• Citer

Involutions and Representations for Reduced Quantum Algebras

par Gutt, Simone ;Waldmann, Stefan
Référence Advances in mathematics, 224, 6, page (2583-2644)
Publication Publié, 2010

Article révisé par les pairs

• Accès en ligne
• Détails
• Contenu
• Statistiques

Résumé :

In the context of deformation quantization, there exist various procedures to deal with the quantization of a reduced space Mred. We shall be concerned here mainly with the classical Marsden-Weinstein reduction, assuming that we have a proper action of a Lie group G on a Poisson manifold M, with a moment map J for which zero is a regular value. For the quantization, we follow Bordemann et al. (2000) [6] (with a simplified approach) and build a star product *red on Mred from a strongly invariant star product * on M. The new questions which are addressed in this paper concern the existence of natural *-involutions on the reduced quantum algebra and the representation theory for such a reduced *-algebra.We assume that * is Hermitian and we show that the choice of a formal series of smooth densities on the embedded coisotropic submanifold C=J-1(0), with some equivariance property, defines a *-involution for *red on the reduced space. Looking into the question whether the corresponding *-involution is the complex conjugation (which is a *-involution in the Marsden-Weinstein context) yields a new notion of quantized modular class.We introduce a left (C∞(M){left open bracket}λ{right open bracket},*)-submodule and a right (C∞(Mred){left open bracket}λ{right open bracket},*red)-submodule Ccf∞(C){left open bracket}λ{right open bracket} of C∞(C){left open bracket}λ{right open bracket}; we define on it a C∞(Mred){left open bracket}λ{right open bracket}-valued inner product and we establish that this gives a strong Morita equivalence bimodule between C∞(Mred){left open bracket}λ{right open bracket} and the finite rank operators on Ccf∞(C){left open bracket}λ{right open bracket}. The crucial point is here to show the complete positivity of the inner product. We obtain a Rieffel induction functor from the strongly non-degenerate *-representations of (C∞(Mred){left open bracket}λ{right open bracket},*red) on pre-Hilbert right D-modules to those of (C∞(M){left open bracket}λ{right open bracket},*), for any auxiliary coefficient *-algebra D over C{left open bracket}λ{right open bracket}.