Classification of bicovariant differential calculi on quantum groups of type \(A, B, C\) and \(D\)
type: Item
instance of: scholarly article
title: Classification of bicovariant differential calculi on quantum groups of type \(A, B, C\) and \(D\)
zbMATH Open document ID: 0835.17012
DOI: 10.1007/BF02101539
author: Konrad Schmüdgen
author: Axel Schüler
published in: Communications in Mathematical Physics
publication date: 1995-06-22T00:00:00Z
review text: This paper is devoted to the study of bicovariant differential calculi on quantum groups of the types \(A_{n-1}\), \(B_n\), \(D_n\), and \(C_n\) in the sense of Faddeev, Reshetikhin and Takhtajan, in the case where the deformation parameter \(q\) is not a root of unity. The paper gives a classification of certain bicovariant calculi under the assumption that the differentials of the basic matrix entries generate the module of one forms. The main technical step is to classify certain ad-invariant right ideals in the kernel of the counit, which is done using representation theory. In an introductory section, the authors review the definitions of bicovariant differential calculi and their relation to ad-invariant right ideals of the kernel of the counit. The next part of the paper contains the classification for \(A_{n-1}\) \((n\geq 3)\), the result being that apart from one-dimensional calculi and finitely many values of \(q\), there are precisely \(2n\) calculi satisfying the conditions. The authors also analyze the classical limits of these calculi, showing that precisely two of these \(2n\) calculi have the usual (commutative) calculus on \(\mathrm{SL}(n)\) as a quotient. The second part of the paper treats simultaneously the cases of \(B_n\), \(C_n\) and \(D_n\). In this case, the authors show that except for finitely many values of \(q\), there are precisely two calculi of dimension \(N^2\), where \(N= 2n+1\) for \(B_n\) and \(N=2n\) for \(C_n\) and \(D_n\). In the classical limit, one of these two calculi has the usual calculus on the classical group as a quotient.
Mathematics Subject Classification ID: 46L85
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 16T20
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 763765
zbMATH Keywords: bicovariant differential calculus
zbMATH Keywords: quantum groups
zbMATH Keywords: ad-invariant right ideals
zbMATH Keywords: classical limits
reviewed by: Andreas Cap
MaRDI profile type: Publication
cites work: Q4273719
cites work: Braids, Link Polynomials and a New Algebra
cites work: A class of bicovariant differential calculi on Hopf algebras
cites work: Bicovariant differential calculus on quantum groups \(SU_ q(N)\) and \(SO_ q(N)\)
cites work: Non-commutative differential geometry
cites work: Q3202270
cites work: Differential calculus on quantized simple Lie groups
cites work: Quantum deformations of certain simple modules over enveloping algebras
cites work: Q3998175
cites work: Differential calculi on the quantum group GLp,q(2)
cites work: Q2640687
cites work: Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra
cites work: Q3136272
cites work: Bicovariant differential calculi on \(S_ \mu U(2)\)
cites work: Covariant differential calculus on the quantum hyperplane
cites work: Twisted \(\text{SU}(2)\) group. An example of a non-commutative differential calculus
cites work: Differential calculus on compact matrix pseudogroups (quantum groups)
full work available at URL: https://doi.org/10.1007/bf02101539
OpenAlex ID: W2057316046
description: scientific article; zbMATH DE number 763765