多微分李双代数与图复形
Polydifferential Lie bialgebras and graph complexes
AI总结：
本研究分析李双代数properad到其多微分版本的典范态射的形变复形，证明其与有向图复形拟同构，关联Grothendieck-Teichmüller李代数，并借助纠缠图复形描述完整形变复形的结构。
摘要：
我们研究了从（次数平移的）李双代数的properad $\text{\textbf{Lieb}}_{c,d}$ 到其多微分版本 $\text{\textbf{D}}(\text{\textbf{Lieb}}_{c,d})$ 的典范态射 $i$ 的形变复形，证明它在相差一个重标度类的意义下，与有向图复形 $\text{\textbf{GC}}^{\text{or}}_{c+d+1}$ 拟同构。由于后者与原图复形 $\text{\textbf{GC}}_{c+d}$ 拟同构，我们得出结论：典范映射 $i$ 的同伦非平凡无穷小形变空间可与Grothendieck-Teichmüller李代数 $\text{\textfrak{grt}}$ 等同；此外，每个此类无穷小形变都可延拓为从 $\text{\textbf{Lieb}}_{c,d}$ 到 $\text{\textbf{D}}(\text{\textbf{Lieb}}_{c,d})$ 的典范态射 $i$ 的真实形变。我们借助一种名为纠缠图的新型图复形描述了完整的形变复形，并证明其适当的商复形包含Kontsevich图复形 $\text{\textbf{GC}}_c \text{\textotimes} \text{\textbf{GC}}_d$ 的上同调张量积 $H(\text{\textbf{GC}}_c) \text{\textotimes} H(\text{\textbf{GC}}_d)$。
英文摘要：
We study the deformation complex of a canonical morphism $i$ from the properad of (degree shifted) Lie bialgebras $\mathbf{Lieb}_{c,d}$ to its polydifferential version $\mathcal{D}(\mathbf{Lieb}_{c,d})$ and show that it is quasi-isomorphic to the oriented graph complex $\mathbf{GC}^{\text{or}}_{c+d+1}$, up to one rescaling class. As the latter complex is quasi-isomorphic to the original graph complex $\mathbf{GC}_{c+d}$, we conclude that the space of homotopy non-trivial infinitesimal deformations of the canonical map $i$ can be identified with the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}$; moreover, every such an infinitesimal deformation extends to a genuine deformation of the canonical morphism $i$ from $\mathbf{Lieb}_{c,d}$ to $\mathcal{D}(\mathbf{Lieb}_{c,d})$. The full deformation complex is described with the help of a new graph complex of so called entangled graphs, whose suitable quotient complex is shown to contain the tensor product $H(\mathbf{GC}_c) \otimes H(\mathbf{GC}_d)$ of cohomologies of Kontsevich graph complexes $\mathbf{GC}_c \otimes \mathbf{GC}_d$.
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