Free braided nonassociative Hopf algebras and Sabinin τ-algebras

Let V be a linear space over a field k with a braiding τ:V⊗V→V⊗V. We prove that the braiding τ has a unique extension on the free nonassociative algebra k{V} freely generated by V so that k{V} is a braided algebra. Moreover, we prove that the free braided algebra k{V} has a natural structure of a braided nonassociative Hopf algebra such that every element of the space of generators V is primitive. In the case of involutive braidings, τ2=id, we describe braided analogues of Shestakov-Umirbaev operations and prove that these operations are primitive operations. We introduce a braided version of Sabinin algebras and prove that the set of all primitive elements of a nonassociative τ-algebra is a Sabinin τ-algebra.