Abstract
In this paper, we investigate weak Hopf algebras introduced in Li (J Algebra 208:72–100, 1998; Commun Math Phys 225:191–217, 2002) corresponding to quantum algebras U q (f (K, H)) (see Wang et al. in Commun Algebra 30:2191–2211, 2002). A new class of algebras is defined, which is denoted by \({\mathfrak{w}U^{d}_q.}\) For d = ((1, 1) | (1, 1)), denote \({\mathfrak{w}U^{d}_q}\) briefly by \({\mathfrak{w}_{1}U_q}\) ; for d = ((0, 0) | (0, 0)), denote \({\mathfrak{w}U^{d}_q}\) briefly by \({\mathfrak{w}_{2}U_q.}\) In some cases, the necessary and sufficient conditions for \({\mathfrak{w}_{1}U_q}\) and \({\mathfrak{w}_{2}U_q}\) to be weak Hopf algebras are given. The PBW bases of \({\mathfrak{w}_{1}U_q}\) and \({\mathfrak{w}_{2}U_q}\) are presented. Finally, representations and the center of \({\mathfrak{w}_{1}U_q}\) are characterized over \({\mathbb{C}}\) with \({q \in \mathbb{C}}\) which is not a root of unity.
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The authors would like to thank the referees for careful reading and useful comments on this paper. Project supported by the National Natural Science Foundation of China (No. 10871170) and the Zhejiang Provincial Natural Science Foundation of China (No. D7080064).
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Hong, Y., Li, F. Weak Hopf algebras corresponding to quantum algebras U q (f (K, H)). Arab. J. Math. 1, 195–218 (2012). https://doi.org/10.1007/s40065-012-0014-5
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DOI: https://doi.org/10.1007/s40065-012-0014-5