On \(UHL\) and \(HUL\). (English) Zbl 1073.17502

Summary: Let \(R\) be a principal ideal domain of characteristic zero, containing 1/2, and let \(\rho=\rho(R)<\infty\) be the least non-invertible prime in \(R\). Our main result is the following: Let \((L,d)\) be a connected differential non-negatively graded Lie algebra over \(R\) whose underlying module is \(R\)-free of finite type. If \(\text{ad}^{\rho-1}(x)(dx)=0\), for homogeneous \(x\) in \(L_{\text{even}}\), then the natural morphism \(UFHL\to FHUL\) is an isomorphism of graded Hopf algebras; as usual, \(F\) stands for free part, \(H\) for homology, and \(U\) for universal enveloping algebra.
Related facts and examples are also considered.


17B35 Universal enveloping (super)algebras
16S30 Universal enveloping algebras of Lie algebras
16W30 Hopf algebras (associative rings and algebras) (MSC2000)