Quantum affine algebras at roots of unity

Authors:
Vyjayanthi Chari and Andrew Pressley

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

MSC (1991):
Primary 17B67

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

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.**J. Beck,*Braid group action and quantum affine algebras*, Comm. Math. Phys.**165**(1994), 555-568. MR**95i:17011****2.**J. Beck and V. G. Kac,*Finite-dimensional representations of quantum affine algebras at roots of unity*, J. Amer. Math. Soc.**9**(1996), 391-423. MR**96f:17015****3.**V. Chari and A. N. Pressley,*New unitary representations of loop groups*, Math. Ann.**275**(1986), 87-104. MR**88f:17029****4.**V. Chari and A. N. Pressley,*Quantum affine algebras*, Comm. Math. Phys.**142**(1991), 261-283. MR**93d:17017****5.**V. Chari and A. N. Pressley,*A Guide to Quantum Groups*, Cambridge University Press, Cambridge, UK, 1994. MR**96h:17014****6.**V. Chari and A. N. Pressley,*Quantum affine algebras and their representations*, Canadian Math. Soc. Conf. Proc., Vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 59-78. MR**96j:17009****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.**V. Chari and A. N. Pressley,*Minimal affinizations of representations of quantum groups: the simply-laced case*, J. Algebra**184**(1996), 1-30. MR**97f:17012****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.**C. De Concini and V. G. Kac,*Representations of quantum groups at roots of 1*, Operator Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory, A. Connes, M. Duflo, A. Joseph and R. Rentschler (eds.), Progress in Mathematics, Vol. 92, Birkhäuser, Boston, pp. 471-506. MR**92g:17012****11.**C. De Concini, V. G. Kac and C. Procesi,*Quantum coadjoint action*, J. Amer. Math. Soc.**5**(1992), 151-189. MR**93f:17020****12.**V. G. Drinfel'd,*A new realization of Yangians and of quantum affine algebras*, Soviet Math. Dokl**36**, 212-216. MR**88j:17020****13.**H. Garland,*The arithmetic theory of loop algebras*, J. Algebra**53**(1978), 480-551. MR**80a:17012**; MR**81d:17008****14.**N.-H. Jing,*On Drinfeld realization of quantum affine algebras*, preprint q-alg/9610035.**15.**G. Lusztig,*Introduction to Quantum Groups*, Birkhäuser, Boston, MA, 1993. MR**94m:17016**

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:
https://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