A semidirect product decomposition for certain Hopf algebras over an algebraically closed field

Author:
Richard K. Molnar

Journal:
Proc. Amer. Math. Soc. **59** (1976), 29-32

MSC:
Primary 17B50

DOI:
https://doi.org/10.1090/S0002-9939-1976-0430009-X

MathSciNet review:
0430009

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Abstract: Let be a finite dimensional Hopf algebra over an algebraically closed field. We show that if is commutative and the coradical is a sub Hopf algebra, then the canonical inclusion has a Hopf algebra retract; or equivalently, if is cocommutative and the Jacobson radical is a Hopf ideal, then the canonical projection has a Hopf algebra section.

**[1]**R. G. Heyneman and M. E. Sweedler,*Affine Hopf algebras*. I, J. Algebra**13**(1969), 192-241. MR**39**#6876. MR**0245570 (39:6876)****[2]**S. Mac Lane,*Homology*, Grundlehren math. Wiss., Band 114, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR**28**#122. MR**1344215 (96d:18001)****[3]**R. K. Molnar,*Semi-direct products of Hopf algebras*, J. Algebra (to appear).**4.**-,*On the coradical of a Hopf algebra*(to appear). MR**0498612 (58:16700)****[5]**J. B. Sullivan,*Affine group schemes with integrals*, J. Algebra**22**(1972), 546-558. MR**46**#3553. MR**0304418 (46:3553)****[6]**-,*A decomposition theorem for pro-affine solvable algebraic groups over algebraically closed fields*, Amer. J. Math.**95**(1973), 221-228. MR**50**#13061. MR**0360614 (50:13061)****[7]**M. E. Sweedler,*Connected fully reducible affine group schemes in positive characteristic are abelian*, J. Math. Kyoto Univ.**11**(1971), 51-70. MR**43**#6219. MR**0280499 (43:6219)****[8]**-,*Hopf algebras*, Benjamin, New York, 1969. MR**40**#5705. MR**0252485 (40:5705)****[9]**M. Takeuchi,*A correspondence between Hopf ideals and sub Hopf algebras*, Manuscripta Math.**7**(1972), 251-270. MR**48**#328. MR**0321963 (48:328)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0430009-X

Keywords:
Hopf algebra,
coradical,
semidirect product

Article copyright:
© Copyright 1976
American Mathematical Society