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# 数学代写|黎曼几何代写Riemannian geometry代考|MAT00052M Higher Order Differential Operators

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## 数学代写|黎曼几何代写Riemannian geometry代考|Higher Order Differential Operators

In the last section, we considered the action of a single vector field on an algebra, and later we will also have actions on modules with connection. However, to have an action we should have an algebra acting, and this requires actions of multiple vector fields giving higher order differential operators. It is reasonable to wish to view these operators as actions of tensor products of vector fields, but such an identification is not entirely trivial. In particular, we will require a connection (called $\bigcirc$ here) on vector fields and 1-forms to achieve this. As we will require several copies of 1-forms and vector fields on $A$, we define
\begin{aligned} &\mathfrak{X}^{\otimes 0}=A, \quad \mathfrak{X}^{\otimes n}=\mathfrak{X} \otimes_A \mathfrak{X} \otimes_A \cdots \otimes_A \mathfrak{X}, \ &\Omega^{1 \otimes 0}=A, \quad \Omega^{1 \otimes n}=\Omega^1 \otimes_A \Omega^1 \otimes_A \cdots \otimes_A \Omega^1, \end{aligned}

where we have $n$ copies of $\mathfrak{X}$ and $\Omega^1$. Note that the definition of $\mathfrak{X}^{\otimes n}$ and $\Omega^{1 \otimes n}$ uses $\otimes_A$, the tensor product over the algebra. We will sometimes use id ${ }^{\otimes n}$ as the identity on $\mathfrak{X}^{\otimes n}$ or $\Omega^{1 \otimes n}$. As for the duality of braided tensor algebras in $\S 2.6$, we define $n$-fold evaluation $\mathrm{ev}^{(n)}: \mathfrak{X}^{\otimes n} \otimes_A \Omega^{1 \otimes n} \rightarrow A$ and $n$-fold coevaluation $\operatorname{coev}^{(n)}: A \rightarrow \Omega^{1 \otimes n} \otimes_A \mathfrak{X}^{\otimes n}$ in a nested way, which we specify recursively by
$$\begin{gathered} \mathrm{ev}^{\langle 1\rangle}=\mathrm{ev}, \quad \mathrm{ev}^{\langle n+1\rangle}=\mathrm{ev}\left(\mathrm{id} \otimes \mathrm{ev}^{(n)} \otimes \mathrm{id}\right) \ \mathrm{coev}^{(1)}=\operatorname{coev}, \quad \operatorname{coev}^{(n+1)}=\left(\text { id } \otimes \operatorname{coev}^{(n)} \otimes \text { id }\right) \operatorname{coev} \end{gathered}$$
In diagrammatic terms, the first of these is
and there is a similar upside down version of this for coevaluation.

## 数学代写|黎曼几何代写Riemannian geometry代考|The Sheaf of Differential Operators DA

So far we have no relations for differential operators corresponding classically to commutativity of partial derivatives. When $T \mathfrak{X}{\bullet}$ is represented on itself, the commutator of covariant derivatives gives curvature, so our first task is to write this as a differential operator, which we do with the help of the torsion $T{\odot}=\mathrm{d}+\wedge \bigcirc$ : $\Omega^1 \rightarrow \Omega^2$, a right module map.

Proposition 6.21 Let $\left(A, \Omega^1\right)$ be an algebra with fgp calculus. There is a central element $\mathcal{R} \in \Omega^2 \otimes_A$ TX๋ given by
$$\mathcal{R}=\mathrm{d} e_i \otimes f_i-e_i \wedge e_j \otimes f_j \bullet f_i=T_{\odot}\left(e_i\right) \otimes f_i-e_i \wedge e_j \otimes\left(f_j \otimes f_i\right)$$
where $\operatorname{coev}=e_i \otimes f_i \in \Omega^1 \otimes_A \mathfrak{X}$ are dual bases and $e_j \otimes f_j$ is another, such that the curvature $R_{\nabla}$ on $T X_{\bullet}$ in Lemma $6.16$ is given by $R_{\nabla}(\underline{v})=\mathcal{R} \bullet \underline{v}$.

Proof The connection $\nabla$ on $T \mathfrak{X} \bullet$ in Lemma $6.16$ is $\nabla(\underline{v})=e_i \otimes\left(f_i \bullet \underline{v}\right)$ with curvature $R_{\nabla}: T \mathfrak{X}{\bullet} \rightarrow \Omega^2 \otimes_A T$X${\bullet}$ given by
$$R_{\nabla}(\underline{v})=\mathrm{d} e_i \otimes\left(f_i \bullet \underline{v}\right)-e_i \wedge \nabla\left(f_i \bullet \underline{v}\right)=\mathrm{d} e_i \otimes\left(f_i \bullet \underline{v}\right)-e_i \wedge e_j \otimes\left(f_j \bullet f_i \bullet \underline{v}\right) .$$
Now, from the formula for $\bullet$,
$$\begin{gathered} e_i \otimes e_j \otimes f_j \bullet f_i=e_i \otimes e_j \otimes f_j \otimes f_i+e_i \otimes e_j \otimes(\mathrm{ev} \otimes \mathrm{id})\left(f_j \otimes \otimes f_i\right) \ =e_i \otimes e_j \otimes f_j \otimes f_i+\left(\mathrm{id}^{\otimes 2} \otimes \mathrm{ev} \otimes \mathrm{id}\right)\left(\mathrm{id} \otimes \mathrm{coev} \otimes \mathrm{id}^{\otimes 2}\right)\left(e_i \otimes \nabla f_i\right) \ =e_i \otimes e_j \otimes f_j \otimes f_i+e_i \otimes \nabla f_i=e_i \otimes e_j \otimes f_j \otimes f_i-\nabla e_i \otimes f_i, \end{gathered}$$
where we have used the usual equations for the evaluation and coevaluation. Using the torsion $T_{\odot}$, we can rewrite $\mathcal{R}$ as
$$\mathcal{R}=\mathrm{d} e_i \otimes f_i-e_i \wedge e_j \otimes\left(f_j \otimes f_i\right)+\wedge \otimes e_i \otimes f_i=T_{\bigcirc}\left(e_i\right) \otimes f_i-e_i \wedge e_j \otimes\left(f_j \otimes f_i\right) .$$
Since $R_{\nabla}$ is a left module map, applying $R_{\nabla}$ to $a \in A$ gives
$$\mathcal{R} \bullet a=R_{\nabla}(a)=R_{\nabla}(a \cdot 1)=a \cdot R_{\nabla}(1)=a \cdot \mathcal{R} \bullet 1=a \cdot \mathcal{R} .$$

## 数学代写|黎曼几何代写黎曼几何代考|高阶微分算子

\begin{aligned} &\mathfrak{X}^{\otimes 0}=A, \quad \mathfrak{X}^{\otimes n}=\mathfrak{X} \otimes_A \mathfrak{X} \otimes_A \cdots \otimes_A \mathfrak{X}, \ &\Omega^{1 \otimes 0}=A, \quad \Omega^{1 \otimes n}=\Omega^1 \otimes_A \Omega^1 \otimes_A \cdots \otimes_A \Omega^1, \end{aligned}

，其中我们有$\mathfrak{X}$和$\Omega^1$的$n$副本。注意$\mathfrak{X}^{\otimes n}$和$\Omega^{1 \otimes n}$的定义使用了$\otimes_A$，即代数上的张量积。我们有时会使用id ${ }^{\otimes n}$作为$\mathfrak{X}^{\otimes n}$或$\Omega^{1 \otimes n}$上的标识。对于编织张量代数$\S 2.6$中的对偶性，我们以嵌套的方式定义了$n$ -fold求值$\mathrm{ev}^{(n)}: \mathfrak{X}^{\otimes n} \otimes_A \Omega^{1 \otimes n} \rightarrow A$和$n$ -fold共求值$\operatorname{coev}^{(n)}: A \rightarrow \Omega^{1 \otimes n} \otimes_A \mathfrak{X}^{\otimes n}$，我们递归指定为
$$\begin{gathered} \mathrm{ev}^{\langle 1\rangle}=\mathrm{ev}, \quad \mathrm{ev}^{\langle n+1\rangle}=\mathrm{ev}\left(\mathrm{id} \otimes \mathrm{ev}^{(n)} \otimes \mathrm{id}\right) \ \mathrm{coev}^{(1)}=\operatorname{coev}, \quad \operatorname{coev}^{(n+1)}=\left(\text { id } \otimes \operatorname{coev}^{(n)} \otimes \text { id }\right) \operatorname{coev} \end{gathered}$$

，还有一个类似的倒挂版本用于共求值

## 数学代写|黎曼几何代写黎曼几何代考|The Sheaf of Differential Operators DA

. The Sheaf of Differential Operators DA

$$\mathcal{R}=\mathrm{d} e_i \otimes f_i-e_i \wedge e_j \otimes f_j \bullet f_i=T_{\odot}\left(e_i\right) \otimes f_i-e_i \wedge e_j \otimes\left(f_j \otimes f_i\right)$$

$$\begin{gathered} e_i \otimes e_j \otimes f_j \bullet f_i=e_i \otimes e_j \otimes f_j \otimes f_i+e_i \otimes e_j \otimes(\mathrm{ev} \otimes \mathrm{id})\left(f_j \otimes \otimes f_i\right) \ =e_i \otimes e_j \otimes f_j \otimes f_i+\left(\mathrm{id}^{\otimes 2} \otimes \mathrm{ev} \otimes \mathrm{id}\right)\left(\mathrm{id} \otimes \mathrm{coev} \otimes \mathrm{id}^{\otimes 2}\right)\left(e_i \otimes \nabla f_i\right) \ =e_i \otimes e_j \otimes f_j \otimes f_i+e_i \otimes \nabla f_i=e_i \otimes e_j \otimes f_j \otimes f_i-\nabla e_i \otimes f_i, \end{gathered}$$

$$\mathcal{R}=\mathrm{d} e_i \otimes f_i-e_i \wedge e_j \otimes\left(f_j \otimes f_i\right)+\wedge \otimes e_i \otimes f_i=T_{\bigcirc}\left(e_i\right) \otimes f_i-e_i \wedge e_j \otimes\left(f_j \otimes f_i\right) .$$

$$\mathcal{R} \bullet a=R_{\nabla}(a)=R_{\nabla}(a \cdot 1)=a \cdot R_{\nabla}(1)=a \cdot \mathcal{R} \bullet 1=a \cdot \mathcal{R} .$$

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