Algebra universalis

, 79:18 | Cite as

Hopf monoids in varieties

Article

Abstract

Commutative varieties provide a natural setting for generalizing Hopf algebra theory over commutative rings, since they satisfy the various conditions identified in the category theoretical analysis of this theory to guarantee for example the existence of all naturally occurring forgetful functors in this context, the existence of universal measuring comonoids and the existence of generalized finite duals. It will be shown in addition, that crucial properties of the latter, known from the case of Hopf algebra theory over commutative rings, can be generalized Hopf algebra theory over a commutative variety. The attempt to generalize its construction leads to a couple of questions concerning properties of the monoidal structure of a commutative variety, which seem to be of a more general interest.

Keywords

Commutative varieties Hopf monoids Free and cofree Hopf monoids Universal measuring comonoids Group algebras Finite dual 

Mathematics Subject Classification

08B99 16T05 

Notes

Acknowledgements

I am grateful to the anonymous referee for his of her constructive criticism, which lead to a considerable improvement of the presentation.

References

  1. 1.
    Abuhlail, J.Y., Al-Sulaiman, N.: Hopf semialgebras. Commun. Algebra 43, 1241–1278 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Abuhlail, J.Y., Gómez-Torrecillas, J., Wisbauer, R.: Dual coalgebras of algebras over commutative rings. J. Pure Appl. Algebra 153, 107–120 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. Wiley, New York (1990)MATHGoogle Scholar
  4. 4.
    Aguiar, M., Mahajan, S.: Monoidal functors, species and Hopf algebras. In: CRM Monograph Series , vol. 29. American Mathematical Society, Providence (2010)Google Scholar
  5. 5.
    Anel, M., Joyal, A.: Sweedler theory of (co)algebras and the bar–cobar constructions. arXiv:1309.6952 (2013)
  6. 6.
    Banaschewski, B., Nelson, E.: Tensor products and bimorphisms. Can. Math. Bull. 19, 385–402 (1976)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bonsangue, M.M., Milius, S., Silva, A.: Sound and complete axiomatizations of coalgebraic language equivalence. ACM Trans. Comput. Log. 14.1(article 7) (2013)Google Scholar
  8. 8.
    Borceux, F.: Handbook of Categorical Algebra 2. Cambridge University Press, Cambridge (1994)CrossRefMATHGoogle Scholar
  9. 9.
    Chen, C.Y., Nichols, W.D.: A duality theorem for Hopf module algebras over Dedekind rings. Commun. Algebra 18(10), 3209–3221 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Davey, B.A., Davis, G.: Tensor products and entropic varieties. Algebra Univ. 21, 68–88 (1985)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ésik, Z., Maletti, A.: Simulations of weighted tree automata. In: Domaratzki, M., Salomaa, K. (eds.) Proceedings of the 15th International Conference on Implementation and Application of Automata (CIAA). Lecture Notes in Computer Science, vol. 6482, pp. 321–330. Springer, Berlin (2011)Google Scholar
  12. 12.
    Hyland, M., López Franco, I., Vasilakopoulou, C.: Hopf measuring comonoids and enrichment. arXiv:1509.07632 (2015)
  13. 13.
    Kelly, G.M.: Doctrinal adjunction. Lect. Notes Math. 420, 257–280 (1974). (Springer)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kelly, G.M., Street, R.: Review of the elements of 2-categories. Lect. Notes Math. 420, 75–103 (1974). Springer-VerlagMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Klukovits, L.: On commutative universal algebras. Acta. Sci. Math. (Szeged) 37, 11–15 (1973)MathSciNetMATHGoogle Scholar
  16. 16.
    Linton, F.E.J.: Autonomous equational categories. J. Math. Mech. 15, 637–642 (1966)MathSciNetMATHGoogle Scholar
  17. 17.
    Manes, E.G.: Algebraic Theories. Springer, New York (1976)CrossRefMATHGoogle Scholar
  18. 18.
    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1998)MATHGoogle Scholar
  19. 19.
    Neumann, W.D.: Mal’cev conditions, spectra, and the Kronecker product. J. Aust. Math. Soc. (Ser. A) 25, 103–117 (1978)CrossRefMATHGoogle Scholar
  20. 20.
    Pareigis, B.: Non-additive ring and module theory I. Publ. Math. Debr. 24, 189–204 (1977)MATHGoogle Scholar
  21. 21.
    Porst, H.-E.: On categories of monoids, comonoids, and bimonoids. Quaest. Math. 31, 127–139 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Porst, H.-E.: The formal theory of Hopf algebras part I. Quaest. Math. 38, 631–682 (2015)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Porst, H.-E.: The formal theory of Hopf algebras part II. Quaest. Math. 38, 683–708 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Porst, H.-E.: Hopf monoids in semi-additive varieties. Log. Methods Comput. Sci. 13, 1–13 (2017)MathSciNetGoogle Scholar
  25. 25.
    Porst, H.-E., Street, R.: Generalizations of the Sweedler dual. Appl. Categ. Struct. 24, 619–647 (2016)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Romanowska, A., Smith, J.D.H.: On Hopf algebras in entropic Jónsson–Tarski varieties. Bull. Korean Math. Soc. 52, 1587–1606 (2015)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Sweedler, M.E.: Hopf Algebras. Benjamin, New York (1969)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of StellenboschStellenboschSouth Africa
  2. 2.Department of MathematicsUniversity of BremenBremenGermany

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