Involutions and representations for reduced quantum algebras

Abstract

In the context of deformation quantization, there exist various procedures to deal with the quantization of a reduced space M red . 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 [6] (with a simplified approach) and build a star product ⋆ red on M red 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 unimodular class.We introduce a left (C ∞ (M ) [[λ]], ⋆)-submodule and a right (]-valued inner product and we establish that this gives a strong Morita equivalence bimodule between C ∞ (M red ) [[λ]] and the finite rank operators on C ∞ cf (C) [[λ]]. The crucial point is here to show the complete positivity of the inner product. We obtain a Rieffel induction functor from the strongly nondegenerate * -representations of (C ∞ (M red ) [[λ]], ⋆ red ) on pre-Hilbert right D-modules to those of (C ∞ (M ) [[λ]], ⋆) , for any auxiliary coefficient * -algebra D over [[λ]].

Twisted Quadrics and Algebraic Submanifolds in $\mathbb {R}^{n}$

“…As argued in [32], these conditions are fulfilled if 2 , for all α ∈ X , a = 1, .., k, α f a = αf a = f a α ⇔ B l (α, f a ) = 0 = B l (f a , α) ∀l ∈ N (8) (this implies that the f a are central in X , again). The quotient (7) also appears in the context of deformation quantization of Marsden-Weinstein reduction [10,37]. A more algebraic approach to deformation quantization of reduced spaces is given in the recent article [18].…”

Characteristic classes of star products on Marsden–Weinstein reduced symplectic manifolds

“…However, just as with the multitude of quantization schemes developed over time, even in one such scheme, there is typically no "universal" reduction process. Here we will investigate only one quantum reduction scheme in the context of deformation quantization [2] proposed by Bordemann, Herbig and Waldmann in [5] and further developed in [20] by Gutt and Waldmann, which is based on BRST cohomology. We will provide a brief recap to the extent needed later on in Section 2.…”

“…being the commutator with respect to . We will start by introducing the quantized Koszul operator [20]:…”

Morita theory in deformation quantization

“…Concerning the reduction aspect, one has by now a rather good understanding, starting from the BRST approach in [7], see also [5,17]. In [22] the representation theory of the reduced algebras was studied in detail, including some aspects of Morita theory. The usage of invariant star products (and even better: invariant star products with a quantum momentum map) will hopefully allow also to treat the Morita theory of star products on singular quotients, see e.g.…”