Nakajima monomials, Young walls and Kashiwara embedding for \(U_q(A_n^{(1)})\)
type: Item
instance of: scholarly article
title: Nakajima monomials, Young walls and Kashiwara embedding for \(U_q(A_n^{(1)})\)
zbMATH Open document ID: 1270.17006
author: Jeong-Ah Kim
author: Dong-Uy Shin
published in: Journal of Algebra
publication date: 2011-07-25T00:00:00Z
review text: In this paper, the authors investigate three kinds of realizations of the level 1 crystal \(B(\Lambda)\) of the quantum affine algebras \(U_q(A_n^{(1)})\), and construct explicit crystal isomorphisms among them. The three realizations are polyhedral realizations (see [\textit{T. Nakashima} and \textit{A. Zelevinsky}, Adv. Math. 131, No. 1, 253--278 (1997; Zbl 0897.17014); \textit{T. Nakashima}, J. Algebra 219, No. 2, 571--597 (1999; Zbl 0967.17011)]), Nakajima monomial realizations (see [\textit{H. Nakajima}, Physics and Combinatorics, Proc. Nagoya 2000 Internat. Workshop. Singapore: World Scientific. 196--219 (2001; Zbl 1011.17013); \textit{M. Kashiwara}, Contemp. Math. 325, 133--139 (2003; Zbl 1066.17006)]), and Young wall realizations (see [\textit{S.-J. Kang}, Proc. Lond. Math. Soc., III. Ser. 86, No. 1, 29--69 (2003; Zbl 1030.17013)]). They first give an explicit description of the connected component of the Nakajima monomial realization with special parameters, and show that this is isomorphic to a certain polyhedral realization. Using this isomorphism, they finally show the connections to Young wall realizations.
reviewed by: Euiyong Park
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 5931645
zbMATH Keywords: Nakajima monomials
zbMATH Keywords: Young walls
zbMATH Keywords: Kashiwara embeddings
zbMATH Keywords: crystal bases
zbMATH Keywords: polyhedral realizations
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.11.010
OpenAlex ID: W2094125128
cites work: Crystal bases and Young tableaux
cites work: Q3715293
cites work: Q4537716
cites work: Crystals and Nakajima monomials for quantum generalized Kac-Moody algebras
cites work: A q-difference analogue of \(U({\mathfrak g})\) and the Yang-Baxter equation
cites work: CRYSTAL BASES FOR QUANTUM AFFINE ALGEBRAS AND COMBINATORICS OF YOUNG WALLS
cites work: Q4301633
cites work: AFFINE CRYSTALS AND VERTEX MODELS
cites work: Perfect crystals of quantum affine Lie algebras
cites work: Young wall realization of crystal bases for classical Lie algebras
cites work: Monomial realization of crystal bases for special linear Lie algebras
cites work: Crystal bases for quantum classical algebras and Nakajima's monomials
cites work: Modified Nakajima monomials and the crystal \(B(\infty )\)
cites work: Higher level affine crystals and Young walls
cites work: On crystal bases of the \(q\)-analogue of universal enveloping algebras
cites work: The crystal base and Littelmann's refined Demazure character formula
cites work: Q4422485
cites work: Crystal graphs for representations of the \(q\)-analogue of classical Lie algebras
cites work: Monomial realization of crystal graphs for \(U_q(A_n^{(1)})\)
cites work: Correspondence between Young walls and Young tableaux and its application
cites work: Monomial realization of crystal bases \(B(\infty)\) for the quantum finite algebras
cites work: Monomial Realization of the Crystals OverUq(A∞) and the Quantum Monster Algebra
cites work: A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras
cites work: Paths and root operators in representation theory
cites work: Quiver varieties and tensor products
cites work: Q4422486
cites work: Polyhedral realizations of crystal bases for integrable highest weight modules
cites work: Polyhedral realizations of crystal bases for quantized Kac-Moody algebras
cites work: Crystal Bases and Monomials forUq(G2)-Modules
cites work: Polyhedral realization of crystal bases for generalized Kac-Moody algebras
cites work: Polyhedral realization of the highest weight crystals for generalized Kac-Moody algebras
DOI: 10.1016/J.JALGEBRA.2010.11.010
description: scientific article; zbMATH DE number 5931645