Finite-dimensional representations of quantum affine algebras at roots of unity

Beck, Jonathan; Kac, Victor

doi:10.1090/S0894-0347-96-00183-X

Finite-dimensional representations of quantum affine algebras at roots of unity
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by Jonathan Beck and Victor G. Kac PDF

J. Amer. Math. Soc. 9 (1996), 391-423 Request permission

Abstract:

We describe explicitly the canonical map $\chi :$ Spec $U_{\varepsilon }(\tilde {\mathfrak {g}}) \rightarrow$ Spec $Z_{\varepsilon }$, where $U_{\varepsilon } (\tilde {\mathfrak {g}})$ is a quantum loop algebra at an odd root of unity $\varepsilon$. Here $Z_{\varepsilon }$ is the center of $U_{\varepsilon }(\tilde {\mathfrak {g}})$ and Spec $R$ stands for the set of all finite–dimensional irreducible representations of an algebra $R$. We show that Spec $Z_{\varepsilon }$ is a Poisson proalgebraic group which is essentially the group of points of $G$ over the regular adeles concentrated at $0$ and $\infty$. Our main result is that the image under $\chi$ of Spec $U_{\varepsilon }(\tilde {\mathfrak {g}})$ is the subgroup of principal adeles.

References

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 $\mathfrak {sl}_{2}$, J. Algebra 161 (1993), 291–310.
C. De Concini C. and V. G. Kac, Representations of quantum groups at roots of $1$, 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 $GL_{n}$) at roots of unity, Internat. Math. Res. Notes 4 (1994), 215–217.
M. Jimbo, A q-difference analog of $U({\mathfrak {g}})$ and the Yang–Baxter equation, Lett. Math. Phys. 10 (1985), 63–69.
—, A $q$-analog of $U(gl(N+1))$, 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.
R. Steinberg, Lectures on Chevalley groups, Yale University, 1967.
V. Tarasov, Cyclic monodromy matrices for $\mathfrak {s}l_{n}$ trigonometric $R$–matrices, Comm. Math. Phys. 158 (1993), 459–483.