Universal enveloping algebras and loop space homology
type: Item
instance of: scholarly article
title: Universal enveloping algebras and loop space homology
zbMATH Open document ID: 0769.57025
DOI: 10.1016/0022-4049(92)90046-I
author: Stephen Halperin
published in: Journal of Pure and Applied Algebra
publication date: 1993-05-16T00:00:00Z
review text: If \(R\) is a principal ideal domain such that \(H_ *(\Omega X; R)\) is torsion free then \(H_ *(\Omega X; R)\) is a graded cocommutative Hopf algebra. The submodule \(P \subset H_ *(\Omega X;R)\) of primitive elements is a subgraded Lie algebra and the inclusion extends to a Hopf algebras homomorphism \[ \Phi: UP \to H_ *(\Omega X; R). \] In [Ann. Math., II. Ser. 81, 211-264 (1965; Zbl 0163.282)], \textit{J. W. Milnor} and \textit{J. C. Moore} show that if \(X\) is simply connected and \(R = Q\) then \(\Phi\) is an isomorphism. The first result of this paper is to extend this result to more general coefficient rings, at the cost of restricting the class of spaces \(X\). The second result consists to associate functorially to each space \(X\) in a certain class of spaces a graded Lie algebra \(L\) over \(\mathbb{F}_ p\) such that there exists a natural isomorphism of Hopf algebras \[ UL @>\cong>> H_ *(\Omega X;\mathbb{F}_ p). \] This theorem represents a considerable strengthening of a result of \textit{D. J. Anick} [J. Am. Math. Soc. 2, No. 3, 417-453 (1989; Zbl 0681.55006)] which asserts that \(H_ *(\Omega X;\mathbb{F}_ p)\) is primitively generated.
Mathematics Subject Classification ID: 57T25
Mathematics Subject Classification ID: 57T05
Mathematics Subject Classification ID: 55P35
zbMATH DE Number: 166156
zbMATH Keywords: Milnor-Moore theorem
zbMATH Keywords: \(\bmod p\) homology of loop spaces
zbMATH Keywords: graded cocommutative Hopf algebra
zbMATH Keywords: primitive elements
zbMATH Keywords: graded Lie algebra
reviewed by: Jean-Claude Thomas
MaRDI profile type: Publication
cites work: ON THE COBAR CONSTRUCTION
cites work: Hopf algebras with divided powers
cites work: Hopf Algebras up to Homotopy
cites work: Q3344077
cites work: Q3715232
cites work: Q3240881
cites work: Cohomology Theory of Lie Groups and Lie Algebras
cites work: Homology and fibrations I: Coalgebras, cotensor product and its derived functors
cites work: The Radical of the Homotopy Lie Algebra
cites work: Sur certaines algèbres de Lie de dérivations
cites work: Gorenstein spaces
cites work: Adams' Cobar Equivalence
cites work: On the chain-complex of a fibration
cites work: Q5610428
cites work: Le complexe de Koszul en algèbre et topologie. (The Koszul complex in algebra and topology)
cites work: Cohomology of Lie algebras
cites work: Differential homological algebra and homogeneous spaces
cites work: A Lower Central Series for Split Hopf Algebras with Involution
cites work: Homologie et cohomologie des algèbres de Lie
cites work: On the structure of Hopf algebras
cites work: Q5641276
cites work: Rational homotopy theory
cites work: Hopf algebras and derivations
cites work: Infinitesimal computations in topology
cites work: Homology of Noetherian rings and local rings
description: scientific article; zbMATH DE number 166156