Majid, Shahn More examples of bicrossproduct and double cross product Hopf algebras. (English) Zbl 0725.17015 Isr. J. Math. 72, No. 1-2, 133-148 (1990). The author had previously defined bicrossproducts and double cross products of Hopf algebras. Here he constructs examples where the factors are certain quantum groups coming from solutions of the quantum Yang-Baxter equations. He also describes iterated double cross products of quantum groups. These particular quantum groups are constructed using a suitable notion of mutually dual Hopf algebras and a certain dual quantum group to the Hopf algebra given directly by a solution of the quantum Yang-Baxter equation. This dual pairing leads to the notion of a weak Hopf algebra with weak antipode. The algebraic ideas here are motivated by the author’s approach to quantum mechanics combined with gravity. Reviewer: Earl J. Taft (New Brunswick) Cited in 4 ReviewsCited in 36 Documents MSC: 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B38 Yang-Baxter equations and Rota-Baxter operators 16T20 Ring-theoretic aspects of quantum groups 16T25 Yang-Baxter equations Keywords:bicrossproducts; double cross products; Hopf algebras; quantum groups; quantum Yang-Baxter equations PDFBibTeX XMLCite \textit{S. Majid}, Isr. J. Math. 72, No. 1--2, 133--148 (1990; Zbl 0725.17015) Full Text: DOI References: [1] Blattner, R. J.; Cohen, M.; Montgomery, S., Crossed products and inner actions of Hopf algebras, Trans. Am. Math. Soc., 298, 2, 671-671 (1986) · Zbl 0619.16004 · doi:10.2307/2000643 [2] V. G. Drinfeld,Quantum groups, inProc. ICM, Berkeley (A. Gleason, ed.), AMS, 1987. · Zbl 0667.16003 [3] L. D. Faddeev, N. Yu. Reshetikhin and L. A. Takhtajan,Quantization of Lie groups and Lie algebras, LOMI preprint, 1987. [4] Majid, S.; Majid, S.; Majid, S.; Majid, S., Hopf-von Neumann algebra bicrossproducts, Kac algebra bicrossproducts and the CYBE, J. Classical and Quantum Gravity, 141, 1587-1606 (1988) · Zbl 0672.16009 [5] Majid, S., Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra, 130, 17-64 (1990) · Zbl 0694.16008 · doi:10.1016/0021-8693(90)90099-A [6] Majid, S.; Chau, L.-L.; Nahm, W., Quantum group duality in vertex models, Proc. XVIII DGM, Tahoe City (1989), New York: Plenum Press, New York [7] N. Yu. Reshetikhin,Quantized universal enveloping algebras, the Yang-Baxter Equations and invariants of links, I and II, LOMI preprints, 1988. [8] Singer, W., Extension theory for connected Hopf algebras, J. Algebra, 21, 1-16 (1972) · Zbl 0269.16011 · doi:10.1016/0021-8693(72)90031-2 [9] M. E. Sweedler,Hopf Algebras, Benjamin, 1969. · Zbl 0194.32901 [10] Takeuchi, M., Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra, 9, 841-841 (1981) · Zbl 0456.16011 · doi:10.1080/00927878108822621 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.