## Quantum loop modules

HTML articles powered by AMS MathViewer

- by Vyjayanthi Chari and Jacob Greenstein
- Represent. Theory
**7**(2003), 56-80 - DOI: https://doi.org/10.1090/S1088-4165-03-00168-7
- Published electronically: February 26, 2003
- PDF | Request permission

## Abstract:

We classify the simple infinite-dimensional integrable modules with finite-dimensional weight spaces over the quantized enveloping algebra of an untwisted affine algebra. We prove that these are either highest (lowest) weight integrable modules or simple submodules of a loop module of a finite-dimensional simple integrable module and describe the latter class. Their characters and crystal basis theory are discussed in a special case.## References

- Tatsuya Akasaka and Masaki Kashiwara,
*Finite-dimensional representations of quantum affine algebras*, Publ. Res. Inst. Math. Sci.**33**(1997), no. 5, 839–867. MR**1607008**, DOI 10.2977/prims/1195145020 - Jonathan Beck,
*Braid group action and quantum affine algebras*, Comm. Math. Phys.**165**(1994), no. 3, 555–568. MR**1301623**, DOI 10.1007/BF02099423 - Jonathan Beck, Vyjayanthi Chari, and Andrew Pressley,
*An algebraic characterization of the affine canonical basis*, Duke Math. J.**99**(1999), no. 3, 455–487. MR**1712630**, DOI 10.1215/S0012-7094-99-09915-5 - Vyjayanthi Chari,
*Integrable representations of affine Lie-algebras*, Invent. Math.**85**(1986), no. 2, 317–335. MR**846931**, DOI 10.1007/BF01389093 - Vyjayanthi Chari,
*Braid group actions and tensor products*, Int. Math. Res. Not.**7**(2002), 357–382. MR**1883181**, DOI 10.1155/S107379280210612X - Vyjayanthi Chari and Andrew Pressley,
*New unitary representations of loop groups*, Math. Ann.**275**(1986), no. 1, 87–104. MR**849057**, DOI 10.1007/BF01458586 - Vyjayanthi Chari and Andrew Pressley,
*Quantum affine algebras*, Comm. Math. Phys.**142**(1991), no. 2, 261–283. MR**1137064**, DOI 10.1007/BF02102063 - Vyjayanthi Chari and Andrew Pressley,
*A guide to quantum groups*, Cambridge University Press, Cambridge, 1995. Corrected reprint of the 1994 original. MR**1358358** - Vyjayanthi Chari and Andrew Pressley,
*Weyl modules for classical and quantum affine algebras*, Represent. Theory**5**(2001), 191–223. MR**1850556**, DOI 10.1090/S1088-4165-01-00115-7 - V. G. Drinfel′d,
*A new realization of Yangians and of quantum affine algebras*, Dokl. Akad. Nauk SSSR**296**(1987), no. 1, 13–17 (Russian); English transl., Soviet Math. Dokl.**36**(1988), no. 2, 212–216. MR**914215**
EM P. Etingof and A. Moura, - Edward Frenkel and Evgeny Mukhin,
*Combinatorics of $q$-characters of finite-dimensional representations of quantum affine algebras*, Comm. Math. Phys.**216**(2001), no. 1, 23–57. MR**1810773**, DOI 10.1007/s002200000323 - Edward Frenkel and Nicolai Reshetikhin,
*The $q$-characters of representations of quantum affine algebras and deformations of $\scr W$-algebras*, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998) Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 163–205. MR**1745260**, DOI 10.1090/conm/248/03823
G1 J. Greenstein, - M. Kashiwara,
*On crystal bases of the $Q$-analogue of universal enveloping algebras*, Duke Math. J.**63**(1991), no. 2, 465–516. MR**1115118**, DOI 10.1215/S0012-7094-91-06321-0 - Masaki Kashiwara,
*The crystal base and Littelmann’s refined Demazure character formula*, Duke Math. J.**71**(1993), no. 3, 839–858. MR**1240605**, DOI 10.1215/S0012-7094-93-07131-1 - Masaki Kashiwara,
*On level-zero representations of quantized affine algebras*, Duke Math. J.**112**(2002), no. 1, 117–175. MR**1890649**, DOI 10.1215/S0012-9074-02-11214-9 - Naihuan Jing,
*On Drinfeld realization of quantum affine algebras*, The Monster and Lie algebras (Columbus, OH, 1996) Ohio State Univ. Math. Res. Inst. Publ., vol. 7, de Gruyter, Berlin, 1998, pp. 195–206. MR**1650669**, DOI 10.1515/9783110801897.195 - Anthony Joseph,
*A completion of the quantized enveloping algebra of a Kac-Moody algebra*, J. Algebra**214**(1999), no. 1, 235–275. MR**1684872**, DOI 10.1006/jabr.1998.7677 - A. Joseph,
*The admissibility of simple bounded modules for an affine Lie algebra*, Algebr. Represent. Theory**3**(2000), no. 2, 131–149. MR**1769606**, DOI 10.1023/A:1009985328440
JT A. Joseph and D. Todorić, - G. Lusztig,
*Quantum deformations of certain simple modules over enveloping algebras*, Adv. in Math.**70**(1988), no. 2, 237–249. MR**954661**, DOI 10.1016/0001-8708(88)90056-4 - George Lusztig,
*Introduction to quantum groups*, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR**1227098** - Hiraku Nakajima,
*$T$-analogue of the $q$-characters of finite dimensional representations of quantum affine algebras*, Physics and combinatorics, 2000 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, pp. 196–219. MR**1872257**, DOI 10.1142/9789812810007_{0}009
N2 —,

*Elliptic Central Characters and Blocks of Finite Dimensional Representations of Quantum Affine Algebras*, Preprint math.QA/0204302.

*Characters of simple bounded modules over an untwisted affine Lie algebra*, Algebr. Represent. Theory (to appear). G —,

*Littelmann’s path crystal and combinatorics of certain integrable $\widehat {\mathfrak {sl}}_{l+1}$ modules of level zero*. J. Algebra (to appear).

*On the quantum KPRV determinants for semisimple and affine Lie algebras*, Algebr. Represent. Theory

**5**(2002), no. 1, 57–99.

*Extremal weight modules of quantum affine algebras*, Preprint math.QA/0204183. VV M. Varagnolo and E. Vasserot,

*Standard modules of quantum affine algebras*, Duke Math. J.

**111**(2002), no. 3, 509–533.

## Bibliographic Information

**Vyjayanthi Chari**- Affiliation: Department of Mathematics, University of California, Riverside, California 92521
- Email: chari@math.ucr.edu
**Jacob Greenstein**- Affiliation: Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 175 rue du Chevaleret, Plateau 7D, F-75013 Paris, France
- Email: greenste@math.jussieu.fr
- Received by editor(s): June 28, 2002
- Received by editor(s) in revised form: October 25, 2002
- Published electronically: February 26, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Represent. Theory
**7**(2003), 56-80 - MSC (2000): Primary 17B67
- DOI: https://doi.org/10.1090/S1088-4165-03-00168-7
- MathSciNet review: 1973367

Dedicated: Dedicated to Anthony Joseph on the occasion of his 60th birthday