On the homology of free Lie algebras. (English) Zbl 1059.17503

Popescu, Calin

Commentat. Math. Univ. Carol. 39, No. 4, 661-669 (1998).

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.

Reviewer: Jiří Vanžura (Brno)

MSC:

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

Keywords:

differential graded Lie algebra; free Lie algebra; universal enveloping algebra; Quillen’s model

Citations:

Zbl 0191.53702