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.


17B55 Homological methods in Lie (super)algebras
17B01 Identities, free Lie (super)algebras
17B35 Universal enveloping (super)algebras


Zbl 0191.53702
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