Skip to main content
SpringerLink
Log in

Epimorphic Quantum Subgroups and Coalgebra Codominions

  • Research
  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic subgroup be epimorphically embedded; (b) the fact that a morphism in the category of (cocommutative) coalgebras, (cocommutative) bialgebras, and a host of categories of Hopf algebras has the same codominion in any of these categories which contain it; (c) the invariance of the Hopf algebra or bialgebra (co)dominion construction under field extension, again mimicking the well-known corresponding algebraic-group result; (d) the fact that surjections of coalgebras, bialgebras or Hopf algebras are regular epimorphisms (i.e. coequalizers) provided the codomain is cosemisimple; (e) in particular, the fact that embeddings of compact quantum groups are equalizers in the category thereof, generalizing analogous results on (plain) compact groups; (f) coalgebra-limit preservation results for scalar-extension functors (e.g. extending scalars along a field extension \(\Bbbk \le \Bbbk '\) is a right adjoint on the category of \(\Bbbk \)-coalgebras).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Availability of data and materials

Not applicable

References

  1. Adámek, J., Herrlich, H., Strecker, George E.: Abstract and concrete categories. Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York (1990) The joy of cats, A Wiley-Interscience Publication

  2. Adámek, J., Rosický, J.: Locally presentable and accessible categories, volume 189 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1994)

  3. Agore, A.L.: Monomorphisms of coalgebras. Colloq Math 120(1), 149–155 (2010)

    Article  Google Scholar 

  4. Agore, A.L.: Limits of coalgebras, bialgebras and Hopf algebras. Proc Amer Math Soc 139(3), 855–863 (2011)

    Article  MathSciNet  Google Scholar 

  5. Barr, M., Wells, C.: Toposes, triples and theories. In: Grundlehren der mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences, vol. 278. Springer-Verlag, New York, (1985)

  6. Bergman, George M.: Epimorphisms of lie algebras (1973). available at https://math.berkeley.edu/~gbergman/papers/unpub/LieEpi.pdf Accessed 02 Sept 2023

  7. Bergman, George M.: The diamond lemma for ring theory. Adv in Math 29(2), 178–218 (1978)

    Article  MathSciNet  Google Scholar 

  8. Białynicki-Birula, A., Hochschild, A., Mostow, A.: Extensions of representations of algebraic linear groups. Amer J Math 85, 131–144 (1963)

    Article  MathSciNet  Google Scholar 

  9. Bichon, J.: Faithful flatness of hopf algebras over coideal subalgebras with a conditional expectation (2023). arXiv:2301.05480v1

  10. Bien, F., Borel, A.: Sous-groupes épimorphiques de groupes linéaires algébriques. I CR Acad Sci Paris Sér I Math 315(6), 649–653 (1992)

    Google Scholar 

  11. Bien, F., Armand Borel, A.: Sous-groupes épimorphiques de groupes linéaires algébriques. II CR Acad Sci Paris Sér I Math 315(13), 1341–1346 (1992)

    Google Scholar 

  12. Borel, A.: Linear algebraic groups. In: Graduate Texts in Mathematics, vol. 278. Springer-Verlag, New York, second edition (1991)

  13. Brion, M.: Epimorphic subgroups of algebraic groups. Math Res Lett 24(6), 1649–1665 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  14. Bröcker, T., Dieck, Tammo tom.: Representations of compact Lie groups, volume 98 of Graduate Texts in Mathematics Springer-Verlag, New York (1985)

  15. Brzezinski, T., Wisbauer, R.: Corings and comodules. In: London Mathematical Society Lecture Note Series, vol. 309. Cambridge University Press, Cambridge (2003)

  16. Chirvasitu, A.: Cosemisimple hopf algebras are faithfully flat over hopf subalgebras (2011) arXiv:1110.6701v1

  17. Chirvasitu, A.: Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras. Algebra Number Theory 8(5), 1179–1199 (2014)

    Article  MathSciNet  Google Scholar 

  18. Chirvasitu, A.: Categorical aspects of compact quantum groups. Appl Categ Structures 23(3), 381–413 (2015)

    Article  MathSciNet  Google Scholar 

  19. Chirvasitu, A.: Non-degeneracy results for (multi-)pushouts of compact groups (2023) arXiv:2301.09847v2

  20. Chirvăsitu, A.: On epimorphisms and monomorphisms of Hopf algebras. J. Algebra 323(5), 1593–1606 (2010)

    Article  MathSciNet  Google Scholar 

  21. Cipolla, M.: Discesa fedelmente piatta dei moduli. Rend Circ Mat Palermo (2) 25(1-2), 43–46 (1976)

  22. Deligne, P.: Catégories tannakiennes. In: The Grothendieck Festschrift, Vol. II. Progr Math, vol. 87. 111–195. Birkhäuser Boston, Boston, MA, (1990)

  23. Dijkhuizen, M.S., Koornwinder, T.H.: CQG algebras: a direct algebraic approach to compact quantum groups. Lett. Math. Phys. 32(4), 315–330 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  24. Dăscălescu, S., Năstăsescu, C., Raianu, Ş.: Hopf algebras. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 235. Marcel Dekker, Inc., New York, An introduction (2001)

  25. Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories. In: Mathematical Surveys and Monographs, vol. 205. American Mathematical Society, Providence, RI, (2015)

  26. Evens, L.: The cohomology of groups. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, Oxford Science Publications (1991)

    Book  Google Scholar 

  27. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous lattices and domains. In: Encyclopedia of Mathematics and its Applications, vol. 93. Cambridge University Press, Cambridge, (2003)

  28. Grosshans, F.D.: Algebraic homogeneous spaces and invariant theory. In: Lecture Notes in Mathematics, vol. 1673. Springer-Verlag, Berlin, (1997)

  29. Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, (1977)

  30. Hofmann, K.H., Neeb, K.H.: Epimorphisms of \(C^*\)-algebras are surjective. Arch Math (Basel) 65(2), 134–137 (1995)

    Article  MathSciNet  Google Scholar 

  31. Howie, J.M., Isbell, J.R.: Epimorphisms and dominions. II. In: J. Algebra 6, 7–21 (1967)

    MathSciNet  Google Scholar 

  32. Humphreys, J.E.: Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, Vol. 9. Springer-Verlag, New York-Berlin, (1972)

  33. Isbell, J.R.: Epimorphisms and dominions, V. Algebra Universalis 3, 318–320 (1973)

  34. Isbell, J.R.: Epimorphisms and dominions. In: Proc Conf Categorical Algebra (La Jolla, Calif., 1965) pages 232–246. Springer, New York, (1966)

  35. Isbell, J.R.: Epimorphisms and dominions. III. In: Amer J Math 90, 1025–1030 (1968)

    MathSciNet  Google Scholar 

  36. Isbell, J.R.: Epimorphisms and dominions. IV. In: J London Math Soc 2(1), 265–27 (1969)

    MathSciNet  Google Scholar 

  37. Jacobson, N.: Lie algebras. Dover Publications, Inc., New York. Republication of the 1962 Original (1979)

  38. Krause, H.: Deriving Auslander’s formula. Doc Math 20, 669–688 (2015)

    Article  Google Scholar 

  39. Lam, T.Y.: Lectures on modules and rings. In: Graduate Texts in Mathematics, vol. 189. Springer-Verlag, New York, (1999)

  40. Mac Lane, S.: Categories for the working mathematician. Graduate Texts in Mathematics, vol. 5. Springer-Verlag, New York, second edition, (1998)

  41. Mesablishvili, B.: Monads of effective descent type and comonadicity. Theory Appl Categ 16(1), 1–45 (2006)

    MathSciNet  Google Scholar 

  42. Montgomery, S.: Hopf algebras and their actions on rings. In: CBMS Regional Conference Series in Mathematics, vol. 82. Published for the Conference Board of the Mathematical Sciences, Washington, DC, (1993)

  43. Montgomery, S.: Hopf algebras and their actions on rings. In: CBMS Regional Conference Series in Mathematics, vol. 82. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, (1993)

  44. Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34. Springer-Verlag, Berlin, third edition, (1994)

  45. Năstăsescu, C., Torrecillas, B.: Torsion theories for coalgebras. In: J Pure Appl Algebra 97(2),203–220 (1994)

  46. Nuss, P.: Noncommutative descent and non-abelian cohomology. \(K\)-Theory 12(1):23–74 (1997)

  47. Petukhov, A.V.: A geometric description of epimorphic subgroups. Uspekhi Mat. Nauk 65(5(395)), 193–194 (2010)

  48. Poguntke, D.: Epimorphisms of compact groups are onto. Proc Amer Math Soc 26, 503–504 (1970)

    Article  MathSciNet  Google Scholar 

  49. Popescu, N.: Abelian categories with applications to rings and modules. Academic Press, London-New York, London Mathematical Society Monographs, No. 3 (1973)

  50. Porst, H.E.: On categories of monoids, comonoids, and bimonoids. Quaest Math 31(2), 127–139 (2008)

    Article  MathSciNet  Google Scholar 

  51. Porst, H.E.: Universal constructions for Hopf algebras. In: J Pure Appl Algebra 212(11), 2547–2554 (2008)

  52. Porst, H.E.: The formal theory of Hopf algebras Part II: The case of Hopf algebras. Quaest Math 38(5), 683–708 (2015)

    Article  MathSciNet  Google Scholar 

  53. Radford, D.E.: Hopf algebras. In: Series on Knots and Everything, vol. 49. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2012)

  54. Reid, G.A.: Epimorphisms and surjectivity. Invent Math 9, 295–307 (1969/70)

  55. Schauenburg, P.: Tannaka duality for arbitrary Hopf algebras. Algebra Berichte [Algebra Reports], vol. 66. Verlag Reinhard Fischer, Munich, (1992)

  56. Schauenburg, P.: Faithful flatness over Hopf subalgebras: counterexamples. In: Interactions between ring theory and representations of algebras (Murcia), vol. 210. Lecture Notes in Pure and Appl Math pages 331–344. Dekker, New York, (2000)

  57. Stenström, B.: Rings of quotients. Springer-Verlag, New York, Die Grundlehren der Mathematischen Wissenschaften, Band 217, An introduction to methods of ring theory (1975)

  58. Sweedler, M.E.: Hopf algebras. In: Mathematics Lecture Note Series. W. A. Benjamin, Inc., New York, (1969)

  59. Takeuchi, M.: Relative Hopf modules–equivalences and freeness criteria. In: J Algebra 60(2), 452–471 (1979)

  60. Thǎńg, N.Q., Bǎć, D.P.: Some rationality properties of observable groups and related questions. In: Illinois J Math 49(2), 431–444 (2005)

  61. Wang, S.: Free products of compact quantum groups. Comm Math Phys 167(3), 671–692 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  62. Waterhouse, W.C.: Introduction to affine group schemes. In: Graduate Texts in Mathematics, vol. 66. Springer-Verlag, New York-Berlin, (1979)

Download references

Acknowledgements

This work was partially supported through NSF grant DMS-2001128. I am grateful for assorted comments and pointers to the literature from A. Agore, M. Brion, T. Brzezinski, G. Militaru and R. Wisbauer

Funding

NSF grant DMS-2001128

Author information

Authors and Affiliations

  1. Department of Mathematics, University at Buffalo, Buffalo, 14260-2900, NY, USA

    Alexandru Chirvasitu

Authors
  1. Alexandru Chirvasitu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Not applicable (single author)

Corresponding author

Correspondence to Alexandru Chirvasitu.

Ethics declarations

Competing interests

Not applicable

Ethical Approval

Not applicable

Additional information

Presented by: Michel Brion.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chirvasitu, A. Epimorphic Quantum Subgroups and Coalgebra Codominions. Algebr Represent Theor 27, 219–244 (2024). https://doi.org/10.1007/s10468-023-10219-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-023-10219-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation