A monoidal approach to splitting morphisms of bialgebras

Authors:
A. Ardizzoni, C. Menini and D. Ştefan

Journal:
Trans. Amer. Math. Soc. **359** (2007), 991-1044

MSC (2000):
Primary 16W30; Secondary 16S40

DOI:
https://doi.org/10.1090/S0002-9947-06-03902-X

Published electronically:
October 17, 2006

MathSciNet review:
2262840

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. Let us consider a Hopf algebra $A$ such that its Jacobson radical $J$ is a nilpotent Hopf ideal and $H:=A/J$ is a semisimple algebra. We prove that the canonical projection of $A$ on $H$ has a section which is an $H$–colinear algebra map. Furthermore, if $H$ is cosemisimple too, then we can choose this section to be an $(H,H)$–bicolinear algebra morphism. This fact allows us to describe $A$ as a ‘generalized bosonization’ of a certain algebra $R$ in the category of Yetter–Drinfeld modules over $H$. As an application we give a categorical proof of Radford’s result about Hopf algebras with projections. We also consider the dual situation. Let $A$ be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of $H$ into $A$ which is an $H$–linear coalgebra morphism. Furthermore, if $H$ is semisimple too, then we can choose this retraction to be an $(H,H)$–bilinear coalgebra morphism. Then, also in this case, we can describe $A$ as a ‘generalized bosonization’ of a certain coalgebra $R$ in the category of Yetter–Drinfeld modules over $H$.

- N. Andruskiewitsch and J. Devoto,
*Extensions of Hopf algebras*, Algebra i Analiz**7**(1995), no. 1, 22–61; English transl., St. Petersburg Math. J.**7**(1996), no. 1, 17–52. MR**1334152** - Nicolás Andruskiewitsch and Hans-Jürgen Schneider,
*Hopf algebras of order $p^2$ and braided Hopf algebras of order $p$*, J. Algebra**199**(1998), no. 2, 430–454. MR**1489920**, DOI https://doi.org/10.1006/jabr.1997.7175 - N. Andruskiewitsch and H.-J. Schneider,
*Lifting of quantum linear spaces and pointed Hopf algebras of order $p^3$*, J. Algebra**209**(1998), no. 2, 658–691. MR**1659895**, DOI https://doi.org/10.1006/jabr.1998.7643 - Nicolás Andruskiewitsch and Hans-Jürgen Schneider,
*Finite quantum groups and Cartan matrices*, Adv. Math.**154**(2000), no. 1, 1–45. MR**1780094**, DOI https://doi.org/10.1006/aima.1999.1880 - Nicolás Andruskiewitsch and Hans-Jürgen Schneider,
*Pointed Hopf algebras*, New directions in Hopf algebras, Math. Sci. Res. Inst. Publ., vol. 43, Cambridge Univ. Press, Cambridge, 2002, pp. 1–68. MR**1913436**, DOI https://doi.org/10.2977/prims/1199403805 - Nicolás Andruskiewitsch and Hans-Jürgen Schneider,
*On the coradical filtration of Hopf algebras whose coradical is a Hopf subalgebra*, Bol. Acad. Nac. Cienc. (Córdoba)**65**(2000), 45–50 (English, with English and Spanish summaries). Colloquium on Homology and Representation Theory (Spanish) (Vaquerías, 1998). MR**1840438** - A. Ardizzoni,
*Separable Functors and Formal Smoothness*, submitted (arXiv:math.QA/0407095). - A. Ardizzoni, C. Menini, D. Ştefan,
*Hochschild Cohomology and “Smoothness” in Monoidal Categories*, J. Pure Appl. Algebra, in press, available online at doi:10.1016/j.jpaa.2005.12.003. - M. Beattie, S. Dăscălescu, and L. Grünenfelder,
*On the number of types of finite-dimensional Hopf algebras*, Invent. Math.**136**(1999), no. 1, 1–7. MR**1681117**, DOI https://doi.org/10.1007/s002220050302 - C. Călinescu, S. Dăscălescu, A. Masuoka, and C. Menini,
*Quantum lines over non-cocommutative cosemisimple Hopf algebras*, J. Algebra**273**(2004), no. 2, 753–779. MR**2037722**, DOI https://doi.org/10.1016/j.jalgebra.2003.08.006 - V. G. Drinfel′d,
*Quantum groups*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR**934283** - Christian Kassel,
*Quantum groups*, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR**1321145** - Shahn Majid,
*Cross products by braided groups and bosonization*, J. Algebra**163**(1994), no. 1, 165–190. MR**1257312**, DOI https://doi.org/10.1006/jabr.1994.1011 - Shahn Majid,
*Foundations of quantum group theory*, Cambridge University Press, Cambridge, 1995. MR**1381692** - Akira Masuoka,
*Hopf cohomology vanishing via approximation by Hochschild cohomology*, Noncommutative geometry and quantum groups (Warsaw, 2001) Banach Center Publ., vol. 61, Polish Acad. Sci. Inst. Math., Warsaw, 2003, pp. 111–123. MR**2024425**, DOI https://doi.org/10.4064/bc61-0-8 - Susan Montgomery,
*Hopf algebras and their actions on rings*, 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. MR**1243637** - David E. Radford,
*The structure of Hopf algebras with a projection*, J. Algebra**92**(1985), no. 2, 322–347. MR**778452**, DOI https://doi.org/10.1016/0021-8693%2885%2990124-3 - David E. Radford,
*Minimal quasitriangular Hopf algebras*, J. Algebra**157**(1993), no. 2, 285–315. MR**1220770**, DOI https://doi.org/10.1006/jabr.1993.1102 - David E. Radford and Jacob Towber,
*Yetter-Drinfel′d categories associated to an arbitrary bialgebra*, J. Pure Appl. Algebra**87**(1993), no. 3, 259–279. MR**1228157**, DOI https://doi.org/10.1016/0022-4049%2893%2990114-9 - M. D. Rafael,
*Separable functors revisited*, Comm. Algebra**18**(1990), no. 5, 1445–1459. MR**1059740**, DOI https://doi.org/10.1080/00927879008823975 - Peter Schauenburg,
*Hopf modules and Yetter-Drinfel′d modules*, J. Algebra**169**(1994), no. 3, 874–890. MR**1302122**, DOI https://doi.org/10.1006/jabr.1994.1314 - Peter Schauenburg,
*The structure of Hopf algebras with a weak projection*, Algebr. Represent. Theory**3**(2000), no. 3, 187–211. MR**1783799**, DOI https://doi.org/10.1023/A%3A1009993517021 - Peter Schauenburg,
*Turning monoidal categories into strict ones*, New York J. Math.**7**(2001), 257–265. MR**1870871** - D. Ştefan and F. Van Oystaeyen,
*The Wedderburn-Malcev theorem for comodule algebras*, Comm. Algebra**27**(1999), no. 8, 3569–3581. MR**1699590**, DOI https://doi.org/10.1080/00927879908826648 - Neantro Saavedra Rivano,
*Catégories Tannakiennes*, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin-New York, 1972 (French). MR**0338002** - Moss E. Sweedler,
*Hopf algebras*, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR**0252485** - Moss Eisenberg Sweedler,
*Cohomology of algebras over Hopf algebras*, Trans. Amer. Math. Soc.**133**(1968), 205–239. MR**224684**, DOI https://doi.org/10.1090/S0002-9947-1968-0224684-2 - Earl J. Taft and Robert Lee Wilson,
*On antipodes in pointed Hopf algebras*, J. Algebra**29**(1974), 27–32. MR**338053**, DOI https://doi.org/10.1016/0021-8693%2874%2990107-0 - S. L. Woronowicz,
*Differential calculus on compact matrix pseudogroups (quantum groups)*, Comm. Math. Phys.**122**(1989), no. 1, 125–170. MR**994499**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
16W30,
16S40

Retrieve articles in all journals with MSC (2000): 16W30, 16S40

Additional Information

**A. Ardizzoni**

Affiliation:
Department of Mathematics, University of Ferrara, Via Machiavelli 35, Ferrara, I-44100, Italy

Email:
alessandro.ardizzoni@unife.it

**C. Menini**

Affiliation:
Department of Mathematics, University of Ferrara, Via Machiavelli 35, I-44100, Ferrara, Italy

Email:
men@dns.unife.it

**D. Ştefan**

Affiliation:
Faculty of Mathematics, University of Bucharest, Strada Academiei 14, Bucharest, RO-70109, Romania

Email:
dstefan@al.math.unibuc.ro

Keywords:
Hopf algebras,
bialgebras,
smash (co)products,
monoidal categories

Received by editor(s):
July 1, 2004

Received by editor(s) in revised form:
November 3, 2004, and November 17, 2004

Published electronically:
October 17, 2006

Additional Notes:
This paper was written while the first two authors were members of G.N.S.A.G.A. with partial financial support from M.I.U.R. The third author was partially supported by I.N.D.A.M., while he was a visiting professor at the University of Ferrara.

Article copyright:
© Copyright 2006
American Mathematical Society