×
Jansen, Stefan; Waldmann, Stefan

The \(H\)-covariant strong Picard groupoid. (English) Zbl 1108.53056

J. Pure Appl. Algebra 205, No. 3, 542-598 (2006).
The authors introduce the notion of \(H\)-covariant strong Morita equivalence for *-algebras. Basic notions of an \(H\)-covariant *-representation are etablished and various groupoid morphisms between Picard groupoids are discussed. Finally, the authors realize several Morita invariants from actions of the \(H\)-covariant strong Picard groupoid.
Reviewer: Angela Gammella (Creil)

Cited in 1 Review
Cited in 7 Documents

MSC:

53D55 Deformation quantization, star products
81S10 Geometry and quantization, symplectic methods
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16D90 Module categories in associative algebras
17B63 Poisson algebras

Keywords:

groupoids; *-algebras; *-representations; Morita equivalence
PDF BibTeX XML Cite
\textit{S. Jansen} and \textit{S. Waldmann}, J. Pure Appl. Algebra 205, No. 3, 542--598 (2006; Zbl 1108.53056)
Full Text: DOI arXiv
  • logo of hbz
  • logo of WorldCat

References:

[1] Ara, P., Morita equivalence for rings with involution, Algebra rep. theory, 2, 227-247, (1999) · Zbl 0944.16005
[2] Ara, P., Morita equivalence and pedersen ideals, Proc. AMS, 129, 4, 1041-1049, (2000) · Zbl 1058.46038
[3] 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
[4] Bass, H., Algebraic \(K\)-theory, (1968), W.A. Benjamin New York, Amsterdam · Zbl 0174.30302
[5] Bayen, F.; Flato, M.; Frønsdal, C.; Lichnerowicz, A.; Sternheimer, D., Deformation theory and quantization, Ann. phys., 111, 61-151, (1978) · Zbl 0377.53025
[6] J. Bénabou, Introduction to Bicategories, in: J. Benabou et al. (Eds.), Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1-77. · Zbl 1375.18001
[7] Bursztyn, H., Semiclassical geometry of quantum line bundles and Morita equivalence of star products, Internat. math. res. notices, 2002, 16, 821-846, (2002) · Zbl 1031.53120
[8] Bursztyn, H.; Waldmann, S., \({}^*\)-ideals and formal Morita equivalence of \({}^*\)-algebras, Internat. J. math., 12, 5, 555-577, (2001) · Zbl 1111.46303
[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] H. Bursztyn, S. Waldmann, Completely positive inner products and strong Morita equivalence, preprint (FR-THEP 2003/12) math.QA/0309402, September 2003, 36p; Pacific J. Math., to appear.
[12] Bursztyn, H.; Waldmann, S., Bimodule deformations, Picard groups and contravariant connections, K-theory, 31, 1-37, (2004) · Zbl 1054.53101
[13] Bursztyn, H.; Weinstein, A., Picard groups in Poisson geometry, Moscow math. J., 4, 39-66, (2004) · Zbl 1068.53055
[14] H. Bursztyn, A. Weinstein, Poisson geometry and Morita equivalence, in: S. Gutt, J. Rawnsley, D. Sternheimer (Eds.), Poisson Geometry, Deformation Quantisation and Group Representations. London Mathematical Society, Lecture Note Series 323, pages 1-78. · Zbl 1075.53082
[15] A. Cannas da Silva, A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, American Mathematical Society, Providence, RI, 1999. · Zbl 1135.58300
[16] Combes, F., Crossed products and Morita equivalence, Proc. London math. soc. (3), 49, 2, 289-306, (1984) · Zbl 0521.46058
[17] Connes, A., Noncommutative geometry, (1994), Academic Press San Diego, New York, London · Zbl 1106.58004
[18] Curto, R.E.; Muhly, P.S.; Williams, D.P., Cross products of strongly Morita equivalent \(C^*\)- algebras, Proc. amer. math. soc., 90, 4, 528-530, (1984) · Zbl 0508.22012
[19] Dito, G.; Sternheimer, D., Deformation quantization: genesis, developments and metamorphoses, (), 9-54 · Zbl 1014.53054
[20] S. Echterhoff, S. Kaliszewski, J. Quigg, I. Raeburn, A categorical approach to imprimitivity theorems for \(C^*\)-dynamical systems, preprint math.OA/0205322, 2002, 152p. · Zbl 1097.46042
[21] Gerstenhaber, M.; Schack, S.D., Algebraic cohomology and deformation theory, (), 13-264 · Zbl 0544.18005
[22] S. Gutt, Variations on deformation quantization, in: G. Dito, D. Sternheimer (Eds.), Conférence Moshé Flato 1999, Quantization, Deformations, and Symmetries, Mathematical Physics Studies, vol. 21, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000, pp. 217-254.
[23] Gutt, S.; Rawnsley, J., Natural star products on symplectic manifolds and quantum moment maps, Lett. math. phys., 66, 123-139, (2003) · Zbl 1064.53061
[24] Jacobson, N., Basic algebra I, (1985), Freeman and Company New York · Zbl 0557.16001
[25] S. Jansen, N. Neumaier, S. Waldmann, Covariant Morita Equivalence of Star Products, in preparation.
[26] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. I: Elementary Theory, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Providence, RI, 1997. · Zbl 0888.46039
[27] Kassel, C., Quantum groups, graduate texts in mathematics, vol. 155, (1995), Springer New York, Berlin, Heidelberg
[28] Klimyk, A.; Schmüdgen, K., Quantum groups and their representations: texts and monographs in physics, (1997), Springer Heidelberg, Berlin, New York
[29] Kustermans, J., Induced corepresentations of locally compact quantum groups, J. funct. anal., 194, 410-459, (2002) · Zbl 1038.46057
[30] Lam, T.Y., Lectures on modules and rings, graduate texts in mathematics, vol. 189, (1999), Springer Berlin, Heidelberg, New York · Zbl 0911.16001
[31] E.C. Lance, Hilbert \(C^*\)-modules: a toolkit for operator algebraists, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995.
[32] Landsman, N.P., Mathematical topics between classical and quantum mechanics, Springer monographs in mathematics, (1998), Springer Berlin, Heidelberg, New York
[33] N.P. Landsman, Bicategories of operator algebras and Poisson manifolds, in: R. Longo (Ed.), Mathematical Physics in Mathematics and Physics, Fields Institute Communications, American Mathematical Society, Providence, RI, 2001, pp. 271-286. Proceedings of the Conference, dedicated to Sergio Doplicher and John E. Roberts on the occasion of their 60th birthday, held in Siena, June 20-24, 2000. · Zbl 1023.18009
[34] Landsman, N.P., Quantized reduction as a tensor product, (), 137-180 · Zbl 1026.53051
[35] Majid, S., Foundations of quantum group theory, (1995), Cambridge University Press Cambridge · Zbl 0857.17009
[36] Moerdijk, I.; Mrčun, J., Introduction to foliations and Lie groupoids, Cambridge studies in advanced mathematics, vol. 91, (2003), Cambridge University Press Cambridge, UK · Zbl 1029.58012
[37] Morita, K., Duality for modules and its applications to the theory of rings with minimum condition, Sci. rep. Tokyo kyoiku daigaku section A, 6, 83-142, (1958) · Zbl 0080.25702
[38] Müller-Bahns, M.F.; Neumaier, N., Some remarks on \(\mathfrak{g}\)-invariant Fedosov star products and quantum momentum mappings, J. geom. phys., 50, 257-272, (2004) · Zbl 1078.53100
[39] Raeburn, I.; Williams, D.P., Morita equivalence and continuous-trace \(C^*\)-algebras, mathematical surveys and monographs, vol. 60, (1998), American Mathematical Society Providence, RI
[40] Rieffel, M.A., Induced representations of \(C^*\)-algebras, Adv. math., 13, 176-257, (1974) · Zbl 0284.46040
[41] Rieffel, M.A., Morita equivalence for \(C^*\)-algebras and \(W^*\)-algebras, J. pure. appl. math., 5, 51-96, (1974) · Zbl 0295.46099
[42] Schmüdgen, K., Unbounded operator algebras and representation theory, operator theory: advances and applications, vol. 37, (1990), Birkhäuser Verlag Basel, Boston, Berlin
[43] Schmüdgen, K.; Wagner, E., Hilbert space representations of cross product algebras, J. funct. anal., 200, 451-493, (2003) · Zbl 1029.46110
[44] S. Vaes, A new approach to induction and imprimitivity results, preprint math.QA/0407525, 2004, 39p.
[45] Waldmann, S., The Picard groupoid in deformation quantization, Lett. math. phys., 69, 223-235, (2004) · Zbl 1065.53070
[46] Waldmann, S., States and representation theory in deformation quantization, Rev. math. phys., 17, 15-75, (2005) · Zbl 1138.53316
[47] Xu, P., Morita equivalence of Poisson manifolds, Comm. math. phys., 142, 493-509, (1991) · Zbl 0746.58034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.