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Separable Functors and Formal Smoothness

Published online by Cambridge University Press:  21 December 2007

Alessandro Ardizzoni
Alessandro Ardizzoni
Affiliation:
[email protected] of Ferrara, Department of Mathematics, Via Machiavelli 35, Ferrara, I-44100, Italy
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Abstract

The natural problem we approach in the present paper is to show how the notion of formally smooth (co)algebra inside monoidal categories can substitute that of (co)separable (co)algebra in the study of splitting bialgebra homomorphisms. This is performed investigating the relation between formal smoothness and separability of certain functors and led to other results related to Hopf algebra theory. Between them we prove that the existence of ad-(co)invariant integrals for a Hopf algebra H is equivalent to the separability of some forgetful functors. In the finite dimensional case, this is also equivalent to the separability of the Drinfeld Double D(H) over H. Hopf algebras which are formally smooth as (co)algebras are characterized. We prove that if π : E → H is a bialgebra surjection with nilpotent kernel such that H is a Hopf algebra which is formally smooth as a K-algebra, then π has a section which is a right H-colinear algebra homomorphism. Moreover, if H is also endowed with an ad-invariant integral, then this section can be chosen to be H-bicolinear. We also deal with the dual case.

Keywords

Monoidal categoriesHopf algebrasseparable and formally smooth algebrasseparable functorsad-invariant integrals
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 3 , June 2008 , pp. 535 - 582
Copyright
Copyright © ISOPP 2008

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References

Ar.Ardizzoni, A., The Category of Modules over a Monoidal Category: Abelian or not?, Ann. Univ. Ferrara - Sez. VII - Sc. Mat., Vol L (2004), 167–185CrossRefGoogle Scholar
AMS1.Ardizzoni, A., Menini, C. and Stefan, D.A Monoidal Approach to Splitting Morphisms of Bialgebras, Trans. Amer. Math. Soc., 359 (2007), 991–1044CrossRefGoogle Scholar
AMS2.Ardizzoni, A., Menini, C. and Ştefan, D., Hochschild Cohomology and ‘Smoothness’ in Monoidal Categories, J. Pure Appl. Algebra, Vol. 208 (2007) 297–330.CrossRefGoogle Scholar
Br.Brown, K. S., Cohomology of groups. Graduate Texts in Mathematics, 87. Springer-Verlag, New York-Berlin, 1982Google Scholar
CMZ.Caenepeel, S., Militaru, G. and Zhu, Shenglin, Frobenius Separable Functors for Generalized Module Categories and Nonlinear Equations, LNM 1787 (2002), Springer-Verlag, Berlin - New YorkGoogle Scholar
CQ.Cuntz, J. and Quillen, D., Algebra extensions and nonsingularity, J. of AMS 8 (1995), 251–289Google Scholar
Di.Dicks, W., Groups, trees and projective modules, Lecture Notes in Mathematics, 790. Springer, Berlin, 1980Google Scholar
DNR.Dăacăalescu, S., Năastăsescu, C. and Raianu, Ş., Hopf Algebras, Marcel Dekker, 2001Google Scholar
HS.Hilton, P.J. and Stambach, U., A course in Homological algebra, Graduate Text in Mathematics 4, Springer, New York, 1971Google Scholar
JLMS.Jara, P., Llena, D., Merino, L. and Ştefan, D., Hereditary and formally smooth coalgebras, Algebr. Represent. Theory 8 (2005), 363–374CrossRefGoogle Scholar
Ka.Kassel, , Quantum Groups, Graduate Text in Mathematics 155, Springer, 1995Google Scholar
MO.Masuoka, A. and Oka, T., Unipotent algebraic affine supergroups and nilpotent Lie superalgebras, Algebr. Represent. Theory 8 (2005), no. 3, 397–413CrossRefGoogle Scholar
McL.MacLane, S., Homology. Reprint of the first edition. Die Grundlehren der mathematischen Wissenschaften, Band 114. Springer-Verlag, Berlin-New York, 1967Google Scholar
Maj.Majid, S., Foundations of quantum group theory, Cambridge University Press, 1995CrossRefGoogle Scholar
Mon.Montgomery, S., Hopf Algebras and their actions on rings, CMBS Regional Conference Series in Mathematics 82 (1993)Google Scholar
NTZ.Năstăsescu, C., Torrecillas, B. and Zhang, Y. H., Hereditary coalgebras. Comm. Algebra 24 (1996), 1521–1528Google Scholar
NVV.Năstăsescu, C., Van den Bergh, M. and Van Oystaeyen, F., Separable functors applied to graded rings, J.Algebra 123 (1989), 397–413CrossRefGoogle Scholar
LB.Le Bruyn, L., Qurves and quivers, J. Algebra 290 (2005), no. 2, 447–472CrossRefGoogle Scholar
Rad.Radford, D. E., The structure of Hopf algebras with a projection, J. Algebra 92 (1985), 322–347CrossRefGoogle Scholar
Raf.Rafael, M. D., Separable Functors Revisited, Comm. Algebra 18 (1990), 1445–1459CrossRefGoogle Scholar
Scha1.Schauenburg, P., A generalization of Hopf crossed products, Comm. Algebra 27 1999), 4779–4801CrossRefGoogle Scholar
Scha2.Schauenburg, P., Hopf Modules and Yetter-Drienfel'd Modules, J. Algebra 169 (1994), 874–890CrossRefGoogle Scholar
Scha3.Schauenburg, P., The structure of Hopf algebras with a weak projection, Algebr. Represent. Theory 3 (1999), 187–211CrossRefGoogle Scholar
SVO.Ştefan, D. and Van Oystaeyen, F., The Wedderburn-Malcev theorem for comodule algebras, Comm. Algebra 27 (1999), 3569–3581CrossRefGoogle Scholar
Sw.Sweedler, M., Hopf Algebras, Benjamin, New York, 1969Google Scholar
We.Weibel, C., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, 1994Google Scholar