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Elliptic central characters and blocks of finite dimensional representations of quantum affine algebras


Authors: Pavel I. Etingof and Adriano A. Moura
Journal: Represent. Theory 7 (2003), 346-373
MSC (2000): Primary 20G42
DOI: https://doi.org/10.1090/S1088-4165-03-00201-2
Published electronically: August 26, 2003
MathSciNet review: 2017062
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Abstract: The category of finite dimensional (type 1) representations of a quantum affine algebra $U_q(\widehat{{\mathfrak g}})$ is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition in a direct sum of indecomposable subcategories (blocks). We define the elliptic central character of a finite dimensional (type 1) representation of $U_q(\widehat{{\mathfrak g}})$ and show that the block decomposition of this category is parametrized by these elliptic central characters.


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  • 1. Akasaka, T. and Kashiwara, M., Finite dimensional representations of quantum affine algebras. Publ. Res. Inst. Math. Sci., 33 (5):839-867, 1997. MR 99d:17017
  • 2. Beck, J., Braid group action and quantum affine algebras. Comm. Math. Phys., 165:555-568, 1993. MR 95i:17011
  • 3. Chari, V., Braid group actions and tensor products. Int. Math. Res. Not., 357-382, 2002. MR 2003a:17014
  • 4. Chari, V. and Pressley, A., Fundamental representations of Yangians and singularities of R-matrices. J. Reine Angew. Math. (Crelle), 417:87-128, 1991. MR 92h:17010
  • 5. Chari, V. and Pressley, A., Quantum affine algebras and affine Hecke algebras. Pacific J. Math., 174:295-326, 1996. MR 97i:17011
  • 6. Chari, V. and Pressley, A., A guide to quantum groups. Cambridge University Press, 1994; corrected reprint of the 1994 original. MR 95j:17010; MR 96h:17014
  • 7. Chari, V. and Pressley, A., Quantum affine algebras and their representations, in Representations of groups. CMS Conf. Proc (Banff, AB, 1994), 16:59-78, 1995. hep-th/9411145. MR 96j:17009
  • 8. Chari, V. and Pressley, A., Yangians, integrable systems and Dorey's rule. Comm. Math. Phys., 181:265-302, 1996. hep-th/9505085. MR 92h:17010
  • 9. Chari, V.-Pressley, A., Weyl modules for classical and quantum affine algebras. Represent. Theory 5:191-223, 2001. MR 2002g:17027
  • 10. Drinfeld, V., Hopf algebras and the quantum Yang-Baxter equations. Soviet Math. Dokl., 32:254-258, 1985.
  • 11. Drinfeld, V., A new realization of Yangians and quantized affine algebras. Soviet Math. Dokl., 36:212-216, 1988. MR 88j:17020
  • 12. Etingof, P. and Frenkel, I.-Kirillov Jr., A., Lectures on representation theory and Knizhnik-Zamolodchikov equations, volume 58 of Mathematical Surveys and Monographs. AMS, 1998. MR 2001b:32028
  • 13. Etingof, P. and Moura, A., On the quantum Kazhdan-Luzstig functor. preprint. QA/0203003.
  • 14. Frenkel, E. and Mukhin, E., Combinatorics of $q$-characters of finite-dimensional representations of quantum affine algebras. Comm. Math. Phys., 216:23-57, 2001. MR 2002c:17023
  • 15. Frenkel, E. and Reshetikhin, N., Deformations of ${\mathcal W}$-algebras associated to simple Lie algebras. Comm. Math. Phys., 197:1-32, 1998. MR 99k:17028
  • 16. Frenkel, E. and Reshetikhin, N., The $q$-characters of of representation of quantum affine algebras and deformations of ${\mathcal W}$-algebras. Contemporary Math., 248:163-205, 2000. MR 2002f:17022
  • 17. Frenkel, I. B. and Reshetikhin, N. Yu., Quantum affine algebras and holonomic difference equations. Comm. Math. Phys. 146 (1992), no. 1, 1-60. MR 94c:17024
  • 18. Jimbo, M.A., A $q$-difference analogue of ${U}(\mathfrak{g})$ and the Yang-Baxter equation. Lett. Math. Phys., 10:63-69, 1985. MR 86k:17008
  • 19. Jimbo, M.A., A q-analogue of ${U}(\mathfrak{gl}(n+1))$, Hecke algebra and the Yang-Baxter equation. Lett. Math. Phys., 11:257-252, 1986. MR 87k:17011
  • 20. Kashiwara, M., On level zero representations of quantized affine algebras. Duke Math. J. 112:117-195, 2002. MR 2002m:17013
  • 21. Kazhdan, D. and Soibelman, Y., Representations of quantum affine algebras. Selecta Math. (N.S.), 1 (3):537-595, 1995. MR 96m:17031
  • 22. Khoroshkin, S. and Tolstoy, V., Extremal projector and universal R-matrix for quantized contragredient Lie (super) algebras. Quantum Groups and related topics, pages 23-32, 1992. MR 94b:17006
  • 23. Moura, A., Elliptic dynamical R-matrices from the monodromy of the q-Knizhnik-Zamolodchikov equations for the standard representation of ${U}_q(\tilde{\mathfrak{sl}}_{n+1})$. preprint. rt/0112145.
  • 24. Nakajima, H., Extremal weight modules of quantum affine algebras. preprint. QA/0204183.

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Additional Information

Pavel I. Etingof
Affiliation: Massachussets Institute of Technology, 77 Massachussets Ave., Room 2-176, Cambridge, Massachusetts 02139
Email: etingof@math.mit.edu

Adriano A. Moura
Affiliation: IMECC/UNICAMP, Caixa Postal: 6065, CEP: 13083-970, Campinas SP Brazil
Email: adrianoam@ime.unicamp.br

DOI: https://doi.org/10.1090/S1088-4165-03-00201-2
Received by editor(s): April 24, 2002
Received by editor(s) in revised form: December 10, 2002
Published electronically: August 26, 2003
Dedicated: For Igor Frenkel, on the occasion of his 50th birthday
Article copyright: © Copyright 2003 American Mathematical Society

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