A monoidal approach to splitting morphisms of bialgebras

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

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

Full-text PDF

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 such that its Jacobson radical is a nilpotent Hopf ideal and is a semisimple algebra. We prove that the canonical projection of on has a section which is an -colinear algebra map. Furthermore, if is cosemisimple too, then we can choose this section to be an -bicolinear algebra morphism. This fact allows us to describe as a `generalized bosonization' of a certain algebra in the category of Yetter-Drinfeld modules over . As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. Let be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of into which is an -linear coalgebra morphism. Furthermore, if is semisimple too, then we can choose this retraction to be an -bilinear coalgebra morphism. Then, also in this case, we can describe as a `generalized bosonization' of a certain coalgebra in the category of Yetter-Drinfeld modules over .

**[AD]**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****[AS1]**Nicolás Andruskiewitsch and Hans-Jürgen Schneider,*Hopf algebras of order 𝑝² and braided Hopf algebras of order 𝑝*, J. Algebra**199**(1998), no. 2, 430–454. MR**1489920**, https://doi.org/10.1006/jabr.1997.7175**[AS2]**N. Andruskiewitsch and H.-J. Schneider,*Lifting of quantum linear spaces and pointed Hopf algebras of order 𝑝³*, J. Algebra**209**(1998), no. 2, 658–691. MR**1659895**, https://doi.org/10.1006/jabr.1998.7643**[AS3]**Nicolás Andruskiewitsch and Hans-Jürgen Schneider,*Finite quantum groups and Cartan matrices*, Adv. Math.**154**(2000), no. 1, 1–45. MR**1780094**, https://doi.org/10.1006/aima.1999.1880**[AS4]**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**, https://doi.org/10.2977/prims/1199403805**[AS5]**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****[Ar1]**A. Ardizzoni,*Separable Functors and Formal Smoothness*, submitted (arXiv:math.QA/0407095).**[AMS]**A. Ardizzoni, C. Menini, D. Stefan,*Hochschild Cohomology and ``Smoothness'' in Monoidal Categories*, J. Pure Appl. Algebra, in press, available online at doi:10.1016/j.jpaa.2005.12.003.**[BDG]**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**, https://doi.org/10.1007/s002220050302**[CDMM]**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**, https://doi.org/10.1016/j.jalgebra.2003.08.006**[Dr]**V. G. Drinfeld,*Quantum groups*, ``Proceedings of the International Congress of Mathematicians", Vol.**1, 2**(Berkeley, Calif., 1986), 798-820, Amer. Math. Soc., Providence, RI, 1987. MR**0934283 (89f:17017)****[Ka]**Christian Kassel,*Quantum groups*, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR**1321145****[Maj1]**Shahn Majid,*Cross products by braided groups and bosonization*, J. Algebra**163**(1994), no. 1, 165–190. MR**1257312**, https://doi.org/10.1006/jabr.1994.1011**[Maj2]**Shahn Majid,*Foundations of quantum group theory*, Cambridge University Press, Cambridge, 1995. MR**1381692****[Mas]**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**, https://doi.org/10.4064/bc61-0-8**[Mo]**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****[Ra1]**D.E. Radford,*The structure of Hopf algebras with a projection*, J. Algebra**92**(1985), 322-347. MR**0778452 (86k:16004)****[Ra2]**David E. Radford,*Minimal quasitriangular Hopf algebras*, J. Algebra**157**(1993), no. 2, 285–315. MR**1220770**, https://doi.org/10.1006/jabr.1993.1102**[RT]**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**, https://doi.org/10.1016/0022-4049(93)90114-9**[Raf]**M. D. Rafael,*Separable functors revisited*, Comm. Algebra**18**(1990), no. 5, 1445–1459. MR**1059740**, https://doi.org/10.1080/00927879008823975**[Sch1]**Peter Schauenburg,*Hopf modules and Yetter-Drinfel′d modules*, J. Algebra**169**(1994), no. 3, 874–890. MR**1302122**, https://doi.org/10.1006/jabr.1994.1314**[Sch2]**Peter Schauenburg,*The structure of Hopf algebras with a weak projection*, Algebr. Represent. Theory**3**(2000), no. 3, 187–211. MR**1783799**, https://doi.org/10.1023/A:1009993517021**[Sch3]**Peter Schauenburg,*Turning monoidal categories into strict ones*, New York J. Math.**7**(2001), 257–265. MR**1870871****[SvO]**D. Ştefan and F. Van Oystaeyen,*The Wedderburn-Malcev theorem for comodule algebras*, Comm. Algebra**27**(1999), no. 8, 3569–3581. MR**1699590**, https://doi.org/10.1080/00927879908826648**[SR]**Neantro Saavedra Rivano,*Catégories Tannakiennes*, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin-New York, 1972 (French). MR**0338002****[Sw]**Moss E. Sweedler,*Hopf algebras*, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR**0252485****[Sw2]**Moss Eisenberg Sweedler,*Cohomology of algebras over Hopf algebras*, Trans. Amer. Math. Soc.**133**(1968), 205–239. MR**0224684**, https://doi.org/10.1090/S0002-9947-1968-0224684-2**[TW]**Earl J. Taft and Robert Lee Wilson,*On antipodes in pointed Hopf algebras*, J. Algebra**29**(1974), 27–32. MR**0338053**, https://doi.org/10.1016/0021-8693(74)90107-0**[Wo]**S. L. Woronowicz,*Differential calculus on compact matrix pseudogroups (quantum groups)*, Comm. Math. Phys.**122**(1989), 125-170. MR**0994499 (90g:58010)**

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. Stefan**

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

Email:
dstefan@al.math.unibuc.ro

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

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