论文标题
差分运营商的平面道具
Planar Prop of Differential Operators
论文作者
论文摘要
我们建议在Hochschild共同体的精神上对协会代数$ a $的差分运算符的定义。具体来说，我们将$ d（a）$定义为由hom空间$ \ mathrm {hom}形成的特定双重复合物的零共生，a^{\ otimes q}，a^{\ otimes p}）$。我们证明它具有平面道具的结构，即每个差分运算符具有多个输入和输出，并且可以沿平面图组成。此外，对于正式平滑的代数，我们将符号符号映射从$ d（a）$到多源的空间。我们还考虑了由$ a $的琐碎关联变形产生的另一个平面提案$ e（a）$在完成自由关联代数的情况下产生的。我们构建了一张自然地图，从$ e（a）$到$ d（a）$并确定其图像。
We propose a definition of differential operators of an associative algebra $A$ in the spirit of Hochschild cohomology. Specifically we define $D(A)$ as the zero cohomology of a certain bicomplex formed by Hom-spaces $\mathrm{Hom}(A^{\otimes q}, A^{\otimes p})$. We show that it has a structure of a planar prop, i.e. each differential operator has multiple inputs and outputs and they can be composed along planar graphs. Furthermore, for a formally smooth algebra we have the surjective symbol map from $D(A)$ to the space of poly-derivations. We also consider another planar prop $E(A)$ generated by automorphisms of the trivial associative deformation of $A$ over the completion of a free associative algebra. We construct a natural map from $E(A)$ to $D(A)$ and identify its image.
