Quantum loop modules

Chari, Vyjayanthi; Greenstein, Jacob

doi:10.1090/S1088-4165-03-00168-7

Quantum loop modules
HTML articles powered by AMS MathViewer

by Vyjayanthi Chari and Jacob Greenstein PDF

Represent. Theory 7 (2003), 56-80 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

Elliptic Central Characters and Blocks of Finite Dimensional Representations of Quantum Affine Algebras

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

Characters of simple bounded modules over an untwisted affine Lie algebra

Littelmann’s path crystal and combinatorics of certain integrable $\widehat {\mathfrak {sl}}_{l+1}$ modules of level zero

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

On the quantum KPRV determinants for semisimple and affine Lie algebras

5

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

Extremal weight modules of quantum affine algebras

Standard modules of quantum affine algebras

111