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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
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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.


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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