Free braided nonassociative Hopf algebras and Sabinin $\tau$-algebras

Abstract

Let $V$ be a linear space over a field $\mathbf{k}$ with a braiding $\tau :V\otimes V\to V\otimes V.$ We prove that the braiding $\tau$ has a unique extension on the free nonassociative algebra $\mathbf{k}\{V\}$ freely generated by $V$ so that $\mathbf{k}\{V\}$ is a braided algebra. Moreover, we prove that the free braided algebra $\mathbf{k}\{V\}$ 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, ${\tau }^2=\mathrm{id}$, 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 $\tau$-algebra is a Sabinin $\tau$-algebra.

21 pages