Generalized Young walls and crystal bases for quantum affine algebra of type

Authors:
Jeong-Ah Kim and Dong-Uy Shin

Journal:
Proc. Amer. Math. Soc. **138** (2010), 3877-3889

MSC (2010):
Primary 17B37, 81R50

DOI:
https://doi.org/10.1090/S0002-9939-2010-10428-8

Published electronically:
June 9, 2010

MathSciNet review:
2679610

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Abstract | References | Similar Articles | Additional Information

Abstract: We give a new realization of the crystal of using *generalized Young walls*, a modified notion of Young walls of type . Moreover, by the fact that the irreducible highest weight crystal lies in the crystal , we construct the crystal using generalized Young walls.

**1.**M. Jimbo, K. C. Misra, T. Miwa, M. Okado,*Combinatorics of representations of at*, Comm. Math. Phys.**136**(1991), 543-566. MR**1099695 (93a:17015)****2.**S.-J. Kang,*Crystal bases for quantum affine algebras and combinatorics of Young walls*, Proc. London Math. Soc. (3)**86**(2003), 29-69. MR**1971463 (2004c:17028)****3.**S.-J. Kang, M. Kashiwara, K. C. Misra,*Crystal bases of Verma modules for quantum affine Lie algebras*, Compositio Math.**92**(1994), 299-325. MR**1286129 (95h:17016)****4.**S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, A. Nakayashiki,*Affine crystals and vertex models*, Int. J. Mod. Phys. A.**Suppl. 1A**(1992), 449-484. MR**1187560 (94a:17008)****5.**S.-J. Kang, J.-A. Kim, D.-U. Shin,*Monomial realization of crystal bases for special linear Lie algebras*, J. Algebra**274**(2004), 629-642. MR**2043368 (2005a:17010)****6.**S.-J. Kang, J.-A. Kim, D.-U. Shin,*Modified Nakajima monomials and the crystal*, J. Algebra**308**(2007), 524-535. MR**2295073 (2008b:17020)****7.**M. Kashiwara,*On crystal bases of the -analogue of universal enveloping algebras*, Duke Math. J.**63**(1991), 465-516. MR**1115118 (93b:17045)****8.**M. Kashiwara, T. Nakashima,*Crystal graphs for representations of the -analogue of classical Lie algebras*, J. Algebra**165**(1994), 295-345. MR**1273277 (95c:17025)****9.**P. Littelmann,*Paths and root operators in representation theory*, Ann. of Math. (2)**142**(1995), 499-525. MR**1356780 (96m:17011)****10.**T. Nakashima,*Polyhedral realizations of crystal bases for integrable highest weight modules*, J. Algebra**219**(1999), 571-597. MR**1706829 (2000g:17020)****11.**T. Nakashima, A. Zelevinsky,*Polyhedral realizations of crystal bases for quantized Kac-Moody algebras*, Adv. Math.**131**(1997), 253-278. MR**1475048 (98m:17023)**

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Additional Information

**Jeong-Ah Kim**

Affiliation:
Department of Mathematics, University of Seoul, Seoul, 130-743, Korea

Email:
jakim@uos.ac.kr

**Dong-Uy Shin**

Affiliation:
Department of Mathematics Education, Hanyang University, Seoul 133-791, Korea

Email:
dushin@hanyang.ac.kr

DOI:
https://doi.org/10.1090/S0002-9939-2010-10428-8

Received by editor(s):
August 12, 2009

Received by editor(s) in revised form:
January 20, 2010, and February 3, 2010

Published electronically:
June 9, 2010

Additional Notes:
This work was supported by the research fund of Hanyang University (HY-2009-O)

Communicated by:
Gail R. Letzter

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.