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Representation Theory

ISSN 1088-4165



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
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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

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