论文标题
免费编织的非缔合HOPF代数和sabinin $τ$ -Algebras
Free braided nonassociative Hopf algebras and Sabinin $τ$-algebras
论文作者
论文摘要
让$ v $是字段$ {\ bf k} $的线性空间，并带有编织$τ：v \ otimes v \ rightarrow v \ otimes v. $我们证明，编织$τ$在免费的非cociative norpociative norpociative norpociative norpociative bf k} $ bif bf k} $ a $ a $ a}上具有独特的扩展名。 $ {\ bf k} \ {v \} $是编织的代数。此外，我们证明了免费编织的代数$ {\ bf k} \ {v \} $具有编织的非社交hopf代数的自然结构，使生成器$ v $的每个元素都是原始的。在涉及辫子的情况下，$τ^2 = {\ rm ID} $，我们描述了Shestakov-umirbaev操作的编织类似物，并证明这些操作是原始操作。我们介绍了sabinin代数的编织版本，并证明非缔合$τ$ -Algebra的所有原始元素的集合是sabinin $τ$ -Algebra。
Let $V$ be a linear space over a field ${\bf k}$ with a braiding $τ: V\otimes V\rightarrow V\otimes V.$ We prove that the braiding $τ$ 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, $τ^2={\rm id}$, 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 $τ$-algebra is a Sabinin $τ$-algebra.
