Hopf algebras of polynomial growth. (English) Zbl 0676.16008

Let G be a graded connected cocommutative Hopf algebra, with graded pieces \(G_ i\), \(i\geq 0\), of finite dimension over a field k (arbitrary characteristic). Assume that G has polynomial growth, i.e., for some integer \(r\geq 0\) and constant C, \(\sum^{n}_{i=0}(\dim G_ i)\leq Cn^ r\) for all \(n\geq 1\). Assume also that \(Ext_ G(k,G)\neq 0\). The main theorem of this paper is that G is then nilpotent and finitely- generated (as an algebra). The hypothesis \(Ext_ G(k,G)\neq 0\) is satisfied by \(H_*(\Omega X,k)\), \(\Omega\) X the loop space of a simply connected topological space X which has finite Lusternik-Schnirelmann category, and a topological application is obtained in this setting.
Reviewer: E.J.Taft


16W30 Hopf algebras (associative rings and algebras) (MSC2000)
57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
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