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The cohomology of certain Hopf algebras associated with $p$-groups


Author: Justin M. Mauger
Journal: Trans. Amer. Math. Soc. 356 (2004), 3301-3323
MSC (2000): Primary 16E40; Secondary 16S37, 16S30
DOI: https://doi.org/10.1090/S0002-9947-03-03381-6
Published electronically: November 12, 2003
MathSciNet review: 2052951
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Abstract: We study the cohomology $H^*(A)=\operatorname{Ext}_A^*(k,k)$ of a locally finite, connected, cocommutative Hopf algebra $A$ over $k=\mathbb{F} _p$. Specifically, we are interested in those algebras $A$ for which $H^*(A)$ is generated as an algebra by $H^1(A)$ and $H^2(A)$. We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras $F\rightarrow A\rightarrow B$ with $F$ monogenic and $B$ semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for $A$ to be semi-Koszul. Special attention is given to the case in which $A$ is the restricted universal enveloping algebra of the Lie algebra obtained from the mod-$p$ lower central series of a $p$-group. We show that the algebras arising in this way from extensions by $\mathbb{Z} /(p)$ of an abelian $p$-group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 $p$-groups, and it is shown that these are all semi-Koszul for $p\geq 5$.


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Additional Information

Justin M. Mauger
Affiliation: Department of Mathematics, Whittier College, Whittier, California 90608
Address at time of publication: Department of Mathematics and Computer Science, California State University, Channel Islands, Camarillo, California 93012
Email: jmauger@whittier.edu, justin.mauger@csuci.edu

DOI: https://doi.org/10.1090/S0002-9947-03-03381-6
Keywords: Koszul algebras, cohomology of algebras, Hopf algebras
Received by editor(s): April 30, 2002
Received by editor(s) in revised form: April 2, 2003
Published electronically: November 12, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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