## Quantum affine algebras at roots of unity

HTML articles powered by AMS MathViewer

- by Vyjayanthi Chari and Andrew Pressley
- Represent. Theory
**1**(1997), 280-328 - DOI: https://doi.org/10.1090/S1088-4165-97-00030-7
- Published electronically: August 14, 1997
- PDF | Request permission

## Abstract:

Let $U_{q}(\hat {\mathfrak {g}})$ be the quantized universal enveloping algebra of the affine Lie algebra $\hat {\mathfrak {g}}$ associated to a finite-dimensional complex simple Lie algebra $\mathfrak {g}$, and let $U_{q}^{\mathrm {res}}(\hat {\mathfrak {g}})$ be the $\mathbb {C}[q,q^{-1}]$-subalgebra of $U_{q}(\hat {\mathfrak {g}})$ generated by the $q$-divided powers of the Chevalley generators. Let $U_{\epsilon }^{\mathrm {res}}(\hat {\mathfrak {g}})$ be the Hopf algebra obtained from $U_{q}^{\mathrm {res}}(\hat {\mathfrak {g}})$ by specialising $q$ to a non-zero complex number $\epsilon$ of odd order. We classify the finite-dimensional irreducible representations of $U_{\epsilon }^{\mathrm {res}}(\hat {\mathfrak {g}})$ in terms of highest weights. We also give a “factorisation” theorem for such representations: namely, any finite-dimensional irreducible representation of $U_{\epsilon }^{\mathrm {res}}(\hat {\mathfrak {g}})$ is isomorphic to a tensor product of two representations, one factor being the pull-back of a representation of $\hat {\mathfrak {g}}$ by Lusztig’s Frobenius homomorphism $\hat {\mathrm {Fr}}_{\epsilon }:U_{\epsilon }^{\mathrm {res}}(\hat {\mathfrak {g}})\to U(\hat {\mathfrak {g}})$, 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 $U_{\epsilon }^{\mathrm {res}}(\hat {sl}_{2})$. The proofs make use of several interesting new identities in $U_{q}(\hat {\mathfrak {g}})$.## References

- Jonathan Beck,
*Braid group action and quantum affine algebras*, Comm. Math. Phys.**165**(1994), no. 3, 555–568. MR**1301623**, DOI 10.1007/BF02099423 - 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**, DOI 10.1090/S0894-0347-96-00183-X - Vyjayanthi Chari and Andrew Pressley,
*New unitary representations of loop groups*, Math. Ann.**275**(1986), no. 1, 87–104. MR**849057**, DOI 10.1007/BF01458586 - Vyjayanthi Chari and Andrew Pressley,
*Quantum affine algebras*, Comm. Math. Phys.**142**(1991), no. 2, 261–283. MR**1137064**, DOI 10.1007/BF02102063 - Vyjayanthi Chari and Andrew Pressley,
*A guide to quantum groups*, Cambridge University Press, Cambridge, 1995. Corrected reprint of the 1994 original. MR**1358358** - 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**, DOI 10.1007/BF00750760 - V. Chari and A. N. Pressley,
*Yangians, integrable quantum systems and Dorey’s rule*, Comm. Math. Phys.**181**(1996), 265–302. - 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**, DOI 10.1006/jabr.1996.0247 - 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. - 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** - C. De Concini, V. G. Kac, and C. Procesi,
*Quantum coadjoint action*, J. Amer. Math. Soc.**5**(1992), no. 1, 151–189. MR**1124981**, DOI 10.1090/S0894-0347-1992-1124981-X - 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** - Howard Garland,
*The arithmetic theory of loop algebras*, J. Algebra**53**(1978), no. 2, 480–551. MR**502647**, DOI 10.1016/0021-8693(78)90294-6 - N.-H. Jing,
*On Drinfeld realization of quantum affine algebras*, preprint q-alg/9610035. - George Lusztig,
*Introduction to quantum groups*, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR**1227098**

## Bibliographic 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
- 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) - © Copyright 1997 American Mathematical Society
- Journal: Represent. Theory
**1**(1997), 280-328 - MSC (1991): Primary 17B67
- DOI: https://doi.org/10.1090/S1088-4165-97-00030-7
- MathSciNet review: 1463925