Abstract
The structure of a cotangent bundle is investigated for quantum linear groups GL q (n) and SL q (n). Using a q-version of the Cayley-Hamilton theorem we construct an extension of the algebra of differential operators on SL q (n) (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators—the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive SL q (n) type dynamical R-matrices in a surprisingly simple way. Second, we calculate an evolution operator for the model of the q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. The relation between the two operators is given by a modular functional equation for the Riemann theta function.
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Alekseev A.Yu., Faddeev L.D.: (T*G) t : A Toy Model For Conformal Field Theory. Commun. Math. Phys. 141(2), 413–422 (1991)
Alekseev, A.Yu., Faddeev, L.D.:An involution and dynamics for the q deformed quantum top. Zap. Nauchn. Semin. LOMI 200, 3 (1992) (in Russian); English translation available at http://arxiv.org/abs/hep-th/9406196, 1994
Burroughs N.: Relating the Approaches to Quantized Algebras and Quantum Groups. Commun. Math. Phys. 113, 91–117 (1990)
Chari V., Pressley A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)
Conway J.H., Sloane N.J.A.: Sphere Packings, Lattices and Groups. Springer-Verlag, Berlin-Heidelberg-New York (1993)
Drinfeld, V.G.: Quantum Groups. In: Proceedings of the Intern. Congress of Mathematics, Vol. 1 (Berkeley, 1986), p. 798. For the expanded version see J. Math. Sci. 41(2), 898–915 (1988) (translated from Zap. Nauch. Sem. LOMI 155, 18–49) (1986)
Drinfeld, V.G.: it On almost cocommutative Hopf algebras. (Russian) Algebra i Analiz 1(2), 30–46 (1989); English translation in: Leningrad Math. J. 1(2), 321–342 (1990)
Donin J., Mudrov A.: U q (sl(n))-covariant quantization of symmetric coadjoint orbits via reflection equation algebra. Contemp. Math. 315, 61–79 (2002)
Donin J., Mudrov A.: Explicit equivariant quantization on coadjoint orbits of \({{\rm {GL}}(n,{{\mathbb{C}}})}\) . Lett. Math. Phys. 62(1), 17–32 (2002)
Etingof, P., Schiffmann, O.: Lectures on the dynamical Yang-Baxter equations. In: Quantum groups and Lie theory (Durham 1999), London Math. Soc. LN series 290, Cambridge: Cambridge Univ. Press 2001
Faddeev L.D.: On the exchange matrix for WZNW model. Commun. Math. Phys. 132(1), 131–138 (1990)
Faddeev, L.D.: it Current-like variables in massive nad massless integrable models. Lectures delivered at the International School of Physics it ‘Enrico Fermi’. Varenna, Italy, 1994; available at http://arxiv.org/abs/hep-th/9408041, 1994
Faddeev L.D.: Discrete Heisenberg-Weyl group and modular group. Lett. Math. Phys. 34(3), 249–254 (1995)
Faddeev, L.D.: Modular double of a quantum group. In: Conf’erence Mosh’e Flato 1999, Quantization, Deformation, and Symmetries. Vol. I, Dordrecht: Kluwer Acad. Publ., 2000, pp. 149–156; available at http://arxiv.org/abs/math.QA/9912078, 1999
Furlan P., Hadjiivanov L.K., Isaev A.P., Ogievetsky O.V., Pyatov P.N., Todorov I.T.: Quantum matrix algebra for the SU(n) WZNW model. J. Phys. A: Math. Gen. 36, 5497–5530 (2003)
Faddeev, L.D., Reshetikhin, N.Yu., Takhtajan, L.A.: it Quantization of Lie groups and Lie algebras. (Russian) Algebra i Analiz 1(1), 178–206; (1989) English translation in: Leningrad Math. J. 1(1), 193–225 (1990)
Gerasimov, A., Kharchev, S., Lebedev, D.: it Representation theory and quantum integrability. Progr. Math. 237, Basel: Birkhäuser, 2005, pp. 133–156, available at http://arxiv.org/abs/math.QA/0402112, 2004
Gurevich, D.I.: it Algebraic aspects of the quantum Yang-Baxter equation. (Russian) Algebra i Analiz 2, 119–148 (1990); English translation in: Leningrad Math. J. 2, 801–828 (1991)
Gurevich D.I., Pyatov P.N., Saponov P.A.: Hecke symmetries and characteristic relations on reflection equation algebras. Lett. Math. Phys. 41, 255–264 (1997)
Gurevich, D.I., Pyatov, P.N., Saponov, P.A.: it Cayley-Hamilton Theorem for Quantum Matrix Algebras of GL(m|n) type. Algebra i Analiz 17(1) 160–182 (2005) (in Russian). English translation in: St. Petersburg Math. J. 17(1), 119–135 (2006)
Gurevich, D.I., Pyatov, P.N., Saponov, P.A.: it Quantum matrix algebras of the GL(m–n)type: the structure and spectral parameterization of the characteristic subalgebra. Teor. Matem. Fiz. 147(1), 14–46 (2006) (in Russian). English translation in: Theor. Math. Phys. 147(1), 460–485 (2006)
Gelfand I.M., Retakh V.S.: Determinants of matrices over noncommutative rings. Funct. Anal. Appl. 25, 91–102 (1991)
Gelfand, I.M., Retakh, V.S.: it A theory of noncommutative determinants and characteristic funstions of graphs. Funct. Anal. Appl. 26, 1–20 (1992); Publ. LACIM, Montreal: UQAM, 14, pp. 1–26
Gurevich D., Saponov P.: Quantum line bundles via Cayley-Hamilton identity. J. Phys. A: Math. Gen. 34(21), 4553–4569 (2001)
Gurevich, D., Saponov, P.: it Geometry of non-commutative orbits related to Hecke symmetries. to appear in Contemp. Math.: Joseph Donin memorial volume, available at http://arxiv.org/abs/math.QA/0411579, 2004
Hlavaty L.: Quantized braided groups. J. Math. Phys. 35, 2560–2569 (1994)
Hadjiivanov L.K., Isaev A.P., Ogievetsky O.V., Pyatov P.N., Todorov I.T.: Hecke algebraic properties of dynamical R-matrices: Application to related quantum matrix algebras. J. Math. Phys. 40(1), 427–448 (1999)
Igusa, J.: it Theta Functions. Grund. Math. Wiss. 194, Berlin-Heidelberg-New York: Springer-Verlag, 1972
Isaev A.P.: Twisted Yang-Baxter equations for linear quantum (super) groups. J. Phys. A: Math. Gen. 29, 6903–6910 (1996)
Isaev, A.P.: it Quantum groups and Yang-Baxter equations. MPIM Preprint 2004-132; available at http://www.mpim-bonn.mpg.de/Research/MPIM-Preprint-Series/
Isaev A.P., Ogievetsky O.V., Pyatov P.N.: Generalized Cayley-Hamilton-Newton identities. Czech. J. Phys. 48, 1369–1374 (1998)
Isaev A., Ogievetsky O., Pyatov P.: On quantum matrix algebras satisfying the Cayley-Hamilton-Newton identities. J. Phys. A: Math. Gen. 32, L115–L121 (1999)
Isaev A.P., Pyatov P.N.: Covariant Differential Complexes on Quantum Linear Groups. J. Phys. A: Math. Gen. 28, 2227–2246 (1995)
Jimbo M.: A q-difference analogue of U(g) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63–69 (1985)
Jimbo M.: A q-analogue of U q (gl(N + 1)), Hecke algebra and the Yang-Baxter equation. Lett. Math. Phys. 11, 247–252 (1986)
Krob D., Leclerc B.: Minor identities for quasi-determinants and quantum determinants. Commun. Math. Phys. 169(1), 1–23 (1995)
Kharchev S., Lebedev D., Semenov-Tian-Shansky M.: Unitary Representations of U q (sl(2,R)), the Modular Double and the Multiparticle q-deformed Toda Chains. Commun. Math. Phys. 225(3), 573–609 (2002)
Kulish P.P., Sklyanin E.K.: Algebraic structures related to reflection equations. J. Phys. A: Math. Gen. 25(22), 5963–5975 (1992)
Klimyk A., Schmüdgen K.: Quantum groups and their representations. Springer, Berlin (1997)
Montgomery, S.: it Hopf Algebras and their Actions on Rings. CBMS Lecture Notes Vol. 82, Providence, RI: Amer. Math. Soc., 1993
Mumford, D.: it Tata Lectures on Theta. I. Progress in Mathematics, Vol. 28, Boston, MA: Birkhäuser Boston Inc., 1983
Ogievetsky, O.: Uses of quantum spaces. In: Proc. of School Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), Contemp. Math. 294, Providence, RI: Amer. Math. Soc., 2002 pp. 161–232
Ogievetsky, O., Pyatov, P.: Lecture on Hecke algebras. In: Proc. of the International School “Symmetries and Integrable Systems” (Dubna, Russia, June 8–11, 1999), JINR, Dubna, D2,5-2000-218, pp.39-88; MPIM Preprint 2001-40, available at http://www.mpim-bonn.mpg.de/Research/MPIM-Preprint-Series/
Ogievetsky, O., Pyatov, P.: it Orthogonal and symplectic quantum matrix algebras and Cayley-Hamilton theorem for them. Preprint MPIM2005–53; http://arxiv.org/abs/math.QA/0511618, 2005
Polishchuk, A., Positselski, L.: Quadratic Algebras. University Lecture Series, 37. Providence, RI: Amer. Math. Soc., 2005
Reshetikhin, N.Yu.: it Quasitriangular Hopf algebras and invariants of tangles. (Russian) Algebra i Analiz 1 (2), 169–188 (1989); English translation in: Leningrad Math. J. 1(2), 491–513 (1990)
Reshetikhin N.Yu.: Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Lett. Math. Phys. 20, 331–335 (1990)
Reshetikhin N.Yu., Turaev V.G.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys. 127(1), 1–26 (1990)
Semenov-Tyan-Shanskii, M.A.: it Poisson-Lie groups. The quantum duality principle and the twisted quantum double. (Russian) Teor. Mat. Fiz. 93(2) 302–329 (1992); English translation in: Theor. Math. Phys. 93(2), 1292–1307 (1992)
Schupp P., Watts P., Zumino B.: Differential geometry on linear quantum groups. Lett. Math. Phys. 25(2), 139–147 (1992)
Schupp P., Watts P., Zumino B.: Bicovariant quantum algebras and quantum Lie algebras. Commun. Math. Phys. 157(2), 305–329 (1993)
Tuba I., Wenzl H.: On braided tensor categories of type BCD. J. Reine Angew. Math. 581, 31–69 (2005)
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Isaev, A.P., Pyatov, P. Spectral Extension of the Quantum Group Cotangent Bundle. Commun. Math. Phys. 288, 1137–1179 (2009). https://doi.org/10.1007/s00220-009-0785-5
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DOI: https://doi.org/10.1007/s00220-009-0785-5