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A CLASS OF QUASITRIANGULAR GROUP-COGRADED MULTIPLIER HOPF ALGEBRAS

Part of: Lie algebras and Lie superalgebras Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  20 December 2018

TAO YANG ,
XUAN ZHOU  and
HAIXING ZHU
TAO YANG*
Affiliation:
College of Science, Nanjing Agricultural University, Nanjing 210095, Jiangsu, China e-mail: tao.yang@njau.edu.cn
XUAN ZHOU
Affiliation:
Department of Mathematics, Jiangsu Second Normal University, Nanjing 210013, Jiangsu, China e-mail: 20668964@qq.com
HAIXING ZHU
Affiliation:
College of Economics and Management, Nanjing Forestry University, Nanjing 210037, Jiangsu, China e-mail: zhuhaixing@163.com
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Abstract

For a multiplier Hopf algebra pairing 〈A,B〉, we construct a class of group-cograded multiplier Hopf algebras D(A,B), generalizing the classical construction of finite dimensional Hopf algebras introduced by Panaite and Staic Mihai [Isr. J. Math. 158 (2007), 349–365]. Furthermore, if the multiplier Hopf algebra pairing admits a canonical multiplier in M(B⊗A) we show the existence of quasitriangular structure on D(A,B). As an application, some special cases and examples are provided.

MSC classification

Primary: 16T05: Hopf algebras and their applications
Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 1 , January 2020 , pp. 43 - 57
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

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References

Abd El-Hafez, A. T., Delvaux, L. and Van Daele, A., Group-cograded multiplier Hopf (*-)algebras, Algebra Represent. Theor. 10 (2007), 77–95.CrossRefGoogle Scholar
Delvaux, L., Twisted tensor product of multiplier Hopf (*-) algebras, J. Algebra 269 (2003), 285–316.CrossRefGoogle Scholar
Delvaux, L. and Van Daele, A., The Drinfel’d double versus the Heisenberg double for an algebraic quantum group, J. Pure Appl. Algebra 190 (2004), 59–84.CrossRefGoogle Scholar
Delvaux, L. and Van Daele, A., The Drinfeld double for group-cograded multiplier Hopf algebras, Algebra Represent. Theor. 10(3) (2007), 197–221.CrossRefGoogle Scholar
Delvaux, L., Van Daele, A. and Wang, S. H., Quasitriangular (G-cograded) multiplier Hopf algebras, J. Algebra 289 (2005), 484–514.CrossRefGoogle Scholar
Drabant, B. and Van Daele, A., Pairing and quantum double of multiplier Hopf algebras, Algebra Represent. Theor. 4 (2001), 109–132.CrossRefGoogle Scholar
Drinfeld, V. G., Quantum groups, Zapiski Nauchnykh Seminarov POMI 155 (1986), 18–49.Google Scholar
Panaite, F. and Staic Mihai, D., Generalized (anti) Yetter-Drinfel’d modules as components of a braided T-category, Isr. J. Math. 158 (2007), 349–365.CrossRefGoogle Scholar
Turaev, V. G., Homotopy field theory in dimension 3 and crossed group-categories. (2000). Preprint GT/0005291.Google Scholar
Van Daele, A., Multiplier Hopf algebras, Trans. Am. Math. Soc. 342(2) (1994), 917–932.CrossRefGoogle Scholar
Van Daele, A., An algebraic framework for group duality, Adv. Math. 140(2) (1998), 323–366.CrossRefGoogle Scholar
Van Daele, A., Tools for working with multiplier Hopf algebras, Arab. J. Sci. Eng. 33(2C) (2008), 505–527.Google Scholar
Van Daele, A. and Zhang, Y. H., Corepresentation theory of multiplier Hopf algebras I, Int. J. Math. 10(4) (1999), 503–539.CrossRefGoogle Scholar
Yang, T. and Wang, S. H., A lot of quasitriangular group-cograded multiplier Hopf algebras, Algebra. Represent. Theor. 14(5) (2011), 959–976.CrossRefGoogle Scholar
Yang, T. and Wang, S. H., Constructing new braided T-categories over regular multiplier Hopf algebras, Comm. Algebra, 39(9) (2011), 3073–3089.CrossRefGoogle Scholar
Yang, T., Zhou, X. and Ma, T., On braided T-categories over multiplier Hopf algebras, Comm. Algebra 41 (2013), 2852–2868.CrossRefGoogle Scholar
Zhang, Y. H., The quantum double of a coFrobenius Hopf algebra, Comm. Algebra, 27(3) (1999), 1413–1427.CrossRefGoogle Scholar