×

The Kirillov-Reshetikhin conjecture and solutions of \(T\)-systems. (English) Zbl 1160.17010

Summary: We prove the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras: we prove that the characters of Kirillov-Reshetikhin modules solve the \(Q\)-system and we give an explicit formula for the character of their tensor products. In the proof we show that Kirillov-Reshetikhin modules are special in the sense of monomials and that their \(q\)-characters solve the \(T\)-system (functional relations appearing in the study of solvable lattice models). Moreover we prove that the \(T\)-system can be written in the form of an exact sequence. For simply-laced cases, these results were proved by H. Nakajima [Ann. Math. (2) 160, 1057–1097 (2005; Zbl 1140.17015), Represent. Theory 7, 259–274 (2003; Zbl 1078.17008)] with geometric arguments (main result of [H. Nakajima, 2004, loc. cit.]) which are not available in general. The proof we use is different and purely algebraic, and so can be extended uniformly to non simply-laced cases.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1007/BF01341708 · doi:10.1007/BF01341708
[2] DOI: 10.1155/S1073792801000332 · Zbl 0982.17004 · doi:10.1155/S1073792801000332
[3] DOI: 10.1155/S107379280210612X · Zbl 0990.17009 · doi:10.1155/S107379280210612X
[4] DOI: 10.1007/BF02102063 · Zbl 0739.17004 · doi:10.1007/BF02102063
[5] DOI: 10.1007/s002200000323 · Zbl 1051.17013 · doi:10.1007/s002200000323
[6] Frenkel E., Cont. Math. 248 pp 163– (1999)
[7] Hatayama G., NC pp 248– (1998)
[8] DOI: 10.1016/j.aim.2003.07.016 · Zbl 1098.17009 · doi:10.1016/j.aim.2003.07.016
[9] DOI: 10.1007/s00031-005-1005-9 · Zbl 1102.17009 · doi:10.1007/s00031-005-1005-9
[10] DOI: 10.1007/s00209-005-0762-4 · Zbl 1098.17010 · doi:10.1007/s00209-005-0762-4
[11] DOI: 10.1007/BF00704588 · Zbl 0587.17004 · doi:10.1007/BF00704588
[12] DOI: 10.1215/S0012-9074-02-11214-9 · Zbl 1033.17017 · doi:10.1215/S0012-9074-02-11214-9
[13] DOI: 10.1007/BF01840426 · Zbl 0685.33014 · doi:10.1007/BF01840426
[14] DOI: 10.1007/BF02342935 · Zbl 0900.16047 · doi:10.1007/BF02342935
[15] DOI: 10.1155/S1073792897000135 · Zbl 0897.17022 · doi:10.1155/S1073792897000135
[16] DOI: 10.1006/jabr.1995.1123 · Zbl 0868.17009 · doi:10.1006/jabr.1995.1123
[17] DOI: 10.1006/jabr.2001.8774 · Zbl 1052.17006 · doi:10.1006/jabr.2001.8774
[18] DOI: 10.1142/S0217751X94002119 · Zbl 0985.82501 · doi:10.1142/S0217751X94002119
[19] DOI: 10.1007/s002200200631 · Zbl 1065.81071 · doi:10.1007/s002200200631
[20] DOI: 10.1088/0305-4470/35/6/307 · Zbl 1012.17009 · doi:10.1088/0305-4470/35/6/307
[21] DOI: 10.1090/S0894-0347-00-00353-2 · Zbl 0981.17016 · doi:10.1090/S0894-0347-00-00353-2
[22] DOI: 10.4007/annals.2004.160.1057 · Zbl 1140.17015 · doi:10.4007/annals.2004.160.1057
[23] DOI: 10.1090/S1088-4165-03-00164-X · Zbl 1078.17008 · doi:10.1090/S1088-4165-03-00164-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.