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Algebras and Representation Theory

, Volume 17, Issue 2, pp 519–538 | Cite as

On Multigraded Generalizations of Kirillov–Reshetikhin Modules

  • Angelo Bianchi
  • Vyjayanthi Chari
  • Ghislain FourierEmail author
  • Adriano Moura
Article

Abstract

We study the category of \(\mathbb Z^\ell\)-graded modules with finite-dimensional graded pieces for certain \(\mathbb Z^\ell_+\)-graded Lie algebras. We also consider certain Serre subcategories with finitely many isomorphism classes of simple objects. We construct projective resolutions for the simple modules in these categories and compute the Ext groups between simple modules. We show that the projective covers of the simple modules in these Serre subcategories can be regarded as multigraded generalizations of Kirillov–Reshetikhin modules and give a recursive formula for computing their graded characters.

Keywords

Kirillov–Reshetikhin modules Lie algebras Current algebras Projective resolutions  Ext groups Graded modules 

Mathematics Subject Classifications (2010)

17B10 17B70 16W50 

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References

  1. 1.
    Batra, P., Pal, T.: Representations of graded multiloop Lie algebras. Commun. Algebra 38, 49–67 (2010). doi: 10.1080/00927870902831201 zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bianchi, A.: Representações de hiperálgebras de laços e álgebras de multi-correntes. Ph.D. thesis, Unicamp (2012)Google Scholar
  3. 3.
    Chari, V.: Minimal affinizations of representations of quantum groups: the rank-2 case. Publ. Res. Inst. Math. Sci. 31, 873–911 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Chari, V.: On the fermionic formula and the Kirillov–Reshetikhin conjecture. Int. Math. Res. Not. 2001(12), 629–654 (2001). doi: 10.1155/S1073792801000332 CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chari, V., Greenstein, J.: Current algebras, highest weight categories and quivers. Adv. Math. 216, 811–840 (2007). doi: 10.1016/j.aim.2007.06.006 CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chari, V., Greenstein, J.: A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras. Adv. Math. 220, 1193–1221 (2009). doi: 10.1016/j.aim.2008.11.007 CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Chari, V., Greenstein, J.: Minimal affinizations as projective objects. J. Geom. Phys. 61, 594–609 (2011). doi: 10.1016/j.geomphys.2010.11.008 CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Chari, V., Kleber, M.: Symmetric functions and representations of quantum affine algebras. Contemp. Math. 297, 27–45 (2002)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Chari, V., Moura, A.: The restricted Kirillov–Reshetikhin modules for the current and twisted current algebras. Commun. Math. Phys. 266, 431–454 (2006). doi: 10.1007/s00220-006-0032-2 CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chari, V., Moura, A.: Kirillov–Reshetikhin modules associated to G 2. Contemp. Math. 442, 41–59 (2007). doi: 10.1090/conm/442 CrossRefMathSciNetGoogle Scholar
  11. 11.
    Chari, V., Fourier, G., Khandai, T.: A categorical approach to Weyl modules. Transf. Groups 15, 517–549 (2010). doi: 10.1007/s00031-010-9090-9 CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Chari, V., Khare, A., Ridenour, T.: Faces of polytopes and Koszul algebras. J. Pure Appl. Algebra 216, 1611–1625 (2012). doi: 10.1016/j.jpaa.2011.10.014 CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Di Francesco, P., Kedem, R.: Proof of the combinatorial Kirillov–Reshetikhin conjecture. Int. Math. Res. Not. 2008, 6–57 (2008). doi: 10.1093/imrn/rnn006 MathSciNetGoogle Scholar
  14. 14.
    Feigin, B., Loktev, S.: Multidimensional Weyl modules and symmetric functions. Commun. Math. Phys. 251, 427–445 (2004). doi: 10.1007/s00220-004-1166-8 CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Ferreira, G.: Representações de álgebras de correntes e álgebras de Koszul. M.Sc. Dissertation, Unicamp (2012)Google Scholar
  16. 16.
    Fourier, G., Littelmann, P.: Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions. Adv. Math. 211, 566–593 (2007). doi: 10.1016/j.aim.2006.09.002 CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Fourier, G., Okado, M., Schilling, A.: Kirillov–Reshetikhin crystals for nonexceptional types. Adv. Math. 222, 1080–1116 (2009). doi: 10.1016/j.aim.2009.05.020 CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Yamada, Y.: Remarks on fermionic formula. Contemp. Math. 248, 243–291 (1999)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Tsuboi, Z.: Paths, crystals and fermionic formulae. Prog. Math. Phys. 23, 205–272 (2002)MathSciNetGoogle Scholar
  20. 20.
    Hernandez, D.: Kirillov–Reshetikhin conjecture: the general case. Int. Math. Res. Not. 2010(1), 149–193 (2010). doi: 10.1093/imrn/rnp121 zbMATHGoogle Scholar
  21. 21.
    Khare, A., Ridenour, T.: Faces of weight polytopes and and a generalization of a theorem of Vinberg. Algebr. Represent. Theory 15(3), 593–611 (2012). doi: 10.1007/s10468-010-9261-3 CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Kodera, R.: Extensions between finite-dimensional simple modules over a generalized current Lie algebra. Transf. Groups 15, 371–388 (2010). doi: 10.1007/s00031-010-9088-3 CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Kodera, R., Naoi, K.: Loewy series of Weyl modules and the Poincaré polynomials of quiver varieties. Publ. RIMS 48(3), 477–500 (2012). doi: 10.2977/PRIMS/77 CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Lau, M.: Representations of multiloop algebras. Pac. J. Math. 245, 167–184 (2010). doi: 10.2140/pjm.2010.245.167 CrossRefzbMATHGoogle Scholar
  25. 25.
    Moura, A.: Restricted limits of minimal affinizations. Pac. J. Math. 244, 359–397 (2010). doi: 10.2140/pjm.2010.244.359 CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Moura, A., Pereira, F.: Graded limits of minimal affinizations and beyond: the multiplicity free case for type E 6. Algebra Discrete Math. 12(1), 69–115 (2011)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Naito, S., Sagaki, D.: Construction of perfect crystals conjecturally corresponding to Kirillov–Reshetikhin modules over twisted quantum affine algebras. Commun. Math. Phys. 263, 749–787 (2006). doi: 10.1007/s00220-005-1515-2 CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Nakai, W., Nakanishi, T.: Paths and tableaux descriptions of Jacobi–Trudi determinant associated with quantum affine algebra of type C n. SIGMA 3, 078, 20 p. (2007). doi: 10.3842/SIGMA.2007.078 MathSciNetGoogle Scholar
  29. 29.
    Nakai, W., Nakanishi, T.: Paths and tableaux descriptions of Jacobi–Trudi determinant associated with quantum affine algebra of type D n. J. Algebr. Comb. 26, 253–290 (2007). doi: 10.1007/s10801-007-0057-4 CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Naoi, K.: Fusion products of Kirillov–Reshetikhin modules and the X = M conjecture. Adv. Math. 231, 1546–1571 (2012). doi: 10.1016/j.aim.2012.07.003 CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Naoi, K.: Demazure modules and graded limits of minimal affinizations. (2012). arXiv:1210.0175.
  32. 32.
    Neher, E., Savage, A.: Extensions and block decompositions for finite-dimensional representations of equivariant map algebras. (2011). arXiv:1103.4367
  33. 33.
    Neher, E., Savage, A., Senesi, P.: Irreducible finite-dimensional representations of equivariant map algebras. Trans. Am. Math. Soc. 364(5), 2619–2646 (2012). doi: 10.1090/S0002-9947-2011-05420-6 CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Schilling, A.: Combinatorial structure of Kirillov–Reshetikhin crystals of type \(D_n^{(1)}, B_n^{(1)}, A_{2n-1}^{(2)}\). J. Algebra 319, 2938–2962 (2008). doi: 10.1016/j.jalgebra.2007.10.020 CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Schilling, A., Tingley, P.: Demazure crystals, Kirillov–Reshetikhin crystals, and the energy function. Electron. J. Comb. 19(2), 4 (2012)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Angelo Bianchi
    • 1
  • Vyjayanthi Chari
    • 2
  • Ghislain Fourier
    • 3
    Email author
  • Adriano Moura
    • 1
  1. 1.Department of MathematicsUniversity of CampinasSPBrazil
  2. 2.Department of MathematicsUniversity of CaliforniaRiversideUSA
  3. 3.Mathematisches InstitutUniversität zu KölnKölnGermany

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