Quasi-quantum groups obtained from tensor braided Hopf algebras
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Quasi-quantum groups obtained from tensor braided Hopf algebras Daniel Bulacu1 Received: 4 December 2018 / Accepted: 30 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract H Let H be a quasi-Hopf algebra, H H M H the category of two-sided two-cosided Hopf H modules over H and H Y D the category of left Yetter–Drinfeld modules over H . We H show that H H M H admits a braided monoidal structure for which the strong monoidal H ∼ H equivalence H M H H = H Y D established by the structure theorem for quasi-Hopf bimodules becomes braided monoidal. Using this braided monoidal equivalence, we H prove that Hopf algebras within H H M H can be characterized as quasi-Hopf algebras with a projection or as biproduct quasi-Hopf algebras in the sense of Bulacu and Nauwelaerts (J Pure Appl Algebra 174:1–42, 2002) . A particular class of such (braided, quasi-) Hopf algebras is obtained from a tensor product Hopf algebra type construction. Our arguments rely on general categorical facts. Keywords Braided category · Biproduct · Projection · Braided tensor Hopf algebra · Quantum shuffle quasi-Hopf algebra Mathematics Subject Classification 16T05 · 18D10
1 Introduction The so-called quantum shuffle Hopf algebras are cotensor Hopf algebras of a Hopf bimodule M over a Hopf algebra H . Their importance resides on the fact that all quantized enveloping algebras associated with finite-dimensional simple Lie algebras or with affine Kac–Moody Lie algebras are of this type; see [25]. As the cotensor defines a monoidal structure on the category of Hopf H -bimodules isomorphic to the one determined by the tensor product over H , it follows that quantum shuffle algebras
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Daniel Bulacu [email protected] Faculty of Mathematics and Informatics, University of Bucharest, Str. Academiei 14, 010014 Bucharest 1, Romania
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Journal of Algebraic Combinatorics
can be as well introduced as tensor Hopf algebras within the braided category of Hopf H -bimodules (also known under the name of two-sided two-cosided Hopf modules). The structure of a Hopf algebra H with a projection π : B → H is due to Radford [24]. Up to an isomorphism, B is a biproduct Hopf algebra A × H between a left H -module algebra and left H -comodule coalgebra A and H , satisfying appropriate compatibility relations. Majid [19] observed that all these conditions are equivalent to the fact that A is a Hopf algebra within H H Y D, the braided monoidal category of left Yetter–Drinfeld modules over H . A second characterization of Hopf algebras with a projection is due to Bespalov and Drabant [1], where Hopf algebras with a projection H are identified with Hopf algebras within H H M H , the braided monoidal category of two-sided two-cosided Hopf modules over H introduced by Woronowicz in [28]. The H H H connection with the Hopf algebras in H H Y D becomes clear now, since H M H and H Y D are braided monoidally equivalent. The latest result was proved by Schauenburg in [26]; see also [25]. We should mention that in all this theory a k
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