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

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Abstract | References | Similar Articles | Additional Information

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 $U_{q}(\hat {\mathfrak {g}})$.

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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

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