Free braided nonassociative Hopf algebras and Sabinin τ-algebras

[摘要] Let V be a linear space over a field k with a braiding tau V circle times V -> V circle times V. We prove that the braiding tau. 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, tau(2) = 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. (C) 2017 Elsevier Inc. All rights reserved.

[发布日期] 2017-12-15

[关键词] Nonassociative algebra;Braiding;Sabinin algebra;Primitive polynomial