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Commutator Representations of Covariant Differential Calculi on Quantum Groups

Konrad Schmuedgen
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Abstract

Let (G,d) be a first order differential *-calculus on a *-algebra A. We say that a pair (π,F) of a *-representation πof 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)=π(a)i[F,π(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

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BibTeXRIS

Konrad Schmuedgen. 2001-05-30. Commutator Representations of Covariant Differential Calculi on Quantum Groups. https://arxiv.org/abs/math/0105251

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