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

TL;DR

This paper constructs free braided nonassociative Hopf algebras with primitive generators and introduces braided Sabinin algebras, extending classical algebraic structures to braided settings.

Contribution

It establishes the existence and uniqueness of a braided extension on free nonassociative algebras and introduces braided Sabinin algebras as primitive elements.

Findings

Unique braided extension on free nonassociative algebra

Free braided algebra forms a braided nonassociative Hopf algebra

Primitive elements form a braided Sabinin algebra

Abstract

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