Sequences of divided powers in irreducible, cocommutative Hopf algebras

Author:
Kenneth Newman

Journal:
Trans. Amer. Math. Soc. **163** (1972), 25-34

MSC:
Primary 16A24; Secondary 18H15

DOI:
https://doi.org/10.1090/S0002-9947-1972-0292875-1

MathSciNet review:
0292875

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In Hopf algebras with one grouplike element, M. E. Sweedler showed that over perfect fields, sequences of divided powers in cocommutative, irreducible Hopf algebras can be extended if certain ``coheight'' conditions are met. Here, we show that with a suitable generalization of ``coheight", Sweedler's theorem is true over nonperfect fields. (We also point out, that in one case Sweedler's theorem was false, and additional conditions must be assumed.) In the same paper, Sweedler gave a structure theorem for irreducible, cocommutative Hopf algebras over perfect fields. We generalize this theorem in both the perfect and nonperfect cases. Specifically, in the nonperfect case, while a cocommutative, irreducible Hopf algebra does not, in general, satisfy the structure theorem, the sub-Hopf algebra, generated by all sequences of divided powers, does. Some additional properties of this sub-Hopf algebra are also given, including a universal property.

**[1]**R. Heyneman and M. E. Sweedler,*Affine Hopf algebras*. II, J. Algebra**16**(1970), 271-297. MR**0265433 (42:342)****[2]**K. Newman,*Topics in the theory of irreducible Hopf algebras*, Ph.D. Thesis, Cornell University, Ithaca, New York, 1970.**[3]**M. E. Sweedler,*Hopf algebras*, Math. Lecture Note Series, Benjamin, New York, 1969. MR**40**#5705. MR**0252485 (40:5705)****[4]**-,*Hopf algebras with one grouplike element*, Trans. Amer. Math. Soc.**127**(1967), 515-526. MR**35**#1634. MR**0210748 (35:1634)****[5]**-,*Weakening a theorem on divided powers*, Trans. Amer. Math. Soc.**154**(1971), 427-428. MR**0279162 (43:4888)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
16A24,
18H15

Retrieve articles in all journals with MSC: 16A24, 18H15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0292875-1

Keywords:
Coheight,
irreducible Hopf algebra,
sequence of divided powers

Article copyright:
© Copyright 1972
American Mathematical Society