Commutator representations of covariant differential calculi on quantum groups

Let (Gamma, 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 F if there exists a linear mapping tau: Gamma --> L(D) such that tau(adb) = pi(a)i[F, pi(b)], a, b is an element of A. Among others, it is shown that each left-covariant *-calculus F 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 SLq(n+1), O-q(n), Sp(q)(2n) with real trancendental q, then this commutator representation is faithful.