A nonnilpotent 1-2-presented graded Hopf algebra whose Hilbert series converges in the unit circle
type: Item
instance of: scholarly article
title: A nonnilpotent 1-2-presented graded Hopf algebra whose Hilbert series converges in the unit circle
zbMATH Open document ID: 0903.17012
author: Clas Löfwall
author: Jan-Erik Roos
published in: Advances in Mathematics
publication date: 1999-01-10T00:00:00Z
review text: The author indicates applications of some examples of a graded Lie algebra \(\eta\) to the theory of homotopy groups of finite CW-complexes. The enveloping algebra \(R^!\) of \(\eta\) is a Hopf algebra and has the properties indicated in the title of this paper: \[ R^!= \frac{k\langle X,Y,Z,U,V\rangle} {\binom{X^2,\;[X,Y],\;[X,Z],\;[Y,Z]- [X,U],\;Z^2+[Y,U],} {Z^2+ [X,V],\;[Z,U]- [Y,V],\;[U,V],\;V^2}}, \] where \(k\langle X,Y,Z,U,V\rangle\) is the free associative algebra in variables \(X,Y,Z,U,V\). The above properties are the answer to a 15 year-old existence question by Anick and Löfwall. In the last section, the authors show how the Lie algebra considered makes it possible to construct a finite simply-connected CW-complex with 12 cells: 5 cells of dimension 2 and 7 cells of dimension 4, whose integral loop space homology groups have torsion of all orders.
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B55
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 55Q15
zbMATH DE Number: 1076163
zbMATH Keywords: Whitehead product
zbMATH Keywords: Noetherian ring
zbMATH Keywords: Hopf algebra
zbMATH Keywords: graded Lie algebra
zbMATH Keywords: homotopy groups
zbMATH Keywords: finite CW-complexes
reviewed by: Jan Kubarski
describes a project that uses: GradedLieAlgebras
describes a project that uses: BERGMAN
MaRDI profile type: Publication
full work available at URL: aima.1997.1667
OpenAlex ID: W2036058017
cites work: A counterexample to a conjecture of Serre
cites work: The smallest \(\Omega\)-irrational CW-complex
cites work: A loop space whose homology has torsion of all orders
cites work: Q3344077
cites work: Torsion in loop space homology
cites work: Q3715234
cites work: Q5610428
cites work: Lie superalgebras
cites work: Q3859954
cites work: Algèbres connexes et homologie des espaces de lacets
cites work: Q3726293
cites work: Q3895579
cites work: Some remarks on Koszul algebras and quantum groups
cites work: On Lie gradings. I
cites work: Q3050539
cites work: Q3674791
cites work: Q1313788
cites work: Q3136279
cites work: Q4863560
cites work: On computer-assisted research in homological algebra.
DOI: 10.1006/AIMA.1997.1667
description: scientific article; zbMATH DE number 1076163