Level one perfect crystals for \(B_n^{(1)}\), \(C_n^{(1)}\), and \(D_n^{(1)}\)
type: Item
instance of: scholarly article
title: Level one perfect crystals for \(B_n^{(1)}\), \(C_n^{(1)}\), and \(D_n^{(1)}\)
zbMATH Open document ID: 0941.17010
author: Yoshiyuki Koga
published in: Journal of Algebra
publication date: 2000-08-03T00:00:00Z
review text: Let \({\mathfrak g}\) be an affine Lie algebra and \({\mathcal U}_q({\mathfrak g})\) denote the associated quantum affine algebra. The crystal base for the integrable highest weight representations of \({\mathcal U}_q({\mathfrak g})\) can be realized in terms of certain combinatorial objects called ``paths'' which are elements of the semi-infinite tensor product \(\cdots\oplus B\oplus B\oplus B\), where \(B\) is a perfect crystal of corresponding level associated with a finite-dimensional representation of the derived quantum affine algebra \({\mathcal U}_q'({\mathfrak g})\). For \({\mathfrak g}={\mathfrak A}_n^{(1)}\), it has been shown in [\textit{S. J. Kang} et al., Duke Math. J. 68, 499-607 (1992; Zbl 0774.17017)] that \(B^{k,\ell}= B(\ell\Lambda_k)\) is a perfect crystal of level \(\ell\) where \(\Lambda_k\) denotes the \(k\)th fundamental weight of \({\mathcal U}_q(A_n)\). In this paper the author constructs new examples of level one perfect crystals for the quantum affine algebras \({\mathcal U}_q(B_n^{(1)})\), \({\mathcal U}_q(C_n^{(1)})\), \({\mathcal U}_q(D_n^{(1)})\). It is conjectured that any perfect crystal of level one for these quantum affine algebras is isomorphic to one of these crystals or the tensor product of them.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 1340615
zbMATH Keywords: quantum group
zbMATH Keywords: affine Lie algebra
zbMATH Keywords: representation
zbMATH Keywords: level one perfect crystals
zbMATH Keywords: quantum affine algebras
reviewed by: Kailash C. Misra
MaRDI profile type: Publication
full work available at URL: jabr.1998.7806
OpenAlex ID: W2019061652
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DOI: 10.1006/JABR.1998.7806
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description: scientific article; zbMATH DE number 1340615