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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)
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