拟线性方程的泛包络代数结构群
The structure group for quasi-linear equations via universal enveloping algebras
摘要：
我们考虑用多指标代替树作为抽象模型空间$\mathsf{T}$的指标集的方法，该模型空间是由Otto，Sauer，Smith和Weber提出的，用于处理拟线性奇异随机偏微分方程。我们表明，这种方法是一致的，当涉及到结构群$\mathsf{G}$的规则性结构的假设。特别地，$\mathsf{G}\subset{\rm Aut}(\mathsf{T})$由一个Hopf代数$\mathsf{T}^+$和一个余代数$Δ\colon\mathsf{T}\rightarrow \mathsf{T}^+\otimes\mathsf{T}$产生。事实上，这种方法，其中抽象模型空间$\mathsf{T}$的对偶$\mathsf{T}^*$自然嵌入到形式幂级数代数中，允许将$\mathsf{G}^*\subset{\rm Aut}(\mathsf{T}^*)$解释为从由该幂级数代数上的导子组成的李代数$\mathsf{L} \subset{\rm End}(\mathsf{T}^*)$产生的李群。这些导数又是对空间上两个作用的无穷小生成元（非线性，时空模常数的函数）。这些作用是时空多项式引起的时空平移和时空倾斜。霍普夫代数$\mathsf{T}^+$源自李代数$\mathsf{L}$的通用包络代数${\rm U}(\mathsf{L})$的坐标表示。坐标由$\mathsf{L}$的导出代数的基础前李代数结构确定。强有限性的性质，这是由等级和限制性定义的$\mathsf{T}$，允许这种纯粹的代数结构的$\mathsf{G}$。我们还认为，在分支粗糙路径（Grossman-Larson，Connes-Kreimer）和广义抛物Anderson模型的情况下，我们的结构与树基结构之间存在预李代数和Hopf代数态射.
英文摘要：
We consider the approach of replacing trees by multi-indices as an index set of the abstract model space $\mathsf{T}$ introduced by Otto, Sauer, Smith and Weber to tackle quasi-linear singular SPDEs. We show that this approach is consistent with the postulates of regularity structures when it comes to the structure group $\mathsf{G}$. In particular, $\mathsf{G}\subset{\rm Aut}(\mathsf{T})$ arises from a Hopf algebra $\mathsf{T}^+$ and a comodule $Δ\colon\mathsf{T}\rightarrow \mathsf{T}^+\otimes\mathsf{T}$. In fact, this approach, where the dual $\mathsf{T}^*$ of the abstract model space $\mathsf{T}$ naturally embeds into a formal power series algebra, allows to interpret $\mathsf{G}^*\subset{\rm Aut}(\mathsf{T}^*)$ as a Lie group arising from a Lie algebra $\mathsf{L} \subset{\rm End}(\mathsf{T}^*)$ consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (nonlinearities, functions of space-time mod constants). These actions are shift of space-time and tilt by space-time polynomials. The Hopf algebra $\mathsf{T}^+$ arises from a coordinate representation of the universal enveloping algebra ${\rm U}(\mathsf{L})$ of the Lie algebra $\mathsf{L}$. The coordinates are determined by an underlying pre-Lie algebra structure of the derived algebra of $\mathsf{L}$. Strong finiteness properties, which are enforced by gradedness and the restrictive definition of $\mathsf{T}$, allow for this purely algebraic construction of $\mathsf{G}$. We also argue that there exist pre-Lie algebra and Hopf algebra morphisms between our structure and the tree-based one in the cases of branched rough paths (Grossman-Larson, Connes-Kreimer) and of the generalized parabolic Anderson model.
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