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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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##### References:
 [1] Ara, P., Morita equivalence for rings with involution, Algebr. represent. theory, 2, 227-247, (1999) · Zbl 0944.16005 [2] Arnal, D.; Cortet, J.C.; Molin, P.; Pinczon, G., Covariance and geometrical invariance in ∗-quantization, J. math. phys., 24, 2, 276-283, (1983) · Zbl 0515.22015 [3] Basart, H.; Lichnerowicz, A., Conformal symplectic geometry, deformations, rigidity and geometrical (KMS) conditions, Lett. math. phys., 10, 167-177, (1985) · Zbl 0589.53037 [4] Bayen, F.; Flato, M.; Frønsdal, C.; Lichnerowicz, A.; Sternheimer, D., Deformation theory and quantization, Ann. phys., 111, 61-151, (1978) · Zbl 0377.53025 [5] Bordemann, M., (bi)modules, morphisms, and reduction of star-products: the symplectic case, foliations, and obstructions, Trav. math., 16, 9-40, (2005) · Zbl 1099.53061 [6] Bordemann, M.; Herbig, H.-C.; Waldmann, S., BRST cohomology and phase space reduction in deformation quantization, Comm. math. phys., 210, 107-144, (2000) · Zbl 0961.53046 [7] Bordemann, M.; Neumaier, N.; Waldmann, S., Homogeneous Fedosov star products on cotangent bundles I: Weyl and standard ordering with differential operator representation, Comm. math. phys., 198, 363-396, (1998) · Zbl 0968.53056 [8] Bordemann, M.; Neumaier, N.; Waldmann, S.; Weiß, S., Deformation quantization of surjective submersions and principal fibre bundles, (2007), 32 pages [9] Bursztyn, H.; Waldmann, S., Algebraic Rieffel induction, formal Morita equivalence and applications to deformation quantization, J. geom. phys., 37, 307-364, (2001) · Zbl 1039.46052 [10] Bursztyn, H.; Waldmann, S., The characteristic classes of Morita equivalent star products on symplectic manifolds, Comm. math. phys., 228, 103-121, (2002) · Zbl 1036.53068 [11] Bursztyn, H.; Waldmann, S., Completely positive inner products and strong Morita equivalence, Pacific J. math., 222, 201-236, (2005) · Zbl 1111.53071 [12] Cattaneo, A.S.; Felder, G., Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model, Lett. math. phys., 69, 157-175, (2004) · Zbl 1065.53063 [13] Cattaneo, A.S.; Felder, G., Relative formality theorem and quantisation of coisotropic submanifolds, Adv. math., 208, 521-548, (2007) · Zbl 1106.53060 [14] Dirac, P.A.M., Lectures on quantum mechanics, (1964), Belfer Graduate School of Science, Yeshiva University New York · Zbl 0141.44603 [15] Dito, G.; Sternheimer, D., Deformation quantization: genesis, developments and metamorphoses, (), 9-54 · Zbl 1014.53054 [16] Dolgushev, V.A., Covariant and equivariant formality theorems, Adv. math., 191, 147-177, (2005) · Zbl 1116.53065 [17] Fedosov, B.V., Deformation quantization and index theory, (1996), Akademie Verlag Berlin · Zbl 0867.58061 [18] Gutt, S., An explicit ∗-product on the cotangent bundle of a Lie group, Lett. math. phys., 7, 249-258, (1983) · Zbl 0522.58019 [19] Gutt, S.; Rawnsley, J., Traces for star products on symplectic manifolds, J. geom. phys., 42, 12-18, (2002) · Zbl 1075.53097 [20] Karabegov, A.V., On the canonical normalization of a trace density of deformation quantization, Lett. math. phys., 45, 217-228, (1998) · Zbl 0943.53052 [21] Landsman, N.P., Mathematical topics between classical and quantum mechanics, Springer monogr. math., (1998), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0923.00008 [22] Lu, J.-H., Moment maps at the quantum level, Comm. math. phys., 157, 389-404, (1993) · Zbl 0801.17019 [23] Nest, R.; Tsygan, B., Algebraic index theorem, Comm. math. phys., 172, 223-262, (1995) · Zbl 0887.58050 [24] Neumaier, N., Local ν-Euler derivations and Deligne’s characteristic class of Fedosov star products and star products of special type, Comm. math. phys., 230, 271-288, (2002) · Zbl 1035.53124 [25] Rieffel, M.A., Morita equivalence for $$C^\ast$$-algebras and $$W^\ast$$-algebras, J. pure appl. math., 5, 51-96, (1974) · Zbl 0295.46099 [26] Schmüdgen, K., Unbounded operator algebras and representation theory, Oper. theory adv. appl., vol. 37, (1990), Birkhäuser Verlag Basel, Boston, Berlin [27] Waldmann, S., Poisson-geometrie und deformationsquantisierung. eine einführung, (2007), Springer-Verlag Heidelberg, Berlin, New York · Zbl 1139.53001 [28] Weinstein, A., The modular automorphism group of a Poisson manifold, J. geom. phys., 23, 379-394, (1997) · Zbl 0902.58013 [29] Willwacher, T., A counterexample to the quantizability of modules, Lett. math. phys., 81, 3, 265-280, (2007) · Zbl 1138.53069
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