Integrable representations of \(U_q(\text{osp}(1,2n))\)
type: Item
instance of: scholarly article
title: Integrable representations of \(U_q(\text{osp}(1,2n))\)
zbMATH Open document ID: 0930.17011
DOI: 10.1016/S0022-4049(97)00088-1
author: Yi Ming Zou
published in: Journal of Pure and Applied Algebra
publication date: 2000-01-11T00:00:00Z
review text: Among the contragredient classical Lie superalgebras, \(G=\text{osp}(1,2n)\) is special since it is the only one whose finite-dimensional representations are completely reducible. Since the representation theory of \(G\) is so close to that of Lie algebras, it is interesting to see if this also holds for the \(q\)-deformation \(U_q(G)\). In the present paper, integrable representations of \(U_q(G)\) are studied. If one works over the field \({\mathbb{C}}(\sqrt{q})\), it is shown that \(U_q(G)\) has integrable simple highest weight representations that are not deformations of representations of \(U(G)\). Furthermore, by considering representations of \(U_q(G)\) which are deformations of \(U(G)\)-modules, the author obtains a PBW theorem for \(U_q(G)\). Finally, the complete reducibility theorem is proved for the category of finite dimensional integrable \(U_q(G)\)-modules.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17A70
zbMATH DE Number: 1308207
zbMATH Keywords: integrable representations
zbMATH Keywords: \(q\)-deformation
zbMATH Keywords: \(\text{osp} (1,2n)\)
zbMATH Keywords: contragredient classical Lie superalgebras
zbMATH Keywords: PBW theorem
zbMATH Keywords: complete reducibility theorem
MaRDI profile type: Publication
cites work: Q3819193
cites work: Q3752572
cites work: A q-difference analogue of \(U({\mathfrak g})\) and the Yang-Baxter equation
cites work: Lie superalgebras
cites work: Q4168737
cites work: Q3693664
cites work: On the defining relations of quantum superalgebras
cites work: \(q\)-oscillator realizations of the quantum superalgebras \(sl_ q(m,n)\) and \(osp_ q(m,2n)\)
cites work: Quantum deformations of certain simple modules over enveloping algebras
cites work: The quantum superalgebra Uq(osp(1/2n)): deformed para-Bose operators and root of unity representations
cites work: Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra
cites work: The presentation and q deformation of special linear Lie superalgebras
cites work: Quantized enveloping algebras associated with simple Lie superalgebras and their universal \(R\)-matrices
cites work: Finite-dimensional representations of \(U_ q(osp (1/2n))\) and its connection with quantum \(so(2n+1)\)
full work available at URL: s0022-4049(97)00088-1
OpenAlex ID: W1502695782
description: scientific article; zbMATH DE number 1308207