The equivalence of certain categories of twisted Lie and Hopf algebras over a commutative ring

AbstractThe category of connected, twisted Lie algebras over a commutative ring R, which includes ordinary connected graded Lie algebras over R, is shown to be equivalent via an enveloping construction to a category of ‘connected R∑∗-Hopf algebras with commutative comultiplication’. Along the way, a PBW-like decomposition is obtained for the enveloping object of any connected, twisted Lie algebra, and free twisted Lie algebras are shown to admit an R-module basis of simple (left-normed) brackets of generators