Quantum affine algebras, combinatorics of Young walls, and global bases
Authors:
Seok-Jin Kang and Jae-Hoon Kwon
Journal:
Electron. Res. Announc. Amer. Math. Soc. 8 (2002), 35-46
MSC (2000):
Primary 17B37; Secondary 17B10
DOI:
https://doi.org/10.1090/S1079-6762-02-00103-8
Published electronically:
September 19, 2002
MathSciNet review:
1928500
Full-text PDF Free Access
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Abstract: We construct the Fock space representation of quantum affine algebras using combinatorics of Young walls. We also show that the crystal basis of the Fock space representation can be realized as the abstract crystal consisting of proper Young walls. We then generalize Lascoux-Leclerc-Thibon algorithm to obtain an effective algorithm for constructing the global bases of basic representations.
- Jonathan Brundan and Alexander Kleshchev, Hecke-Clifford superalgebras, crystals of type $A_{2l}^{(2)}$ and modular branching rules for $\hat S_n$, Represent. Theory 5 (2001), 317–403. MR 1870595, DOI https://doi.org/10.1090/S1088-4165-01-00123-6
- Richard Dipper and Gordon James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 52 (1986), no. 1, 20–52. MR 812444, DOI https://doi.org/10.1112/plms/s3-52.1.20
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219
Kang S.-J. Kang, Crystal bases for quantum affine Lie algebras and combinatorics of Young walls, RIM-GARC preprint (2000) 00-2, Seoul National University, to appear in Proc. London Math. Soc.
KK S.-J. Kang, J.-H. Kwon, Fock space representations of quantum affine algebras and generalized Lascoux-Leclerc-Thibon algorithm, math.QA/0208204.
- 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 https://doi.org/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 https://doi.org/10.1215/S0012-7094-93-07131-1
- Masaki Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), no. 2, 383–413. MR 1262212, DOI https://doi.org/10.1215/S0012-7094-94-07317-1
- M. Kashiwara, T. Miwa, J.-U. H. Petersen, and C. M. Yung, Perfect crystals and $q$-deformed Fock spaces, Selecta Math. (N.S.) 2 (1996), no. 3, 415–499. MR 1422203, DOI https://doi.org/10.1007/BF01587950
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BK J. Brundan, A. Kleshchev, Hecke-Clifford superalgebras, crystals of type $A^{(2)}_{2n}$ and modular branching rules for $\hat {S}_n$, Represent. Theory 5 (2001), 317–403 (electronic).
DJ R. Dipper, G. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. 52 (1986), 20–52.
Kac90 V. G. Kac, Infinite-dimensional Lie algebras, Cambridge University Press, 3rd ed., Cambridge, 1990.
Kang S.-J. Kang, Crystal bases for quantum affine Lie algebras and combinatorics of Young walls, RIM-GARC preprint (2000) 00-2, Seoul National University, to appear in Proc. London Math. Soc.
KK S.-J. Kang, J.-H. Kwon, Fock space representations of quantum affine algebras and generalized Lascoux-Leclerc-Thibon algorithm, math.QA/0208204.
Kash M. Kashiwara, On crystal bases of the $q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465–516.
Kash93 M. Kashiwara, Crystal bases and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993), 839–858.
Kash94 M. Kashiwara, Crystal bases of modified quantized enveloping algebras, Duke Math. J. 73 (1994), 383–413.
KMPY M. Kashiwara, T. Miwa, J.-U. H. Petersen, C. M. Yung, Perfect crystals and $q$-deformed Fock space, Selecta Math. 2 (1996), 415–499.
LLT A. Lascoux, B. Leclerc, J.-Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205–263.
Lusz G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447–498.
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Additional Information
Seok-Jin Kang
Affiliation:
School of Mathematics, Korea Institute for Advanced Study, Seoul 130-012, Korea
MR Author ID:
307910
Email:
sjkang@kias.re.kr
Jae-Hoon Kwon
Affiliation:
School of Mathematics, Korea Institute for Advanced Study, Seoul 130-012, Korea
Email:
jhkwon@kias.re.kr
Keywords:
Quantized universal enveloping algebra,
crystal basis,
global basis
Received by editor(s):
December 14, 2001
Published electronically:
September 19, 2002
Communicated by:
Efim Zelmanov
Article copyright:
© Copyright 2002
American Mathematical Society
