Algebra & Number Theory

Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras

Alexandru Chirvasitu

Full-text: Open access


The question of whether or not a Hopf algebra H is faithfully flat over a Hopf subalgebra A has received positive answers in several particular cases: when H (or more generally, just A) is commutative, cocommutative, or pointed, or when K contains the coradical of H. We prove the statement in the title, adding the class of cosemisimple Hopf algebras to those known to be faithfully flat over all Hopf subalgebras. We also show that the third term of the resulting “exact sequence” AHC is always a cosemisimple coalgebra, and that the expectation HA is positive when H is a CQG algebra.

Article information

Algebra Number Theory, Volume 8, Number 5 (2014), 1179-1199.

Received: 11 August 2013
Revised: 6 March 2014
Accepted: 21 April 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16T20: Ring-theoretic aspects of quantum groups [See also 17B37, 20G42, 81R50]
Secondary: 16T15: Coalgebras and comodules; corings 16T05: Hopf algebras and their applications [See also 16S40, 57T05] 20G42: Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]

cosemisimple Hopf algebra CQG algebra faithfully flat right coideal subalgebra quotient left module coalgebra expectation


Chirvasitu, Alexandru. Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras. Algebra Number Theory 8 (2014), no. 5, 1179--1199. doi:10.2140/ant.2014.8.1179.

Export citation


  • N. Andruskiewitsch and J. Devoto, “Extensions of Hopf algebras”, Algebra i Analiz 7:1 (1995), 22–61.
  • S. Arkhipov and D. Gaitsgory, “Another realization of the category of modules over the small quantum group”, Adv. Math. 173:1 (2003), 114–143.
  • T. Brzezinski and R. Wisbauer, Corings and comodules, London Mathematical Society Lecture Note Series 309, Cambridge University Press, 2003.
  • S. Caenepeel, E. De Groot, and J. Vercruysse, “Galois theory for comatrix corings: descent theory, Morita theory, Frobenius and separability properties”, Trans. Amer. Math. Soc. 359:1 (2007), 185–226.
  • A. Chirvasitu, “On epimorphisms and monomorphisms of Hopf algebras”, J. Algebra 323:5 (2010), 1593–1606.
  • M. Demazure and P. Gabriel, Groupes algébriques, I: Géométrie algébrique, généralités, groupes commutatifs, Masson, Paris, 1970.
  • M. S. Dijkhuizen and T. H. Koornwinder, “CQG algebras: a direct algebraic approach to compact quantum groups”, Lett. Math. Phys. 32:4 (1994), 315–330.
  • Y. Doi, “On the structure of relative Hopf modules”, Comm. Algebra 11:3 (1983), 243–255.
  • Y. Doi, “Hopf extensions of algebras and Maschke type theorems”, Israel J. Math. 72:1-2 (1990), 99–108.
  • J. Fogarty, Invariant theory, W. A. Benjamin, New York, 1969.
  • I. Kaplansky, Bialgebras, University of Chicago, 1975.
  • A. Klimyk and K. Schmüdgen, Quantum groups and their representations, Springer, Berlin, 1997.
  • S. Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics 5, Springer, New York, 1998.
  • A. Masuoka and D. Wigner, “Faithful flatness of Hopf algebras”, J. Algebra 170:1 (1994), 156–164.
  • B. Mesablishvili, “Monads of effective descent type and comonadicity”, Theory Appl. Categ. 16:1 (2006), 1–45.
  • S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics 82, American Mathematical Society, Providence, RI, 1993.
  • P. Nuss, “Noncommutative descent and non-abelian cohomology”, $K$-Theory 12:1 (1997), 23–74.
  • U. Oberst and H.-J. Schneider, “Untergruppen formeller Gruppen von endlichem Index”, J. Algebra 31 (1974), 10–44.
  • P. Podleś and S. L. Woronowicz, “Quantum deformation of Lorentz group”, Comm. Math. Phys. 130:2 (1990), 381–431.
  • D. E. Radford, “Operators on Hopf algebras”, Amer. J. Math. 99:1 (1977), 139–158.
  • D. E. Radford, “Pointed Hopf algebras are free over Hopf subalgebras”, J. Algebra 45:2 (1977), 266–273.
  • W. Rudin, Functional analysis, 2nd ed., McGraw-Hill, New York, 1991.
  • P. Salmi and A. Skalski, “Idempotent states on locally compact quantum groups”, Q. J. Math. 63:4 (2012), 1009–1032.
  • P. Schauenburg, “Faithful flatness over Hopf subalgebras: counterexamples”, pp. 331–344 in Interactions between ring theory and representations of algebras (Murcia, 1998), edited by F. Van Oystaeyen and M. Saorin, Lecture Notes in Pure and Appl. Math. 210, Dekker, New York, 2000.
  • P. Schauenburg and H.-J. Schneider, “On generalized Hopf Galois extensions”, J. Pure Appl. Algebra 202:1-3 (2005), 168–194.
  • H.-J. Schneider, “Principal homogeneous spaces for arbitrary Hopf algebras”, Israel J. Math. 72:1-2 (1990), 167–195.
  • M. E. Sweedler, Hopf algebras, W. A. Benjamin, New York, 1969.
  • M. Takesaki, Theory of operator algebras, I, Encyclopaedia of Mathematical Sciences 124, Springer, Berlin, 2002.
  • M. Takeuchi, “A correspondence between Hopf ideals and sub-Hopf algebras”, Manuscripta Math. 7 (1972), 251–270.
  • M. Takeuchi, “Relative Hopf modules: equivalences and freeness criteria”, J. Algebra 60:2 (1979), 452–471.
  • R. Tomatsu, “A characterization of right coideals of quotient type and its application to classification of Poisson boundaries”, Comm. Math. Phys. 275:1 (2007), 271–296.
  • A. Van Daele, “Discrete quantum groups”, J. Algebra 180:2 (1996), 431–444.
  • A. Van Daele, “The Haar measure on finite quantum groups”, Proc. Amer. Math. Soc. 125:12 (1997), 3489–3500.
  • A. Van Daele, “An algebraic framework for group duality”, Adv. Math. 140:2 (1998), 323–366.
  • S. Wang, “Simple compact quantum groups, I”, J. Funct. Anal. 256:10 (2009), 3313–3341.
  • N. E. Wegge-Olsen, $K$-theory and $C^*$-algebras: a friendly approach, Clarendon, New York, 1993.
  • S. L. Woronowicz, “Compact matrix pseudogroups”, Comm. Math. Phys. 111:4 (1987), 613–665.
  • S. L. Woronowicz, “Tannaka–Krein duality for compact matrix pseudogroups. Twisted $SU(N)$ groups”, Invent. Math. 93:1 (1988), 35–76.