Commutator Representations of Covariant Differential Calculi on Quantum Groups

Abstract

Let (Γ, d) be a first-order differential *-calculus on a *-algebra ${\mathcal{A}}$. We say that a pair (π, F) of a *-representation π of ${\mathcal{A}}$ on a dense domain ${\mathcal{D}}$ of a Hilbert space and a symmetric operator F on ${\mathcal{D}}$ gives a commutator representation of Γ if there exists a linear mapping τ: Γ → L(${\mathcal{D}}$) such that τ(adb) = π(a)i[F, π(b) ], a, b ε ${\mathcal{A}}$. Among others, it is shown that each left-covariant *-calculus Γ of a compact quantum group Hopf *-algebra ${\mathcal{A}}$ has a faithful commutator representation. For a class of bicovariant *-calculi on ${\mathcal{A}}$, there is a commutator representation such that F is the image of a central element of the quantum tangent space. If ${\mathcal{A}}$ 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 trancendental q, then this commutator representation is faithful.