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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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Quantum affine algebras at roots of unity
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by Vyjayanthi Chari and Andrew Pressley
Represent. Theory 1 (1997), 280-328
Published electronically: August 14, 1997


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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Bibliographic Information
  • Vyjayanthi Chari
  • Affiliation: Department of Mathematics, University of California, Riverside, California 92521
  • Email:
  • Andrew Pressley
  • Affiliation: Department of Mathematics, King’s College, Strand, London WC2R 2LS, UK
  • Email:
  • 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:
  • MathSciNet review: 1463925