AI 中文总结
本文通过稳定子群的交集解释了Grothendieck-Teichmüller群GRT₁(k)的结构。
AI 中文摘要
若$\mathfrak u$和$\mathfrak v$为李代数，其外自同构群的乘积$\mathrm{Out}(\mathfrak u)\times \mathrm{Out}(\mathfrak v)$自然作用于从$\mathfrak u$到$\mathfrak v$的外李代数态射集；给定此类态射的外类的稳定子是$\mathrm{Out}(\mathfrak u)\times \mathrm{Out}(\mathfrak v)$的子群。我们表明这导致了对格罗滕迪克 - 泰希米勒群$\mathsf{GRT}_1(\mathbf k)$的两种相关解释，其中$\mathfrak u$、$\mathfrak v$分别是平面上具有3股和4股（或球面上具有4股和5股）的无穷小辫子的李代数（或带框无穷小辫子的李代数）：即它可表示为某些股加倍态射$\phi$和$\psi$的外类的稳定子群与$\mathrm{Out}^*(\mathfrak u)\times\mathrm{Out}(\mathfrak v)$的联合交集，其中$\mathrm{Out}^*(\mathfrak u)$是$\mathrm{Out}(\mathfrak u)$中保持$\mathfrak u$惯性的自同构的外类的子群。
英文摘要
If $\mathfrak u$ and $\mathfrak v$ are Lie algebras, then the product $\mathrm{Out}(\mathfrak u)\times \mathrm{Out}(\mathfrak v)$ of their outer automorphism groups naturally acts on the set of outer Lie algebra morphisms from $\mathfrak u$ to $\mathfrak v$; the stabilizer of the outer class of a given such morphism is then a subgroup of $\mathrm{Out}(\mathfrak u)\times \mathrm{Out}(\mathfrak v)$. We show that this leads to two related interpretation of the Grothendieck-Teichmüller group $\mathsf{GRT}_1(\mathbf k)$, where $\mathfrak u,\mathfrak v$ are the Lie algebra of infinitesimal braids on the plane (resp. framed infinitesimal braids on the sphere) with 3 and 4 (resp. 4 and 5) strands: namely, it can be expressed as the joint intersection of the stabilizer groups of the outer classes of certain strand doubling morphisms $ϕ$ and $ψ$ with $\mathrm{Out}^*(\mathfrak u)\times\mathrm{Out}(\mathfrak v)$, where $\mathrm{Out}^*(\mathfrak u)$ is a subgroup of $\mathrm{Out}(\mathfrak u)$ of outer classes of inertia-preserving automorphisms of $\mathfrak u$.
Comments22 pages
