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1

Foundations of Quantum Theory : From Classical Concepts to Operator Algebras

Resource Type: eBook.

Subjects: Algebra, Mathematical physics, Physics, Matrix theory, Quantum theory

Categories: SCIENCE / Physics / Quantum Theory, MATHEMATICS / Algebra / Linear, SCIENCE / Physics / Mathematical & Computational, SCIENCE / Physics / General, SCIENCE / Philosophy & Social Aspects

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2

Quantized semisimple Lie groups

Authors: Fioresi, Rita Yuncken, Robert

Subject Terms: keyword:quantum groups, keyword:representation theory, keyword:semisimple Lie algebras, msc:16T20, msc:17B37, msc:46L67

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Relation: mr:MR4840181; zbl:Zbl 07980756; reference:[1] Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Differential operators on the base affine space and a study of ${\mathfrak{g}}$-modules.Halsted Press [John Wiley & Sons, Inc.], New York-Toronto, Ont., 1975, 21–64. MR 0578996; reference:[2] Buffenoir, E., Roche, Ph.: Harmonic analysis on the quantum Lorentz group.Comm. Math. Phys. 207 (3) (1999), 499–555. MR 1727241, 10.1007/s002200050736; reference:[3] Čap, A., Slovák, J., Souček, V.: Bernstein-Gelfand-Gelfand sequence.Ann. of Math. (2) 154 (1) (2001), 97–113. MR 1847589, 10.2307/3062111; reference:[4] De Commer, K.: On a correspondence between ${\rm SU}_q(2),\ \widetilde{E}_q(2)$ and $\widetilde{\rm SU}_q(1,1)$.Comm. Math. Phys. 304 (1) (2011), 187–228. MR 2793934, 10.1007/s00220-011-1208-y; reference:[5] De Commer, K., Dzokou Talla, J.R.: Invariant integrals on coideals and their drinfeld doubles.arXiv:2112.07476 [math.QA], 2021. MR 4776189; reference:[6] De Commer, K., Dzokou Talla, J.R.: Quantum $sl(2,\mathbb{R})$ and its irreducible representations.arXiv:2107.04258 [math.QA], 2021. MR 4750927; reference:[7] Dixmier, J.: Enveloping algebras.Grad. Stud. Math., vol. 11, Providence, RI: AMS, American Mathematical Society, 1996. MR 1393197, 10.1090/gsm/011/02; reference:[8] Drinfel’d, V.G.: Quantum groups.Proc. Int. Congr. Math., vol. 1, Berkeley/Calif 1986, 1987, pp. 798–820. MR 0934283; reference:[9] Drinfel’d, V.G.: Quantum groups.Journal of Soviet Mathematics 41 92) (1988), 898–915. MR 0869575, 10.1007/BF01247086; reference:[10] Faddeev, L.D., Reshetikhin, N.Yu., Takhtadzhan, L.A.: Quantization of Lie groups and Lie algebras.Algebraic Analysis 1 (1989), 129–139, Dedicated to Prof. Mikio Sato on the Occas. of his Sixtieth Birthday. MR 1015339; reference:[11] Fioresi, R., Lledó, M.A.: The Minkowski and conformal superspaces. The classical and quantum descriptions.Hackensack, NJ: World Scientific, 2015. MR 3328668; reference:[12] Gavarini, F.: The global quantum duality principle.J. Reine Angew. Math. 612 (2007), 17–33. MR 2364072; reference:[13] Heckenberger, I., Kolb, S.: On the Bernstein-Gelfand-Gelfand resolution for Kac-Moody algebras and quantized enveloping algebras.Transform. Groups 12 (4) (2007), 647–655. MR 2365438, 10.1007/s00031-007-0059-2; reference:[14] Heckenberger, I., Kolb, S.: Differential forms via the Bernstein-Gelfand-Gelfand resolution for quantized irreducible flag manifolds.J. Geom. Phys. 57 (11) (2007), 2316–2344. MR 2360244, 10.1016/j.geomphys.2007.07.005; reference:[15] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces.Grad. Stud. Math., vol. 34, Providence, RI: American Mathematical Society (AMS), 2001, reprint with corrections of the 1978 original edition. MR 1834454, 10.1090/gsm/034; reference:[16] Jimbo, M.: A $q$-analogue of $U(\mathfrak{gl}(N+1))$, Hecke algebra, and the Yang-Baxter equation.Lett. Math. Phys. 11 (1986), 247–252. MR 0841713, 10.1007/BF00400222; reference:[17] Joseph, A.: Quantum groups and their primitive ideals.Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 29, Springer Verlag, Berlin, 1995. MR 1315966; reference:[18] Kassel, Ch.: Quantum groups.Grad. Texts Math., vol. 155, New York, NY: Springer Verlag, 1995. MR 1321145; reference:[19] Klimyk, A., Schmüdgen, K.: Quantum groups and their representations.Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. MR 1492989; reference:[20] Knapp, A.W.: Lie groups beyond an introduction.2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389; reference:[21] Koelink, E., Kustermans, J.: A locally compact quantum group analogue of the normalizer of $\rm SU(1,1)$ in ${\rm SL}(2,\mathbb{C})$.Comm. Math. Phys. 233 (2) (2003), 231–296. MR 1962042, 10.1007/s00220-002-0736-x; reference:[22] Kulish, P.P., Reshetikhin, N.Yu.: Quantum linear problem for the sine-gordon equation and higher representations.Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 101 (1981), 101–110. MR 0623928; reference:[23] Montgomery, S.: Hopf algebras and their actions on rings.Reg. Conf. Ser. Math., vol. 82, Providence, RI: American Mathematical Society, 1993, Expanded version of ten lectures given at the CBMS Conference on Hopf algebras and their actions on rings, which took place at DePaul University in Chicago, USA, August 10-14, 1992. MR 1243637; reference:[24] Ó Buachalla, R., Somberg, P.: Lusztig’s quantum root vectors and a Dolbeault complex for the $A$-series full quantum flag manifolds.arXiv:2312.13493 [math.QA], 2023.; reference:[25] Podleś, P., Woronowicz, S.L.: Quantum deformation of Lorentz group.Comm. Math. Phys. 130 (2) (1990), 381–431. MR 1059324, 10.1007/BF02473358; reference:[26] Sklyanin, E.K.: On an algebra generated by quadratic relations.Uspekhi Mat. Nauk 40 (1985), 214.; reference:[27] Van Daele, A.: Multiplier Hopf algebras.Trans. Amer. Math. Soc. 342 (2) (1994), 917–932. MR 1220906, 10.1090/S0002-9947-1994-1220906-5; reference:[28] Van Daele, A.: An algebraic framework for group duality.Adv. Math. 140 (2) (1998), 323–366. MR 1658585, 10.1006/aima.1998.1775; reference:[29] Varadarajan, V.S.: Lie groups, Lie algebras, and their representations.Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0376938; reference:[30] Voigt, Ch., Yuncken, R.: The plancherel formula for complex semisimple quantum groups.Ann. Sci. Éc. Norm. Sup. (to appear), https://arxiv.org/abs/1906.02672, 2019. MR 4637134; reference:[31] Voigt, Ch., Yuncken, R.: Equivariant Fredholm modules for the full quantum flag manifold of ${\rm SU}_q(3)$.Doc. Math. 20 (2015), 433–490. MR 3398718, 10.4171/dm/495; reference:[32] Voigt, Ch., Yuncken, R.: Complex semisimple quantum groups and representation theory.Lect. Notes in Math., Springer, Cham, 2020. MR 4162277; reference:[33] Woronowicz, S.L.: Compact matrix pseudogroups.Comm. Math. Phys. 111 (4) (1987), 613–665. MR 0901157, 10.1007/BF01219077; reference:[34] Woronowicz, S.L.: Twisted ${\rm SU}(2)$ group. An example of a noncommutative differential calculus.Publ. Res. Inst. Math. Sci. 23 (1) (1987), 117–181. MR 0890482, 10.2977/prims/1195176848

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3

Quantised $\mathfrak{sl}_2$-differential algebras

Authors: Krutov, Andrey Pandžić, Pavle

Subject Terms: keyword:quantum group, keyword:Clifford algebra, keyword:quantised exterior algebra, keyword:$\mathfrak{g}$-differential algebra, msc:16T20, msc:17B37, msc:81R50

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Relation: mr:MR4840182; zbl:Zbl 07980757; reference:[1] Alekseev, A., Krutov, A.: Group-valued moment maps on even and odd simple $\mathfrak{g}$-modules.In preparations.; reference:[2] Alekseev, A., Meinrenken, E.: The non-commutative Weil algebra.Invent. Math. 139 (1) (2000), 135–172, arXiv:math/9903052 [math.DG]. MR 1728878, 10.1007/s002229900025; reference:[3] Alekseev, A., Meinrenken, E.: Lie theory and the Chern-Weil homomorphism.Ann. Sci. Éc. Norm. Supér. (4) 38 (2) (2005), 303–338, arXiv:math/0308135 [math.RT]. MR 2144989; reference:[4] Aschieri, P., Castellani, L.: An introduction to noncommutative differential geometry on quantum groups.Internat. J. Modern Phys. A 8 (10) (1993), 1667–1706. MR 1216230, 10.1142/S0217751X93000692; reference:[5] Aschieri, P., Schupp, P.: Vector fields on quantum groups.Internat. J. Modern Phys. A 11 (6) (1996), 1077–1100. MR 1376230, 10.1142/S0217751X9600050X; reference:[6] Berenstein, A., Zwicknagl, S.: Braided symmetric and exterior algebras.Trans. Amer. Math. Soc. 360 (71) (2008), 3429–3472, arXiv:math/0504155 [math.QA]. MR 2386232, 10.1090/S0002-9947-08-04373-0; reference:[7] Bouarroudj, S., Krutov, A., Leites, D., Shchepochkina, I.: Non-degenerate invariant (super) symmetric bilinear forms on simple Lie (super)algebras.Algebr. Represent. Theory 21 (5) (2018), 897–941, arXiv:1806.05505 [math.RT]. MR 3855668, 10.1007/s10468-018-9802-8; reference:[8] Cartan, H.: La transgression dans un groupe de Lie et dans un espace fibré principal.Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège, 1951, pp. 57–71. MR 0042427; reference:[9] Cartan, H.: Notions d’algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie.Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson $\&$ Cie, Paris, 1951, pp. 15–27. MR 0042426; reference:[10] Cheng, S.-J.: Differentiably simple Lie superalgebras and representations of semisimple Lie superalgebras.J. Algebra 173 (1) (1995), 1–43. MR 1327359, 10.1006/jabr.1995.1076; reference:[11] Drinfeld, V. G.: Quasi-Hopf algebras.Algebra i Analiz 1 (6) (1989), 114–148. MR 1047964; reference:[12] Drinfeld, V.G.: Quantum groups,.Proceedings of the International Congress of Mathematicians, Amer. Math. Soc., Providence, RI, Berkeley, Calif., 1986, 1987, pp. 798–820. MR 0934283; reference:[13] Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories.Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2015. MR 3242743; reference:[14] Guillemin, V.W., Sternberg, S.: Supersymmetry and equivariant de Rham theory.Mathematics Past and Present, Springer-Verlag, Berlin, 1999, With an appendix containing two reprints by Henri Cartan [MR0042426 (13,107e); MR0042427 (13,107f)]. MR 1689252; reference:[15] Henriques, A., Kamnitzer, J.: Crystals and coboundary categories.Duke Math. J. 132 (2) (2006), 191–216. MR 2219257, 10.1215/S0012-7094-06-13221-0; reference:[16] Huang, J.-S., Pandžić, P.: Dirac cohomology, unitary representations and a proof of a conjecture of Vogan.J. Amer. Math. Soc. 15 (1) (2002), 185–202. MR 1862801, 10.1090/S0894-0347-01-00383-6; reference:[17] Huang, J.-S., Pandžić, P.: Dirac operators in representation theory.Mathematics: Theory $\&$ Applications, Birkhäuser Boston, Inc., Boston, MA,, 2006. Zbl 1103.22008, MR 2244116; reference:[18] Jurčo, B.: Differential calculus on quantized simple Lie groups.Lett. Math. Phys. 22 (3) (1991), 177–186. MR 1129172, 10.1007/BF00403543; reference:[19] Klimyk, A., Schmüdgen, K.: Quantum groups and their representations.Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. MR 1492989; reference:[20] Kostant, B.: Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the $\rho $-decomposition $C(\mathfrak{g}) =$ End$V_\rho \otimes C(P)$, and the $\mathfrak{g}$-module structure of $\bigwedge \mathfrak{g}$.Adv. Math. 125 (2) (1997), 275–350. MR 1434113, 10.1006/aima.1997.1608; reference:[21] Krutov, A., Pandžić, P.: Cubic Dirac operator for $U_q({\mathfrak{sl}}_2)$.arXiv:2209.09591 [math.RT].; reference:[22] Krutov, A.O., Ó Buachalla, R., Strung, K.R.: Nichols algebras and quantum principal bundles.Int. Math. Res. Not. 2023 (23) (2023), 20076–20117. MR 4675067, 10.1093/imrn/rnac366; reference:[23] Lu, J.-H.: Moment maps at the quantum level.Comm. Math. Phys. 157 (2) (1993), 389–404. MR 1244874, 10.1007/BF02099767; reference:[24] Meinrenken, E.: Clifford algebras and Lie theory.Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 58, Springer, Heidelberg, 2013. MR 3052646; reference:[25] Schupp, P., Watts, P., Zumino, B.: Cartan calculus on quantum Lie algebras.Differential geometric methods in theoretical physics (Ixtapa-Zihuatanejo, 1993), 1994, pp. 125–134. MR 1337698; reference:[26] Woronowicz, S.L.: Twisted SU(2) group. An example of a noncommutative differential calculus.(2) group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1) (1987), 117–181. MR 0890482, 10.2977/prims/1195176848

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4

Lessons from quantum field theory - Hopf algebras and spacetime geometries

Authors: A. Connes D. Kreimer The Pennsylvania State University CiteSeerX Archives

Contributors: A. Connes D. Kreimer The Pennsylvania State University CiteSeerX Archives

Source: http://arxiv.org/pdf/hep-th/9904044v1.pdf.

Subject Terms: Quantum Field Theory, Noncommutative Geometry, Renormalization, Hopf Algebras, Foliations, ODE. MSC91 81T15, 16W30, 46L87, 58B30, 58F18, 34Axx

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5

Lessons from Quantum Field Theory — Hopf Algebras and Spacetime

Authors: A. Connes D. Kreimer The Pennsylvania State University CiteSeerX Archives

Contributors: A. Connes D. Kreimer The Pennsylvania State University CiteSeerX Archives

Source: http://arxiv.org/pdf/hep-th/9904044v2.pdf.

Subject Terms: Quantum Field Theory, Noncommutative Geometry, Renormalization, Hopf Algebras, Foliations, ODE. MSC91 81T15, 16W30, 46L87, 58B30, 58F18, 34Axx

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Relation: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.263.2586

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6

Cartan geometry, supergravity and group manifold approach

Authors: François, Jordan Ravera, Lucrezia

Subject Terms: keyword:Cartan geometry, keyword:group manifold, keyword:classical gauge field theory of gravity, keyword:Cartan supergeometry, keyword:supergroup manifold, keyword:supergravity, msc:53Z05, msc:58A32, msc:58A50, msc:83-01, msc:83-03, msc:83E50

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On the quantum groups and semigroups of maps between noncommutative spaces

Authors: Sadr, Maysam Maysami

Subject Terms: keyword:Hopf-algebra, keyword:bialgebra, keyword:quantum group, keyword:noncommutative geometry, msc:16T05, msc:16T10, msc:16T20, msc:58B34

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Relation: mr:MR3633001; zbl:Zbl 06738507; reference:[1] Banica, T.: Quantum automorphism groups of small metric spaces.Pac. J. Math. 219 (2005), 27-51. Zbl 1104.46039, MR 2174219, 10.2140/pjm.2005.219.27; reference:[2] Banica, T., Bichon, J., Collins, B.: Quantum permutation groups: a survey.Noncommutative Harmonic Analysis with Applications to Probability Papers presented at the 9th Workshop, Będlewo, Poland, 2006, Banach Center Publications 78, Polish Academy of Sciences, Institute of Mathematics, Warsaw M. Bożejko et al. (2008), 13-34. Zbl 1140.46329, MR 2402345; reference:[3] Baues, H. J.: Algebraic Homotopy.Cambridge Studies in Advanced Mathematics 15, Cambridge University Press, Cambridge (1989). Zbl 0688.55001, MR 0985099; reference:[4] Brzeziński, T., Majid, S.: Quantum group gauge theory on quantum spaces.Commun. Math. Phys. 157 (1993), 591-638. Zbl 0817.58003, MR 1243712, 10.1007/BF02096884; reference:[5] Gersten, S. M.: Homotopy theory of rings.J. Algebra 19 (1971), 396-415. Zbl 0264.18009, MR 0291253, 10.1016/0021-8693(71)90098-6; reference:[6] Hovey, M., Palmieri, J. H., Strickland, N. P.: Axiomatic stable homotopy theory.Mem. Am. Math. Soc. Vol. 128 (1997), 114 pages. Zbl 0881.55001, MR 1388895, 10.1090/memo/0610; reference:[7] Jardine, J. F.: Algebraic homotopy theory.Can. J. Math. 33 (1981), 302-319. Zbl 0444.55018, MR 0617621, 10.4153/CJM-1981-025-9; reference:[8] Lam, T. Y.: Lectures on Modules and Rings.Graduate Texts in Mathematics 189, Springer, New York (1999). Zbl 0911.16001, MR 1653294, 10.1007/978-1-4612-0525-8; reference:[9] Majid, S.: Foundations of Quantum Group Theory.Cambridge Univ. Press, Cambridge (1995). Zbl 0857.17009, MR 1381692; reference:[10] May, J. P.: Picard groups, Grothendieck rings, and Burnside rings of categories.Adv. Math. 163 (2001), 1-16. Zbl 0994.18004, MR 1867201, 10.1006/aima.2001.1996; reference:[11] Milne, J. S.: Basic Theory of Affine Group Schemes.Available online: www.jmilne.org /math/CourseNotes/AGS.pdf (2012).; reference:[12] Podleś, P.: Quantum spaces and their symmetry groups.PhD Thesis, Department of Mathematical Methods in Physics Faculty of Physics, Warsaw University (1989).; reference:[13] Sadr, M. M.: A kind of compact quantum semigroups.Int. J. Math. Math. Sci. 2012 (2012), Article ID 725270, 10 pages. Zbl 1267.46079, MR 3009563, 10.1155/2012/725270; reference:[14] Skalski, A., Sołtan, P. M.: Quantum families of invertible maps and related problems.Can. J. Math. 68 (2016), 698-720. Zbl 06589338, MR 3492633, 10.4153/CJM-2015-037-9; reference:[15] Sołtan, P. M.: Quantum families of maps and quantum semigroups on finite quantum spaces.J. Geom. Phys. 59 (2009), 354-368. Zbl 1160.58007, MR 2501746, 10.1016/j.geomphys.2008.11.007; reference:[16] Sołtan, P. M.: Quantum $\rm SO(3)$ groups and quantum group actions on $M_2$.J. Noncommut. Geom. 4 (2010), 1-28. Zbl 1194.46108, MR 2575388, 10.4171/JNCG/48; reference:[17] Sołtan, P. M.: On quantum maps into quantum semigroups.Houston J. Math. 40 (2014), 779-790. Zbl 1318.46051, MR 3275623; reference:[18] Sweedler, M. E.: Hopf Algebras.Mathematics Lecture Note Series, W. A. Benjamin, New York (1969). Zbl 0194.32901, MR 0252485; reference:[19] Wang, S.: Free products of compact quantum groups.Commun. Math. Phys. 167 (1995), 671-692. Zbl 0838.46057, MR 1316765, 10.1007/BF02101540; reference:[20] Wang, S.: Quantum symmetry groups of finite spaces.Commun. Math. Phys. 195 (1998), 195-211. Zbl 1013.17008, MR 1637425, 10.1007/s002200050385; reference:[21] Woronowicz, S. L.: Pseudospaces, pseudogroups and Pontrjagin duality.Mathematical Problems in Theoretical Physics Proc. Int. Conf. on Mathematical Physics, Lausanne, 1979, Lect. Notes Phys. Vol. 116, Springer, Berlin 407-412 (1980). Zbl 03810280, MR 0582650, 10.1007/3-540-09964-6_354

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8

Homotopy theory of Hopf Galois extensions

Authors: Kassel, Christian Schneider, Hans-Juergen Institut de Recherche Mathématique Avancée (IRMA) et al.

Contributors: Kassel, Christian Schneider, Hans-Juergen Institut de Recherche Mathématique Avancée (IRMA) et al.

Source: ISSN: 0373-0956.

Subject Terms: Galois extension, Hopf algebra, quantum group, homotopy, noncommutative geometry, principal fibre bundle, MSC 16W30, 17B37, 55R10, 58B34, 81R50, 81R60, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA]

Relation: info:eu-repo/semantics/altIdentifier/arxiv/math/0402034; hal-00491119; https://hal.science/hal-00491119; ARXIV: math/0402034

Availability: https://hal.science/hal-00491119
https://doi.org/10.5802/aif.2169

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9

Deformed Field Theory on κ-spacetime

Authors: Marija Dimitrijević Larisa Jonke Lutz Möller et al.

Contributors: Marija Dimitrijević Larisa Jonke Lutz Möller et al.

Source: http://www.esi.ac.at/preprints/esi1523.pdf.

Subject Terms: MSC, 81T75 Noncommutative geometry methods

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Relation: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.148.7011; http://www.esi.ac.at/preprints/esi1523.pdf

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http://www.esi.ac.at/preprints/esi1523.pdf

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10

Quantum Theta Functions and Gabor Frames for Modulation Spaces

Authors: Yuri I. Manin Franz Luef

Source: Letters in Mathematical Physics
Letters in Mathematical Physics: A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics, vol 88, iss 1

Subject Terms: 1010 Mathematics, Physics, Statistical Physics, 1010 Mathematik, 46L89, Mathematics - Operator Algebras, Geometry, Statistical and Nonlinear Physics, Secondary 11F27, 42C15, 46L89, Group Theory and Generalizations, Primary 42C15, Mathematics - Quantum Algebra, Mathematical and Computational Physics, FOS: Mathematics, Quantum Algebra (math.QA), 14K25, Operator Algebras (math.OA), Mathematical Physics