Source: http://www.ams.org/journals/jams/1996-9-02/S0894-0347-96-00183-X/home.html
Timestamp: 2019-04-26 03:42:22+00:00

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Abstract: We describe explicitly the canonical map Spec Spec , where is a quantum loop algebra at an odd root of unity . Here is the center of and Spec stands for the set of all finite--dimensional irreducible representations of an algebra . We show that Spec is a Poisson proalgebraic group which is essentially the group of points of over the regular adeles concentrated at and . Our main result is that the image under of Spec is the subgroup of principal adeles.
[AS] E. Abe and K. Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. 28 (1976), 185--198.
[Be1] J. Beck, Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (1994), 555--568.
[Be2] ------, Convex bases of PBW type for quantum affine algebras, Comm. Math. Phys. 165 (1994), 193--199.
[CP1] V. Chari and A. Pressley, Quantum affine algebras, Comm. Math. Phys. 142 (1991), 261--283.
[CP2] ------, Quantum affine algebras and their representations, Proceedings of the Banff Conference, Canad. Math. Soc., 1994 (to appear).
[Da] I. Damiani, A basis of type Poincaré--Birkhoff--Witt for the quantum algebra of , J. Algebra 161 (1993), 291--310.
[DC--K] C. De Concini C. and V. G. Kac, Representations of quantum groups at roots of , Progr. in Math., vol. 92, Birkhäuser, 1990, pp. 471--506.
[DC--K--P1] C. De Concini, V. G. Kac and C. Procesi, Quantum coadjoint action, J. Amer. Math. Soc. 5 (1992), 151--190.
[DC--K--P2] ------, Some remarkable degenerations of quantum groups, Comm. Math. Phys. 157 (1993), 405--427.
[DC--K--P3] ------, Some quantum analogues of solvable Lie groups, Proceedings of the International Colloquium on Geometry and Analysis (Bombay, 1992), Oxford Univ. Press, London and New York, 1995, pp. 41--66.
[DC--P] C. De Concini and C. Procesi, Quantum groups, Lecture Notes in Math., vol. 1565, Springer-Verlag, Berlin and New York, 1994.
[D1] V. G. Drinfel d, Quantum groups, Proc. ICM Berkeley 1 (1986), 789--820.
[D2] ------, A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl. 36 (1988), 212--216.
[G] I. Grojnowski, Representations of affine Hecke algebras (and affine quantum ) at roots of unity, Internat. Math. Res. Notes 4 (1994), 215--217.
[J1] M. Jimbo, A q-difference analog of and the Yang--Baxter equation, Lett. Math. Phys. 10 (1985), 63--69.
[J2] ------, A -analog of , Hecke algebras, and the Yang--Baxter equation, Lett. Math. Phys. 11 (1986), 247--252.
[K] V. G. Kac, Infinite dimensional Lie algebras,, Third Edition, Cambridge Univ. Press, Cambridge, 1990.
[KP] V. G. Kac and D. H. Peterson, Defining relations of certain infinite--dimensional groups, Astérisque, hors série (1985), 155--208.
[LSS] S. Levendorskii, Y. Soibelman, and V. Stukopin, The quantum Weyl group and the universal quantum R-matrix for affine Lie algebras, Lett. Math. Phys. 27 (1993), 253--264.
[L1] G. Lusztig, Introduction to quantum groups, Birkhäuser, Boston and Basel, 1993.
[L2] ------, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 257--296.
[Pa] P. Papi, Convex orderings in affine root systems, preprint.
[Re] N. Reshetikhin, Quasitriangularity of quantum groups at roots of 1, hep-th/9403105 preprint.
[Ro] M. Rosso, Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys. 117 (1988), 581--593.
[S] R. Stanley, Enumerative combinatorics, Wadsworth, Belmont, CA, 1986.
[St] R. Steinberg, Lectures on Chevalley groups, Yale University, 1967.
[T] V. Tarasov, Cyclic monodromy matrices for trigonometric --matrices, Comm. Math. Phys. 158 (1993), 459--483.
E. Abe and K. Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. 28 (1976), 185--198.
J. Beck, Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (1994), 555--568.
------, Convex bases of PBW type for quantum affine algebras, Comm. Math. Phys. 165 (1994), 193--199.
V. Chari and A. Pressley, Quantum affine algebras, Comm. Math. Phys. 142 (1991), 261--283.
------, Quantum affine algebras and their representations, Proceedings of the Banff Conference, Canad. Math. Soc., 1994 (to appear).
I. Damiani, A basis of type Poincaré--Birkhoff--Witt for the quantum algebra of , J. Algebra 161 (1993), 291--310.
C. De Concini C. and V. G. Kac, Representations of quantum groups at roots of , Progr. in Math., vol. 92, Birkhäuser, 1990, pp. 471--506.
C. De Concini, V. G. Kac and C. Procesi, Quantum coadjoint action, J. Amer. Math. Soc. 5 (1992), 151--190.
------, Some remarkable degenerations of quantum groups, Comm. Math. Phys. 157 (1993), 405--427.
------, Some quantum analogues of solvable Lie groups, Proceedings of the International Colloquium on Geometry and Analysis (Bombay, 1992), Oxford Univ. Press, London and New York, 1995, pp. 41--66.
C. De Concini and C. Procesi, Quantum groups, Lecture Notes in Math., vol. 1565, Springer-Verlag, Berlin and New York, 1994.
V. G. Drinfel d, Quantum groups, Proc. ICM Berkeley 1 (1986), 789--820.
------, A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl. 36 (1988), 212--216.
I. Grojnowski, Representations of affine Hecke algebras (and affine quantum ) at roots of unity, Internat. Math. Res. Notes 4 (1994), 215--217.
M. Jimbo, A q-difference analog of and the Yang--Baxter equation, Lett. Math. Phys. 10 (1985), 63--69.
------, A -analog of , Hecke algebras, and the Yang--Baxter equation, Lett. Math. Phys. 11 (1986), 247--252.
V. G. Kac, Infinite dimensional Lie algebras,, Third Edition, Cambridge Univ. Press, Cambridge, 1990.
V. G. Kac and D. H. Peterson, Defining relations of certain infinite--dimensional groups, Astérisque, hors série (1985), 155--208.
S. Levendorskii, Y. Soibelman, and V. Stukopin, The quantum Weyl group and the universal quantum R-matrix for affine Lie algebras, Lett. Math. Phys. 27 (1993), 253--264.
G. Lusztig, Introduction to quantum groups, Birkhäuser, Boston and Basel, 1993.
------, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 257--296.
P. Papi, Convex orderings in affine root systems, preprint.
N. Reshetikhin, Quasitriangularity of quantum groups at roots of 1, hep-th/9403105 preprint.
M. Rosso, Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys. 117 (1988), 581--593.
R. Stanley, Enumerative combinatorics, Wadsworth, Belmont, CA, 1986.
V. Tarasov, Cyclic monodromy matrices for trigonometric --matrices, Comm. Math. Phys. 158 (1993), 459--483.
Additional Notes: The first author was supported by an NSF Postdoctoral Fellowship.
The second author was supported in part by NSF grant DMS–9103792.

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