Weak Hopf algebras corresponding to quantum algebras \(U_q(f(K,H))\)
type: Item
instance of: scholarly article
title: Weak Hopf algebras corresponding to quantum algebras \(U_q(f(K,H))\)
zbMATH Open document ID: 1270.17005
author: Yanyong Hong
author: Fang Li
published in: Arabian Journal of Mathematics
publication date: 2012-12-18T00:00:00Z
review text: In the paper under review, the authors construct a new class of weak Hopf algebras in the sense of the second author [J. Algebra 208, No. 1, 72--100 (1998; Zbl 0916.16020)]. Let \(U_q(f(K,H))\) be a quantum group introduced by \textit{Q. Ji, D. Wang} and \textit{S. Yang} in [Commun. Algebra 30, No. 5, 2191--2211 (2002; Zbl 1026.17022)]. In the second section, the authors construct a \(\mathbf{k}_q\)-algebra \(\mathfrak{w}U^d_q(f(K,\overline{K},H,\overline{H}))\) corresponding to \(U_q(f(K,H))\). The method, which they use, is similar to that used by \textit{F. Li} and \textit{S. Duplij} [Commun. Math. Phys. 225, No. 1, 191--217 (2002; Zbl 1032.17026)]. In the case when \(d=((1,1)|(1,1))\) (respectively, \(d=((0,0)|(0,0))\)), we abbreviate \(\mathfrak{w}U^d_q(f(K,\overline{K},H,\overline{H}))\) to \(\mathfrak{w}_1U_q\) (respectively, \(\mathfrak{w}_2U_q\)). The authors note that \(\mathfrak{w}_2U_q\) is a homomorphic image of \(\mathfrak{w}_1U_q\), and that both \(\mathfrak{w}_1U_q\) and \(\mathfrak{w}_2U_q\) are Noetherian. They characterize the PBW bases of both these algebras. Next, the authors note that the algebra \(\mathfrak{w}U^d_q(f(K,\overline{K},H,\overline{H}))\) has central idempotents \(P=K\overline{K}=\overline{K}K\neq0,1\) and \(Q=H\overline{H}=\overline{H}H\neq0,1\). In the special case when \(P+Q=1\), they show that the algebra \(\mathfrak{w}_1U_q\) can be decomposed into \(P\mathfrak{w}_1U_q\oplus Q\mathfrak{w}_1U_q\), where every ideal is isomorphic to some quantum group \(U_q(f(K))\) introduced by \textit{Q. Ji, D. Wang} and \textit{X. Zhou} in [East-West J. Math. 2, No. 2, 201--213 (2000; Zbl 1001.17017)]. In the third section, under the assumption that \(f(K,\overline{K},H,\overline{H})\neq0\), the authors characterize all weak Hopf algebra structures of \(\mathfrak{w}_1U_q\), and give a necessary and sufficient condition for \(\mathfrak{w}_2U_q\) to be a weak Hopf algebra. Next, in the special case when \(P=Q\), they show that both algebras \(\…
reviewed by: Małgorzata E. Hryniewicka
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16T20
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 6116595
zbMATH Keywords: (weak) Hopf algebra
zbMATH Keywords: quantum group
MaRDI profile type: Publication
full work available at URL: s40065-012-0014-5
OpenAlex ID: W2066826579
Wikidata QID: Q59289545
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DOI: 10.1007/S40065-012-0014-5
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description: scientific article; zbMATH DE number 6116595