## 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.

### MSC:

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