Crystal bases for \(U_q(osp (1,2r))\)
type: Item
instance of: scholarly article
title: Crystal bases for \(U_q(osp (1,2r))\)
zbMATH Open document ID: 0918.17009
author: Ian M. Musson
author: Yi Ming Zou
published in: Journal of Algebra
publication date: 1999-04-07T00:00:00Z
review text: The authors construct \(\mathbb{Z}_2\)-graded crystal bases for the quantized universal enveloping algebra \(U\) of the Lie superalgebra \(osp(1,2r)\). It is shown that, as in the Lie algebra case, these crystal bases carry a remarkable combinatorial structure. This paper is arranged as follows. In Section 2, the authors first recall the definition of \(U\) and collect some basic facts about its representations; then they define the concept of a crystal base for \(U\) and consider the crystal bases for the simple modules of \(U\) in the case \(r=1\) as an example. In Section 3, two simple \(U\)-modules and their crystal bases are constructed. In Section 4, the behavior of the crystal bases is considered with respect to taking tensor products of \(U\)-modules. It is shown that these crystal bases possess remarkable combinatorial properties, as in the Lie algebra case. In Section 5, the authors first consider the polarization problem, then they apply the result to complete the proof of the existence of the crystal base for any finite-dimensional \(U\)-module. Finally, two examples are used to illustrate how to use the results to decompose tensor products of \(U\)-modules.
reviewed by: Fang Li
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 17A70
zbMATH DE Number: 1251407
zbMATH Keywords: combinatorial structure
zbMATH Keywords: crystal bases
zbMATH Keywords: quantized universal enveloping algebra
zbMATH Keywords: Lie superalgebra \(osp(1,2r)\)
zbMATH Keywords: simple modules
zbMATH Keywords: tensor products
MaRDI profile type: Publication
full work available at URL: jabr.1998.7591
OpenAlex ID: W1987246112
cites work: Q4846800
cites work: Semisimplicity of 2-graded Lie algebras. II
cites work: Q4317871
cites work: Lie superalgebras
cites work: Q4168737
cites work: Crystalizing the q-analogue of universal enveloping algebras
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: Q4865651
cites work: Q3998175
cites work: Integrable representations of \(U_q(\text{osp}(1,2n))\)
DOI: 10.1006/JABR.1998.7591
Recommended article: Crystal bases for $U_q(sl(2,1))$
Recommended article: Crystal bases for the quantum superalgebra $U_q(\mathfrak {gl}(m,n))$
Recommended article: Crystal bases for \(U_{q}(\Gamma(\sigma_{1},\sigma_{2},\sigma_{3}))\)
Recommended article: On the structure ofUq(sl(m,1)): crystal bases
Recommended article: Crystal base and \(q\)-vertex operators
Recommended article: Crystal bases for quantum superalgebras
Recommended article: Crystal base for the basic representation of \(U_ q({\mathfrak sl}^\wedge (n))\)
Recommended article: Fock space representations and crystal bases for \(C_{n}^{(1)}\)
Recommended article: Young tableaux and crystal base for \(U_q(\text{osp}(1|2n))\)
Recommended article: Crystal bases and tensor product decompositions of \(U_ q(G_ 2)\)- modules
description: scientific article; zbMATH DE number 1251407