Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

The crystal $\mathcal{B}(\infty)$ and extended Nakajima monomials

Author(s): Hyeonmi Lee
Journal: Proc. Amer. Math. Soc. 137 (2009), 1915-1924.
MSC (2000): Primary 20G05; Secondary 17B37, 81R50
Posted: January 26, 2009
MathSciNet review: 2480271
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We construct the set of extended Nakajima monomials containing monomials realizing the irreducible highest weight crystals, introduced by Nakajima. We present explicit descriptions of a crystal $\mathcal{B}_{\infty}$ and the crystal $\mathcal{B}(\infty)$ in this language for the case $A_n^{(1)}$ .

References:

1.: J. Hong and H. Lee, Young tableaux and crystal $\mathcal{B}(\infty)$ for finite simple Lie algebras, J. Algebra 320 (2008), no. 10, 3680-3693.
2.: S.-J. Kang, M. Kashiwara, and K. C. Misra, Crystal bases of Verma modules for quantum affine Lie algebras, Compositio Math. 92 (1994), no. 3, 299-325. MR 1286129 (95h:17016)
3.: M. Kashiwara, On crystal bases of the -analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465-516. MR 1115118 (93b:17045)
4.: M. Kashiwara, Realizations of crystals, Combinatorial and geometric representation theory, Contemp. Math., vol. 325, Amer. Math. Soc., Providence, RI, 2003, pp. 133-139. MR 1988989 (2004g:17014)
5.: J.-A. Kim, Monomial realization of crystal graphs for $U_q(A_n^{(1)})$ , Math. Ann. 332 (2005), no. 1, 17-35. MR 2139248 (2006a:17014)
6.: H. Lee, Extended Nakajima monomials and realization of crystal $\mathcal{B}(\infty)$ , KIAS preprint M05008.
7.: H. Lee, Realizations of crystal $\mathcal{B}(\infty)$ using Young tableaux and Young walls, J. Algebra 308 (2007), no. 2, 780-799. MR 2295089 (2008f:17027)
8.: H. Lee, Nakajima monomials and crystals for special linear Lie algebras, Nagoya Math. J. 188 (2007), 31-57. MR 2371768
9.: H. Lee, Extended Nakajima monomials and crystal $\mathcal{B}(\infty)$ for finite simple Lie algebras, J. Algebra 314 (2007), no. 1, 439-470. MR 2331770 (2008f:17028)
10.: H. Nakajima, -analogs of -characters of quantum affine algebras of type $A\sb n,D\sb n$ , Combinatorial and geometric representation theory, Contemp. Math., vol. 325, Amer. Math. Soc., Providence, RI, 2003, pp. 141-160. MR 1988990 (2004h:17018)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|