Quantum affine algebras at roots of unity

Authors:
Vyjayanthi Chari and Andrew Pressley

Journal:
Represent. Theory **1** (1997), 280-328

MSC (1991):
Primary 17B67

Published electronically:
August 14, 1997

MathSciNet review:
1463925

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the quantized universal enveloping algebra of the affine Lie algebra associated to a finite-dimensional complex simple Lie algebra , and let be the -subalgebra of generated by the -divided powers of the Chevalley generators. Let be the Hopf algebra obtained from by specialising to a non-zero complex number of odd order. We classify the finite-dimensional irreducible representations of in terms of highest weights. We also give a ``factorisation'' theorem for such representations: namely, any finite-dimensional irreducible representation of is isomorphic to a tensor product of two representations, one factor being the pull-back of a representation of by Lusztig's Frobenius homomorphism , the other factor being an irreducible representation of the Frobenius kernel. Finally, we give a concrete construction of all of the finite-dimensional irreducible representations of . The proofs make use of several interesting new identities in .

**1.**Jonathan Beck,*Braid group action and quantum affine algebras*, Comm. Math. Phys.**165**(1994), no. 3, 555–568. MR**1301623****2.**Jonathan Beck and Victor G. Kac,*Finite-dimensional representations of quantum affine algebras at roots of unity*, J. Amer. Math. Soc.**9**(1996), no. 2, 391–423. MR**1317228**, 10.1090/S0894-0347-96-00183-X**3.**Vyjayanthi Chari and Andrew Pressley,*New unitary representations of loop groups*, Math. Ann.**275**(1986), no. 1, 87–104. MR**849057**, 10.1007/BF01458586**4.**Vyjayanthi Chari and Andrew Pressley,*Quantum affine algebras*, Comm. Math. Phys.**142**(1991), no. 2, 261–283. MR**1137064****5.**Vyjayanthi Chari and Andrew Pressley,*A guide to quantum groups*, Cambridge University Press, Cambridge, 1995. Corrected reprint of the 1994 original. MR**1358358****6.**Vyjayanthi Chari and Andrew Pressley,*Quantum affine algebras and their representations*, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 59–78. MR**1357195****7.**V. Chari and A. N. Pressley,*Yangians, integrable quantum systems and Dorey's rule*, Comm. Math. Phys.**181**(1996), 265-302. CMP**1 414 834****8.**Vyjayanthi Chari and Andrew Pressley,*Minimal affinizations of representations of quantum groups: the simply laced case*, J. Algebra**184**(1996), no. 1, 1–30. MR**1402568**, 10.1006/jabr.1996.0247**9.**V. Chari and A. N. Pressley,*Quantum affine algebras and rationality*, Proceedings of the NATO Advanced Study Institute, Cargese, 1996, Plenum Press, New York and London, 1997.**10.**Corrado De Concini and Victor G. Kac,*Representations of quantum groups at roots of 1*, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 471–506. MR**1103601****11.**C. De Concini, V. G. Kac, and C. Procesi,*Quantum coadjoint action*, J. Amer. Math. Soc.**5**(1992), no. 1, 151–189. MR**1124981**, 10.1090/S0894-0347-1992-1124981-X**12.**V. G. Drinfel′d,*A new realization of Yangians and of quantum affine algebras*, Dokl. Akad. Nauk SSSR**296**(1987), no. 1, 13–17 (Russian); English transl., Soviet Math. Dokl.**36**(1988), no. 2, 212–216. MR**914215****13.**Howard Garland,*The arithmetic theory of loop algebras*, J. Algebra**53**(1978), no. 2, 480–551. MR**502647**, 10.1016/0021-8693(78)90294-6

Howard Garland,*Erratum: “The arithmetic theory of loop algebras” [J. Algebra 53 (1978), no. 2, 480–551; MR 80a:17012]*, J. Algebra**63**(1980), no. 1, 285. MR**568577**, 10.1016/0021-8693(80)90038-1**14.**N.-H. Jing,*On Drinfeld realization of quantum affine algebras*, preprint q-alg/9610035.**15.**George Lusztig,*Introduction to quantum groups*, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR**1227098**

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (1991):
17B67

Retrieve articles in all journals with MSC (1991): 17B67

Additional Information

**Vyjayanthi Chari**

Affiliation:
Department of Mathematics, University of California, Riverside, California 92521

Email:
chari@math.ucr.edu

**Andrew Pressley**

Affiliation:
Department of Mathematics, King’s College, Strand, London WC2R 2LS, UK

Email:
anp@mth.kcl.ac.uk

DOI:
http://dx.doi.org/10.1090/S1088-4165-97-00030-7

Received by editor(s):
April 30, 1997

Published electronically:
August 14, 1997

Additional Notes:
The first author was partially supported by NATO and EPSRC (GR/K65812)

The second author was partially supported by NATO and EPSRC (GR/L26216)

Article copyright:
© Copyright 1997
American Mathematical Society