Weak Hopf algebras corresponding to quantum algebras \(U_q(f(K,H))\)
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 \(\mathfrak{w}_iU_q\) (\(i=1,2\)) can be decomposed into \(P\mathfrak{w}_iU_q\oplus(1-P)\mathfrak{w}_iU_q\), where \(P\mathfrak{w}_iU_q\) is isomorphic to some quantum group \(U_q(f(K,H))\). In the fourth section, under the assumption that \(\mathbf{k}=\mathbb{C}\) and \(q\in\mathbb{C}\) is not a root of unit, the authors describe all irreducible representations and the center of the algebra \(\mathfrak{w}_1U_q\).
scientific article; zbMATH DE number 6116595
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