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Weyl modules for classical and quantum affine algebras


Authors: Vyjayanthi Chari and Andrew Pressley
Journal: Represent. Theory 5 (2001), 191-223
MSC (2000): Primary 81R50, 17B67
DOI: https://doi.org/10.1090/S1088-4165-01-00115-7
Published electronically: July 5, 2001
MathSciNet review: 1850556
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Abstract:

We introduce and study the notion of a Weyl module for the classical affine algebras, these modules are universal finite-dimensional highest weight modules. We conjecture that the modules are the classical limit of a family of irreducible modules of the quantum affine algebra, and prove the conjecture in the case of $sl_2$. The conjecture implies also that the Weyl modules are the classical limits of the standard modules introduced by Nakajima and further studied by Varagnolo and Vasserot.


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  • [AK] T. Akasaka and M. Kashiwara, Finite-dimensional representations of quantum affine algebras, Publ. Res. Inst. Math. Sci. 33 (1997), no. 5, 839-867. MR 99d:17017
  • [B] J. Beck, Braid group action and quantum affine algebras, Commun. Math. Phys. 165 (1994), 555-568. MR 95i:17011
  • [BCP] J. Beck, V. Chari and A. Pressley, An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (1999), no. 3, 455-487. MR 2000g:17013
  • [C] V. Chari, Integrable representations of affine Lie algebras, Invent. Math 85 (1986), no.2, 317-335. MR 88a:17034
  • [CP1] V. Chari and A. Pressley, New unitary representations of loop groups, Math. Ann. 275 (1986), 87-104. MR 88f:17029
  • [CP2] V. Chari and A. Pressley, A new family of irreducible integrable modules for affine Lie algebras, Math. Ann. 277 (1987), 543-562. MR 88h:17022
  • [CP3] V. Chari, and A. Pressley, Quantum affine algebras, Commun. Math. Phys. 142 (1991), 261-283. MR 93d:17017
  • [CP4] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994; corrected reprint of the 1994 original. MR 95j:17010; MR 96h:17014
  • [CP5] V. Chari and A. Pressley, Quantum affine algebras and their representations, in Representations of Groups, (Banff, AB, 1994), 59-78, CMS Conf. Proc. 16, AMS, Providence, RI 1995. MR 96j:17009
  • [CP6] V. Chari and A. Pressley, Quantum affine algebras at roots of unity, Representation Theory 1 (1997), 280-328. MR 98e:17018
  • [CP7] V. Chari and A. Pressley, Integrable and Weyl modules for quantum affine $sl_2$, preprint, math. qa/007123.
  • [Dr1] V.G. Drinfeld, Hopf Algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985) 254-258.
  • [Dr2] V.G. Drinfeld, A new realization of Yangians and quantum affine algebras. Soviet Math. Dokl. 36 (1988), 212-216. MR 88j:17020
  • [G] H. Garland, The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480-551. MR 80a:17012
  • [GV] V. Ginzburg and E. Vasserot, Langlands reciprocity for affine quantum groups of type $A_n$, Int. Math. Res. Not. 3 (1993), 67-85. MR 94j:17011
  • [J] N. Jing, On Drinfeld realization of quantum affine algebras. The Monster and Lie algebras (Columbus, OH, 1996), pp. 195-206, Ohio State Univ. Math. Res. Inst. Publ., 7, de Gruyter, Berlin, 1998. MR 99j:17021
  • [FM] E. Frenkel and E. Mukhin, Combinatorics of $q$-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phys. 216 (2001), 23-57.
  • [FR] E. Frenkel and N. Reshetikhin, The $q$-characters of representations of quantum affine algebras and deformations of $W$-algebras, Contemp. Math. 248 (1999). CMP 2000:11
  • [K] M. Kashiwara, Crystal bases of the modified quantized enveloping algebra, Duke Math. J. 73 (1994), 383-413. MR 95c:17024
  • [K2] M. Kashiwara, On level zero representations of quantized affine algebras, math.qa/0010293.
  • [KS] D. Kazhdan and Y. Soibelman, Representations of quantum affine algebras, Selecta Math. (NS) 1 (1995), 537-595. MR 96m:17031
  • [L1] G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70 (1988), 237-249. MR 89k:17029
  • [L2] G. Lusztig, Introduction to quantum groups, Progress in Mathematics 110, Birkhäuser, Boston, 1993. MR 94m:17016
  • [N] H. Nakajima, $t$-analogue of the $q$-characters of finite-dimensional representations of quantum affine algebras, math.QA/0009231.
  • [VV] M. Varagnolo and E. Vasserot, Standard modules for quantum affine algebras, math.qa/0006084.

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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, Kings College, London, WC 2R, 2LS, England, United Kingdom
Email: anp@mth.kcl.ac.uk

DOI: https://doi.org/10.1090/S1088-4165-01-00115-7
Received by editor(s): August 23, 2000
Published electronically: July 5, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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