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Universal enveloping algebras and loop space homology. (English) Zbl 0769.57025
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)], J. W. Milnor and 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 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.

##### MSC:
 57T25 Homology and cohomology of $$H$$-spaces 57T05 Hopf algebras (aspects of homology and homotopy of topological groups) 55P35 Loop spaces
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