{\varepsilon} ^{\text{res} +}\). It is proved that the \(U_{\varepsilon} ^{\text{fin} +}\)-coinvariant space of a Weyl module is \(\varepsilon\)-central.
reviewed by: Dražen Adamović
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16W35
zbMATH DE Number: 1251409
zbMATH Keywords: quantum enveloping algebra
zbMATH Keywords: root of unity
zbMATH Keywords: quantum Weyl field
MaRDI profile type: Publication
full work available at URL: jabr.1998.7603
OpenAlex ID: W2062878960
cites work: On the field of fractions of certain quantum algebras
cites work: Q3941610
cites work: Relating the approaches to quantised algebras and quantum groups
cites work: Q4290910
cites work: On the center of \(U_ q({\mathfrak n}^ +)\)
cites work: Study of the \(q\)-commutations in the algebra \(U_ q(n^ +)\)
cites work: Invariants in the enveloping algebra of a semisimple Lie algebra for the adjoint action of a nilpotent Lie subalgebra
cites work: On the Gelfand-Kirillov conjecture for quantum algebras
cites work: Q4321790
cites work: Q3972941
cites work: Quantum Coadjoint Action
cites work: Q5749435
cites work: A Generalization of the Gelfand-Kirillov Conjecture
cites work: Q4317871
cites work: Q4862129
cites work: Some applications of the quantum Weyl groups
cites work: Hall algebras and quantum groups
cites work: KILLING FORMS, HARISH-CHANDRA ISOMORPHISMS, AND UNIVERSAL R-MATRICES FOR QUANTUM ALGEBRAS
DOI: 10.1006/JABR.1998.7603
description: scientific article; zbMATH DE number 1251409