Commutator Representations of Covariant Differential Calculi on Quantum Groups
Abstract
Let (G,d) be a first order differential *-calculus on a *-algebra A. We say that a pair (\pi,F) of a *-representation \pi of A on a dense domain D of a Hilbert space and a symmetric operator F on D gives a commutator representation of G if there exists a linear mapping t:G -> L(D) such that t(adb)=\pi(a)i[F,\pi(b)], a,b in A. Among others, it is shown that each left-covariant *-calculus G of a compact quantum group Hopf *-algebra A has a faithful commutator representation. For a class of bicovariant *-calculi on A there is a commutator representation such that F is the image of a central element of the quantum tangent space. If A is the Hopf *-algebra of the compact form of one of the quantum groups SL_q(n+1), O_q(n), Sp_q(2n) with real transcendental q, then this commutator representation is faithful. Keywords: Quantum Groups; Noncommuative Geometry; Differential Calculus
Keywords
Cite
@article{arxiv.math/0105251,
title = {Commutator Representations of Covariant Differential Calculi on Quantum Groups},
author = {Konrad Schmuedgen},
journal= {arXiv preprint arXiv:math/0105251},
year = {2016}
}
Comments
LaTeX2e, 14 pages