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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Quantum affine algebras at roots of unity
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by Vyjayanthi Chari and Andrew Pressley PDF
Represent. Theory 1 (1997), 280-328 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 $U_{q}(\hat {\mathfrak {g}})$.
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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)
  • © 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