Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 030, 23 pages      arXiv:1307.3307      https://doi.org/10.3842/SIGMA.2014.030
Contribution to the Special Issue on New Directions in Lie Theory

Tilting Modules in Truncated Categories

Matthew Bennett a and Angelo Bianchi b
a) Department of Mathematics, State University of Campinas, Brazil
b) Institute of Science and Technology, Federal University of São Paulo, Brazil

Received September 05, 2013, in final form March 17, 2014; Published online March 26, 2014; Rearrangement of Sections 2, 3 and 7, reference [5] updated, misprints corrected May 02, 2014

Abstract
We begin the study of a tilting theory in certain truncated categories of modules $\mathcal G(\Gamma)$ for the current Lie algebra associated to a finite-dimensional complex simple Lie algebra, where $\Gamma = P^+ \times J$, $J$ is an interval in $\mathbb Z$, and $P^+$ is the set of dominant integral weights of the simple Lie algebra. We use this to put a tilting theory on the category $\mathcal G(\Gamma')$ where $\Gamma' = P' \times J$, where $P'\subseteq P^+$ is saturated. Under certain natural conditions on $\Gamma'$, we note that $\mathcal G(\Gamma')$ admits full tilting modules.

Key words: current algebra; tilting module; Serre subcategory.

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References

  1. Ardonne E., Kedem R., Fusion products of Kirillov-Reshetikhin modules and fermionic multiplicity formulas, J. Algebra 308 (2007), 270-294, math.RT/0602177.
  2. Bennett M., Berenstein A., Chari V., Khoroshkin A., Loktev S., Macdonald polynomials and BGG reciprocity for current algebras, Selecta Math. (N.S.) 20 (2014), 585-607, arXiv:1207.2446.
  3. Bennett M., Chari V., Tilting modules for the current algebra of a simple Lie algebra, in Recent Developments in Lie Algebras, Groups and Representation Theory, Proc. Sympos. Pure Math., Vol. 86, Amer. Math. Soc., Providence, RI, 2012, 75-97, arXiv:1202.6050.
  4. Bennett M., Chari V., Manning N., BGG reciprocity for current algebras, Adv. Math. 231 (2012), 276-305, arXiv:1106.0347.
  5. Bianchi A., Chari V., Fourier G., Moura A., On multigraded generalizations of Kirillov-Reshetikhin modules, Algebr. Represent. Theory, 17 (2014), 519-538, arXiv:1208.3236.
  6. Chari V., Fourier G., Khandai T., A categorical approach to Weyl modules, Transform. Groups 15 (2010), 517-549, arXiv:0906.2014.
  7. Chari V., Greenstein J., Current algebras, highest weight categories and quivers, Adv. Math. 216 (2007), 811-840, math.RT/0612206.
  8. Chari V., Greenstein J., A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras, Adv. Math. 220 (2009), 1193-1221, arXiv:0808.1463.
  9. Chari V., Greenstein J., Minimal affinizations as projective objects, J. Geom. Phys. 61 (2011), 594-609, arXiv:1009.4494.
  10. Chari V., Ion B., BGG reciprocity for current algebras, arXiv:1307.1440.
  11. Chari V., Khare A., Ridenour T., Faces of polytopes and Koszul algebras, J. Pure Appl. Algebra 216 (2012), 1611-1625, arXiv:1105.2840.
  12. Chari V., Loktev S., Weyl, Demazure and fusion modules for the current algebra of ${\mathfrak{sl}}_{r+1}$, Adv. Math. 207 (2006), 928-960, math.QA/0502165.
  13. Chari V., Moura A., The restricted Kirillov-Reshetikhin modules for the current and twisted current algebras, Comm. Math. Phys. 266 (2006), 431-454, math.RT/0507584.
  14. Chari V., Moura A., Kirillov-Reshetikhin modules associated to $G_2$, in Lie Algebras, Vertex Operator Algebras and their Applications, Contemp. Math., Vol. 442, Amer. Math. Soc., Providence, RI, 2007, 41-59, math.RT/0604281.
  15. Chari V., Pressley A., Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001), 191-223, math.QA/0004174.
  16. Cline E., Parshall B., Scott L., Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85-99.
  17. Di Francesco P., Kedem R., Proof of the combinatorial Kirillov-Reshetikhin conjecture, Int. Math. Res. Not. 2008 (2008), no. 7, Art. ID rnn006, 57 pages, arXiv:0710.4415.
  18. Donkin S., Tilting modules for algebraic groups and finite dimensional algebras, in Handbook of Tilting Theory, London Math. Soc. Lecture Note Ser., Vol. 332, Cambridge University Press, Cambridge, 2007, 215-257.
  19. Fourier G., Kus D., Demazure modules and Weyl modules: the twisted current case, Trans. Amer. Math. Soc. 365 (2013), 6037-6064, arXiv:1108.5960.
  20. Fourier G., Littelmann P., Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math. 211 (2007), 566-593, math.RT/0509276.
  21. Khare A., Ridenour T., Faces of weight polytopes and a generalization of a theorem of Vinberg, Algebr. Represent. Theory 15 (2012), 593-611, arXiv:1005.1114.
  22. Kodera R., Naoi K., Loewy series of Weyl modules and the Poincaré polynomials of quiver varieties, Publ. Res. Inst. Math. Sci. 48 (2012), 477-500, arXiv:1103.4207.
  23. Mathieu O., Tilting modules and their applications, in Analysis on homogeneous spaces and Representation Theory of Lie Groups, Okayama-Kyoto (1997), Adv. Stud. Pure Math., Vol. 26, Math. Soc. Japan, Tokyo, 2000, 145-212.
  24. Naoi K., Weyl modules, Demazure modules and finite crystals for non-simply laced type, Adv. Math. 229 (2012), 875-934, arXiv:1012.5480.
  25. Naoi K., Fusion products of Kirillov-Reshetikhin modules and the $X=M$ conjecture, Adv. Math. 231 (2012), 1546-1571, arXiv:1109.2450.

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