The equivalence of certain categories of twisted Lie and Hopf algebras over a commutative ring (Q1210357)

For \(R\) a unital commutative ring, a twisted Lie algebra over \(R\) (or \(R\Sigma_ x\)-Lie algebra) is a graded \(R\)-module \(L = (L_ 0,L_ 1,L_ 2,\dots)\) with a bracket \(L_ p \otimes L_ q \to L_{p+q}\), and an action of the symmetric group \(\Sigma_ n\) on \(L_ n\) such that certain ``Lie identities'' hold. \(L\) is connected if \(L_ 0 = 0\). Similarly, one defines twisted algebras, (here the 0-th component is \(R\)), and twisted Hopf algebras. One also has twisted modules for these various twisted types of algebras. The author generalizes the Poincaré- Birkhoff-Witt theory of Lie algebras to twisted Lie algebras. Every twisted Lie algebra has a universal enveloping twisted Hopf algebra, and the category of twisted connected Lie algebras is shown to be equivalent to a category of connected twisted Hopf algebras with cocommutative comultiplication. A PBW-like decomposition is obtained for the enveloping algebra of a connected twisted Lie algebra, and free twisted Lie algebras are shown to admit a basis of simple (left-normed) brackets of generators.