Rocky Mountain Journal of Mathematics

Smash coproducts of monoidal comonads and Hom-entwining structures

Xiaohui Zhang, Wei Wang, and Xiaofan Zhao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $F$, $G$ be monoidal comonads on a monoidal category $\mathcal {C}$. The aim of this paper is to discuss the smash coproducts of $F$ and $G$. As an application, the smash coproduct of Hom-bialgebras is discussed. Further, the Hom-entwining structure and Hom-entwined modules are investigated.

Article information

Rocky Mountain J. Math., Volume 49, Number 6 (2019), 2063-2105.

First available in Project Euclid: 3 November 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16T99: None of the above, but in this section 16T05: Hopf algebras and their applications [See also 16S40, 57T05] 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]

Monoidal comonad smash coproduct Hom bialgebra Hom-entwining structure


Zhang, Xiaohui; Wang, Wei; Zhao, Xiaofan. Smash coproducts of monoidal comonads and Hom-entwining structures. Rocky Mountain J. Math. 49 (2019), no. 6, 2063--2105. doi:10.1216/RMJ-2019-49-6-2063.

Export citation


  • \labeljjr J.N. Alonso Álvarez, J.M. Fernández Vilaboa and R. González Rodríguez, Cleft extensions and Galois extensions for Hom-associative algebras, Int. J. Math. 27 (2016), 1650025.
  • \labeljb J. Beck, Distributive laws, Lect. Notes Math. 80 (1969), 119–140.
  • \labelABAV A. Bruguières and A. Virelizier, Hopf monads, Adv. Math. 215 (2007), no. 2, 679–733.
  • \labelABA A. Bruguières, S. Lack and A. Virelizier, Hopf monads on monoidal categories, Adv. Math. 227 (2011), no. 2, 745–800.
  • \labelts T. Brzeziński and S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), 467–492.
  • \labelGTR G. Böhm, T. Brzeziński and R. Wisbauer, Monads and comands on module categories, J. Algebra 322 (2009), 1719–1747.
  • \labelG2 G. Böhm, S. Lack and R. Street, Weak bimonads and weak Hopf monads, J. Algebra 328 (2011), no. 1, 1–30.
  • \labelCG S. Caenepeel and I. Goyvaerts, Monoidal Hom-Hopf algebras, Comm. Algebra 39 (2011), 2216–2240.
  • \labelCG1 S. Caenepeel, B. Ion, G. Militaru and S.L. Zhu. The factorization problem and the smash biproduct of algebras and coalgebras, Algebr. Represent. Theory 3 (2000), 19–42.
  • \labelsgz1 S. Caenepeel, G. Militaru and S.L. Zhu. Frobenius and separable functors for generalized module categories and nonlinear equations, Lecture Notes in Mathematics, 1787 (2002), Springer, Berlin.
  • \labelSe S. Eilenberg and J.C. Moore, Adjoint functors and triples, Illinois J. Math. 9 (1965), no. 2, 381–398.
  • \labelAg A. Gohr, On Hom-algebras with surjective twisting, J. Algebra 324 (2010), 1483–1491.
  • \labelgzw S.J. Guo and X.H. Zhang. Separable functors for the category of Doi Hom-Hopf modules, Colloq. Math. 143 (2016), 23–37.
  • \labelgzw2 S.J. Guo, X.H. Zhang and S.X. Wang, Braided monoidal categories and Doi-Hopf modules for monoidal Hom-Hopf algebras, Colloq. Math. 143 (2016), no. 1, 79–103.
  • \labelJDS J. Hartwig, D. Larsson and S. Silvestrov. Deformation of Lie algebras using $\sigma$-derivations, J. Algebra 295 (2006), 314–361.
  • \labelsk S. Karacuha, Hom-entwining structures and Hom-Hopf-type modules (2015), e-Print arXiv: 1412.2002.
  • \labelfl F. Long, The Brauer group of dimodule algebras, J. Algebra 30 (1974), 559–601.
  • \labelSm S. Mac Lane, Categories for the working mathematicians, Graduate Texts in Math., Springer, New York (1971).
  • \labelSMA S. Mac Lane, Homologie des anneaux et des modules, Colloque de topologie algébrique (1956), Louvain.
  • \labelaf2 A. Makhlouf and F. Panaite, Hom-L-R-smash products, Hom-diagonal crossed products and the Drinfeld double of a Hom-Hopf algebra, J. Algebra 441 (2015), no. 1, 314–343.
  • \labelAF A. Makhlouf and F. Panaite, Yetter-Drinfeld modules for Hom-bialgebras, J. Math. Phys. 55 (2014), 013501.
  • \labelAS1 A. Makhlouf and S. Silvestrov, Hom-algebras and Hom-coalgebras, J. Algebra Appl. 9 (2010), 553–589.
  • \labelAS2 A. Makhlouf and S. Silvestrov, Hom-algebras structures, J. Gen. Lie Theory Appl. 2 (2008), 51–64.
  • \labelAS4 A. Makhlouf and S. Silvestrov, Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras, pp. 189–206 in Generalized Lie theory in mathematics, physics and beyond, Springer-Verlag, Berlin (2008).
  • \labelMB B. Mesablishvili, Entwining structures in monoidal categories, J. Algebra 319 (2008), 2496–2517.
  • \labelgm G. Militaru, The Long dimodules and nonlinear equations, Algebr. Represent. Theory 2 (1999), 177–200.
  • \labelMoe I. Moerdijk, Monads on tensor categories, J. Pure Appl. Algebra 168 (2002), no. 2-3, 189–208.
  • \labelJH J. Power and H. Watanabe, Combining a monad and a comonad, Theor. Comput. Sci. 280 (2002), no. 1-2, 137–162.
  • \labelROS R. Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972), no. 2, 149–168.
  • \labelwc D.G. Wang and Q.C. Chen. A Maschke type theorem for weak group entwined modules and applications, Israel J. Math. 204 (2014), no. 1, 329–358.
  • \labelwdg D.G. Wang, J.J. Zhang and G.B. Zhuang. Coassociative Lie algebras, Glasgow Math. J. 55 (2013), 195–215.
  • \labelRw R. Wisbauer, Algebras versus coalgebras, Appl. Categor. Struct. 16 (2008), 255–295.
  • \labelxw Y.J. Xu, D.G. Wang and J.L. Chen, Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups, J. Algebra Appl. 15 (2016), no. 10, 1650179.
  • \labeldy2 D. Yau, Hom-quantum groups I: Quasitriangular Hom-bialgebras, J. Phys. A 45 (2012), no. 6, 065203.
  • \labeldy4 D. Yau, Hom-Yang-Baxter equation, Hom-Lie algebras and quasitriangular bialgebras, J. Phys. A 42 (2009), no. 16, 165202.
  • \labeldy5 D. Yau, The Hom-Yang-Baxter equation and Hom-Lie algebras, J. Math. Phys. 52 (2011), 053502.
  • \labelZD X.H. Zhang and L.H. Dong, Braided mixed datums and their applications on Hom-quantum groups, Glasgow Math. J. 60 (2018), no. 1, 231–251.
  • \labelCoDr X.H. Zhang, S.J. Guo and S.X. Wang, Drinfeld codoubles of Hom-Hopf algebras, Adv. Appl. Clifford Algebras 29 (2019), no. 36.
  • \labelzw X.H. Zhang and S.H. Wang, Weak Hom-Hopf algebras and their (co)representations, J. Geom. Phys. 94 (2015), 50–71.
  • \labelzz X.F. Zhao and X.H. Zhang, Lazy $2$-cocycles over monoidal Hom-Hopf algebras, Colloq. Math. 142 (2016), no. 1, 61–81.