Braided version of Shirshov–Witt theorem

The Shirshov–Witt theorem claims that every subalgebra of a free Lie algebra is free. In characteristic zero this theorem can be restated in terms of a free associative algebra: every Hopf subalgebra of a free algebra k xi with the coproduct δ(xi)=xi 1+1 xi is free, and it is freely generated by primitive elements. Our aim is to extend this result to free algebras with a braided coproduct as far as possible. By means of P.M. Cohn theory we show that if a subalgebra is a right categorical right coideal, then it is free. We consider more thoroughly involutive braidings, τ2=id, over a field of zero characteristic. In this case every braided Hopf subalgebra is generated by primitive elements. Moreover, the space of all primitive elements forms a free Lie τ-algebra. In the context of this result, we discuss the situation that arises around the problem of embedding of a Lie τ-algebra in its associative universal enveloping algebra.