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Involutions and representations for reduced quantum algebras. (English) Zbl 1236.53070
Summary: In the context of deformation quantization, there exist various procedures to deal with the quantization of a reduced space \(M_{\text{red}}\). 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 M. Bordemann, H.-C. Herbig and S. Waldmann [Commun. Math. Phys. 210, No. 1, 107–144 (2000; Zbl 0961.53046)] (with a simplified approach) and build a star product \(*_{\text{red}}\) on \(M_{\text{red}}\) 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 \(*_{\text{red}}\) 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 \(({\mathcal C}^\infty(M)[[\lambda]],*)\)-submodule and a right \(({\mathcal C}^\infty (M_{\text{red}})\)-submodule \({\mathcal C}^\infty_{\text{cf}}(C) [[\lambda]]\) of \(C^\infty(C)[[\lambda]]\); we define on it a \({\mathcal C}^\infty(M_{\text{red}})[[\lambda]]\)-valued inner product and we establish that this gives a strong Morita equivalence bimodule between \({\mathcal C}^\infty(M_{\text{red}})[[\lambda]]\) and the finite rank operators on \({\mathcal C}^\infty_{\text{cf}}(C)[[\lambda]]\). 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 \(({\mathcal C}^\infty(M_{\text{red}})[[\lambda]], *_{\text{red}})\) on pre-Hilbert right \({\mathcal D}\)-modules to those of \(({\mathcal C}^\infty(M)[[\lambda]],*)\), for any auxiliary coefficient \(*\)-algebra \({\mathcal D}\) over \(\mathbb{C}[[\lambda]]\).

MSC:
53D55 Deformation quantization, star products
53D20 Momentum maps; symplectic reduction
16D90 Module categories in associative algebras
81S10 Geometry and quantization, symplectic methods
53D17 Poisson manifolds; Poisson groupoids and algebroids
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