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Demazure modules and Weyl modules: The twisted current case

Authors: Ghislain Fourier and Deniz Kus
Journal: Trans. Amer. Math. Soc. 365 (2013), 6037-6064
MSC (2010): Primary 17B10; Secondary 17B65
Published electronically: June 13, 2013
MathSciNet review: 3091275
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Abstract: We study finite-dimensional respresentations of twisted current algebras and show that any graded twisted Weyl module is isomorphic to level one Demazure modules for the twisted affine Kac-Moody algebra. Using the tensor product property of Demazure modules, we obtain, by analyzing the fundamental Weyl modules, dimension and character formulas. Moreover, we prove that graded twisted Weyl modules can be obtained by taking the associated graded modules of Weyl modules for the loop algebra, which implies that its dimension and classical character are independent of the support and depend only on its classical highest weight. These results were previously known for untwisted current algebras and are new for all twisted types.

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  • 1. Jonathan Beck and Hiraku Nakajima, Crystal bases and two-sided cells of quantum affine algebras, Duke Math. J. 123 (2004), no. 2, 335-402. MR 2066942 (2005e:17020)
  • 2. Roger Carter, Lie algebras of finite and affine type, Cambridge Studies in Advanced Mathematics, vol. 96, Cambridge University Press, Cambridge, 2005. MR 2188930 (2006i:17001)
  • 3. Vyjayanthi Chari, On the fermionic formula and the Kirillov-Reshetikhin conjecture, Internat. Math. Res. Notices (2001), no. 12, 629-654. MR 1836791 (2002i:17019)
  • 4. Vyjayanthi Chari, Ghislain Fourier, and Tanusree Khandai, A categorical approach to Weyl modules, Transform. Groups 15 (2010), no. 3, 517-549. MR 2718936
  • 5. Vyjayanthi Chari, Ghislain Fourier, and Prasad Senesi, Weyl modules for the twisted loop algebras, J. Algebra 319 (2008), no. 12, 5016-5038. MR 2423816 (2009e:17018)
  • 6. Vyjayanthi Chari and Sergei Loktev, Weyl, Demazure and fusion modules for the current algebra of $ \mathfrak{sl}_{r+1}$, Adv. Math. 207 (2006), no. 2, 928-960. MR 2271991 (2008a:17029)
  • 7. Vyjayanthi Chari and Adriano Moura, The restricted Kirillov-Reshetikhin modules for the current and twisted current algebras, Comm. Math. Phys. 266 (2006), no. 2, 431-454. MR 2238884 (2007m:17035)
  • 8. Vyjayanthi Chari and Andrew Pressley, Integrable and Weyl modules for quantum affine $ {\rm sl}\sb 2$, Quantum groups and Lie theory (Durham, 1999), London Math. Soc. Lecture Note Ser., vol. 290, Cambridge Univ. Press, Cambridge, 2001, pp. 48-62. MR 1903959 (2004c:17026)
  • 9. -, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001), 191-223 (electronic). MR 1850556 (2002g:17027)
  • 10. Michel Demazure, Une nouvelle formule des caractères, Bull. Sci. Math. (2) 98 (1974), no. 3, 163-172. MR 0430001 (55:3009)
  • 11. Boris Feigin and Sergei Loktev, On generalized Kostka polynomials and the quantum Verlinde rule, Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, vol. 194, Amer. Math. Soc., Providence, RI, 1999, pp. 61-79. MR 1729359 (2002b:17007)
  • 12. -, Multi-dimensional Weyl modules and symmetric functions, Comm. Math. Phys. 251 (2004), no. 3, 427-445. MR 2102326 (2005m:17005)
  • 13. Ghislain Fourier, Tanusree Khandai, Deniz Kus, and Alistair Savage, Local weyl modules for equivariant map algebras with free abelian group actions, Journal of Algebra, Volume 350, Issue 1, 15 January 2012, Pages 386-404. MR 2859894
  • 14. Ghislain Fourier and Peter Littelmann, Tensor product structure of affine Demazure modules and limit constructions, Nagoya Math. J. 182 (2006), 171-198. MR 2235341 (2007e:17021)
  • 15. -, Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math. 211 (2007), no. 2, 566-593. MR 2323538 (2008k:17005)
  • 16. Ghislain Fourier, Nathan Manning, and Prasad Senesi, Global Weyl modules for the twisted loop algebras, Abh. Math. Semin. Univ. Hambg. 85 (2013), 53-82. MR 3055822
  • 17. William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991, A first course, Readings in Mathematics. MR 1153249 (93a:20069)
  • 18. Goro Hatayama, Atsuo Kuniba, Masato Okado, Taichiro Takagi, and Zengo Tsuboi, Paths, crystals and fermionic formulae, MathPhys odyssey, 2001, Prog. Math. Phys., vol. 23, Birkhäuser Boston, Boston, MA, 2002, pp. 205-272. MR 1903978 (2003e:17020)
  • 19. Victor G. Kac, Infinite-dimensional Lie algebras, third ed., Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
  • 20. Michael Steven Kleber, Finite dimensional representations of quantum affine algebras, ProQuest LLC, Ann Arbor, MI, 1998, Thesis (Ph.D.)-University of California, Berkeley. MR 2697975
  • 21. Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, Birkhäuser Boston Inc., Boston, MA, 2002. MR 1923198 (2003k:22022)
  • 22. Michael Lau, Representations of multiloop algebras, Pacific J. Math. 245 (2010), no. 1, 167-184. MR 2602688
  • 23. Olivier Mathieu, Construction du groupe de Kac-Moody et applications, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 5, 227-230. MR 932325 (89e:17013)
  • 24. Hiraku Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145-238. MR 1808477 (2002i:17023)
  • 25. Katsuyuki Naoi, Weyl modules, Demazure modules and finite crystals for non-simply laced type, arXiv:1012.5480.
  • 26. Erhard Neher, Alistair Savage, and Prasad Senesi, Irreducible finite-dimensional representations of equivariant map algebras, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2619-2646. MR 2888222

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Additional Information

Ghislain Fourier
Affiliation: Mathematisches Institut, Universität zu Köln, 50931 Köln, Germany

Deniz Kus
Affiliation: Mathematisches Institut, Universität zu Köln, 50931 Köln, Germany

Received by editor(s): September 12, 2011
Received by editor(s) in revised form: March 30, 2012
Published electronically: June 13, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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