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On the homology of free Lie algebras. (English) Zbl 1059.17503
Let $$H$$, $$\mathbb L$$, and $$U$$ denote the homology functor, the free Lie algebra functor, and the universal enveloping algebra functor, respectively. Let $$K$$ be a field of characteristic zero. D. G. Quillen has proved [Ann. Math. (2) 90, 205-295 (1969; Zbl 0191.53702)] that if $$V$$ is a differential graded $$K$$-vector space, then the natural homomorphism $$\mathbb L H(V) \rightarrow H\mathbb L(V)$$ is an isomorphism of graded $$K$$-Lie algebras, and if $$L$$ is a differential graded $$K$$-Lie algebra, then the natural homomorphism $$UH(L)\rightarrow HU(L)$$ is an isomorphism of graded cocommutative $$K$$-Hopf algebras. There results are no more valid if we replace $$K$$ by a field of non-zero characteristic or by a ring of characteristic zero containing 1/2. The author has discovered that factoring out the torsion in homology enables to generalize the above Quillen’s results. Let $$F$$ denote the free part. He shows that if $$R$$ is a principal ideal domain of characteristic zero containing 1/2 and $$V$$ is a connected differential non-negatively graded $$R$$-free module of finite type, then the natural homomorphisms $$\mathbb L FH(V) \rightarrow FH \mathbb L(V)$$ of graded $$R$$-Lie algebras, and $$UFH\mathbb L(V) \rightarrow FHU\mathbb L(V)$$ of graded cocommutative $$R$$-Hopf algebras, are both isomorphisms. In the end the author presents several interesting examples showing how these results can be used within the framework of the Quillen’s models.
##### MSC:
 17B55 Homological methods in Lie (super)algebras 17B01 Identities, free Lie (super)algebras 17B35 Universal enveloping (super)algebras
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