Tilting Modules in Truncated Categories

Matthew Bennett ; Angelo Bianchi

Symmetry, integrability and geometry: methods and applications, Tome 10 (2014)

Résumé

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.

Détail

Keywords: current algebra; tilting module; Serre subcategory.

Bibliographie

[1] Ardonne E., Kedem R., “Fusion products of {K}irillov–{R}eshetikhin modules and fermionic multiplicity formulas”, J. Algebra, 308 (2007), 270–294, arXiv: math.RT/0602177 | DOI | MR | Zbl

[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 | DOI | MR

[3] Bennett M., Chari V., “Tilting modules for the current algebra of a simple {L}ie algebra”, Recent Developments in {L}ie Algebras, Groups and Representation Theory, Proc. Sympos. Pure Math., 86, Amer. Math. Soc., Providence, RI, 2012, 75–97, arXiv: 1202.6050 | DOI | MR

[4] Bennett M., Chari V., Manning N., “B{GG} reciprocity for current algebras”, Adv. Math., 231 (2012), 276–305, arXiv: 1106.0347 | DOI | MR | Zbl

[5] Bianchi A., Chari V., Fourier G., Moura A., “On multigraded generalizations of Kirillov–Reshetikhin modules”, Algebr. Represent. Theory (to appear) , arXiv: 1208.3236 | DOI | MR

[6] Chari V., Fourier G., Khandai T., “A categorical approach to {W}eyl modules”, Transform. Groups, 15 (2010), 517–549, arXiv: 0906.2014 | DOI | MR | Zbl

[7] Chari V., Greenstein J., “Current algebras, highest weight categories and quivers”, Adv. Math., 216 (2007), 811–840, arXiv: math.RT/0612206 | DOI | MR | Zbl

[8] Chari V., Greenstein J., “A family of {K}oszul algebras arising from finite-dimensional representations of simple {L}ie algebras”, Adv. Math., 220 (2009), 1193–1221, arXiv: 0808.1463 | DOI | MR | Zbl

[9] Chari V., Greenstein J., “Minimal affinizations as projective objects”, J. Geom. Phys., 61 (2011), 594–609, arXiv: 1009.4494 | DOI | MR | Zbl

[10] Chari V., Ion B., BGG reciprocity for current algebras, arXiv: 1307.1440

[11] Chari V., Khare A., Ridenour T., “Faces of polytopes and {K}oszul algebras”, J. Pure Appl. Algebra, 216 (2012), 1611–1625, arXiv: 1105.2840 | DOI | MR | Zbl

[12] Chari V., Loktev S., “Weyl, {D}emazure and fusion modules for the current algebra of {${\mathfrak{sl}}_{r+1}$}”, Adv. Math., 207 (2006), 928–960, arXiv: math.QA/0502165 | DOI | MR | Zbl

[13] Chari V., Moura A., “The restricted {K}irillov–{R}eshetikhin modules for the current and twisted current algebras”, Comm. Math. Phys., 266 (2006), 431–454, arXiv: math.RT/0507584 | DOI | MR | Zbl

[14] Chari V., Moura A., “Kirillov–{R}eshetikhin modules associated to {$G_2$}”, Lie Algebras, Vertex Operator Algebras and their Applications, Contemp. Math., 442, Amer. Math. Soc., Providence, RI, 2007, 41–59, arXiv: math.RT/0604281 | DOI | MR | Zbl

[15] Chari V., Pressley A., “Weyl modules for classical and quantum affine algebras”, Represent. Theory, 5 (2001), 191–223, arXiv: math.QA/0004174 | DOI | MR | Zbl

[16] Cline E., Parshall B., Scott L., “Finite-dimensional algebras and highest weight categories”, J. Reine Angew. Math., 391 (1988), 85–99 | DOI | MR | Zbl

[17] Di Francesco P., Kedem R., “Proof of the combinatorial {K}irillov–{R}eshetikhin conjecture”, Int. Math. Res. Not., 2008:7 (2008), rnn006, 57 pp., arXiv: 0710.4415 | DOI | MR | Zbl

[18] Donkin S., “Tilting modules for algebraic groups and finite dimensional algebras”, Handbook of Tilting Theory, London Math. Soc. Lecture Note Ser., 332, Cambridge University Press, Cambridge, 2007, 215–257 | DOI | MR

[19] Fourier G., Kus D., “Demazure modules and {W}eyl modules: the twisted current case”, Trans. Amer. Math. Soc., 365 (2013), 6037–6064, arXiv: 1108.5960 | DOI | MR | Zbl

[20] Fourier G., Littelmann P., “Weyl modules, {D}emazure modules, {KR}-modules, crystals, fusion products and limit constructions”, Adv. Math., 211 (2007), 566–593, arXiv: math.RT/0509276 | DOI | MR | Zbl

[21] Khare A., Ridenour T., “Faces of weight polytopes and a generalization of a theorem of {V}inberg”, Algebr. Represent. Theory, 15 (2012), 593–611, arXiv: 1005.1114 | DOI | MR | Zbl

[22] Kodera R., Naoi K., “Loewy series of {W}eyl modules and the {P}oincaré polynomials of quiver varieties”, Publ. Res. Inst. Math. Sci., 48 (2012), 477–500, arXiv: 1103.4207 | DOI | MR | Zbl

[23] Mathieu O., “Tilting modules and their applications”, Analysis on homogeneous spaces and Representation Theory of {L}ie Groups ({O}kayama–{K}yoto, 1997), Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, 2000, 145–212 | MR | Zbl

[24] Naoi K., “Weyl modules, {D}emazure modules and finite crystals for non-simply laced type”, Adv. Math., 229 (2012), 875–934, arXiv: 1012.5480 | DOI | MR | Zbl

[25] Naoi K., “Fusion products of {K}irillov–{R}eshetikhin modules and the {$X=M$} conjecture”, Adv. Math., 231 (2012), 1546–1571, arXiv: 1109.2450 | DOI | MR | Zbl