### 数学代写|表示论代写Representation theory代考|MAST90017

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|表示论代写Representation theory代考|Main Theorem

Let $V$ be a Poisson vertex algebra. By Theorem 3.9, we have an odd element $X \in$ $W_{\mathrm{cl}}(\Pi V)$ such that $[X, X]=0$, which is associated with the PVA structure of $V$ by (3.37). Thus, there is the PVA cohomology complex
$$\left(W_{\mathrm{cl}}(\Pi V), \operatorname{ad} X\right)$$
A classical $n$-cochain is an element $Y \in W_{\mathrm{cl}}^{n-1}(\Pi V)$, namely a map
$$Y: \mathcal{G}(n) \times(\Pi V)^{\otimes n} \longrightarrow(\Pi V)\left[\lambda_{1}, \ldots, \lambda_{n}\right] /\left\langle\partial+\lambda_{1}+\cdots+\lambda_{n}\right\rangle$$
satisfying relations (3.25), (3.26), (3.28), (3.29), and the following symmetry property (by definition (3.20)):
$$Y^{\sigma}=Y, \quad \forall \sigma \in S_{n}$$

Recall the grading of the superoperad $\mathcal{P}{\mathrm{cl}}(\Pi V)$ from $(3.36): \mathrm{gr}^{\prime} W{\mathrm{cl}}^{n-1}(\Pi V)$ is the set of maps $Y$ as in (4.2) such that
$$Y^{\Gamma}=0 \text { unless }|E(\Gamma)|=r .$$
Note that if $\Gamma \in \mathcal{G}(n)$ has $|E(\Gamma)| \geq n$, then necessarily $\Gamma$ contains a cycle. Hence, by the cycle relation $(3.25), Y^{\Gamma}=0$. Therefore the top degree in gr $W_{\mathrm{cl}}^{n-1}(\Pi V)$ is $n-1$, i.e.,
$$\mathrm{gr}^{r} W_{\mathrm{cl}}^{n-1}(\Pi V)=0 \text { if } r \geq n$$
Note that, if $\Gamma \in \mathcal{G}{0}(n)$, then $|E(\Gamma)|=n-1$ if and only if $\Gamma$ is connected. By Remark $3.8$, the top degree subspace gr ${ }^{n-1} W{c l}^{n-1}(\Pi V)$ consists of all collections of maps
$$Y^{\Gamma}:(\Pi V)^{\otimes n} \longrightarrow(\Pi V), \quad \text { for } \Gamma \in \mathcal{G}{0}(n), \quad|E(\Gamma)|=n-1$$ satisfying (3.25), (3.26), (4.3), and $Y^{\Gamma}\left(\partial\left(v{1} \otimes \cdots \otimes v_{n}\right)\right)=\partial Y^{\Gamma}\left(v_{1} \otimes \cdots \otimes v_{n}\right)$. If $\Gamma$ is not connected, then $Y^{\Gamma}=0$.

In addition, as explained in Sect. $2.3$, there is another cohomology complex associated with $V$, viewed as a differential algebra, namely the differential Harrison complex
$$\left(C_{\partial, \operatorname{Har}}(V), d\right)$$
where $C_{\partial, \mathrm{Har}}^{n}(V) \subset \operatorname{Hom}_{\mathbb{F}[\partial]}\left(V^{\otimes n}, V\right)$ is defined by Harrison’s conditions (2.11) and $d$ is the Hochschild differential (2.2).
The main result of this paper is the following:

## 数学代写|表示论代写Representation theory代考|Lines

We say that a graph $\Gamma \in \mathcal{G}(n)$ is a non-connected line if it has the following form:
where $1 \leq k_{1} \leq \cdots \leq k_{s}$ are such that $k_{1}+\cdots+k_{s}=n$, and the set of indices $\left{i_{b}^{a}\right}$ is a permutation of ${1, \ldots, n}$ such that
$$i_{1}^{l}=\min \left{i_{1}^{l}, \ldots, i_{k_{l}}^{l}\right} \quad \forall l=1, \ldots, s$$
If $k_{l}=k_{l+1}$, we also assume that $i_{1}^{l}<i_{1}^{l+1}$. In particular, the connected lines are all of the form
$$\sigma\left(\Lambda_{n}\right), \quad \sigma \in S_{n} \text { such that } \sigma(1)=1$$
where $\Lambda_{n}$ is the $n$-line (4.8). Let $\mathcal{L}(n)$ be the set of $n$-graphs that are non-connected lines. Let also $\mathbb{F} G(n)$ be the vector space with basis the set of graphs $\mathcal{G}(n)$.
Definition 4.2 The cycle relations in $F \mathcal{G}(n)$ are the following elements:
(i) all $\Gamma \in \mathcal{G}(n) \backslash \mathcal{G}{0}(n)$ (i.e., graphs containing a cycle); (ii) all linear combinations $\sum{e \in C} \Gamma \backslash e$, where $\Gamma \in \mathcal{G}(n)$ and $C \subset E(\Gamma)$ is an oriented cycle.

Denote by $R(n) \subset \mathbb{F} G(n)$ the subspace spanned by the cycle relations (4.2) and (4.2).

Note that reversing an arrow in a graph $\Gamma \in \mathcal{G}(n)$ gives us, modulo cycle relations, the element $-\Gamma \in \mathbb{G}(n)$.

## 数学代写|表示论代写Representation theory代考|Relation Between the Symmetry Property and Harrison’s Conditions

Recall that every $Y \in W_{\mathrm{cl}}^{n-1}(\Pi V)$ satisfies the symmetry property (4.3).
Lemma 4.9 If $Y \in W_{\mathrm{cl}}^{n-1}(\Pi V)$, then $Y^{\Lambda_{n}}$ satisfies Harrison’s relations (2.11), hence it lies in the differential Harrison cohomology complex:
$$Y^{\Lambda_{n}} \in C_{\partial, \mathrm{Har}}^{n}(V)$$
Conversely, given $F \in C_{\partial, \mathrm{Har}}^{n}(V)$, there exists a unique $Y \in \mathrm{gr}^{n-1} W_{\mathrm{cl}}^{n-1}(\Pi V)$, such that
$$Y^{\Lambda_{n}}=F$$
Consequently, the linear map
$$\mathrm{gr}^{n-1} W_{\mathrm{cl}}^{n-1}(\Pi V) \stackrel{\sim}{\rightarrow} C_{\partial, \mathrm{Har}}^{n}(V), \quad Y \mapsto Y^{\Lambda_{n}}$$
is bijective.

Proof First, we prove that, if $Y \in W_{\mathrm{cl}}^{n-1}(\Pi V)$ satisfies the symmetry relations (4.3), then $f=Y^{\Lambda_{n}}$ satisfies Harrison’s conditions (2.11). By Lemma $4.8$ (cf. Remark 4.4), we get
$$Y^{\Lambda_{n}}=(-1)^{k-1} \sum_{\pi \in \mathcal{M}{n}^{k}} Y^{\pi\left(\Lambda{n}\right)}$$
Evaluating both sides of this identity on $v_{1} \otimes \cdots \otimes v_{n} \in V^{8 n}$, the left side is simply $Y^{\Lambda_{n}}\left(v_{1} \otimes \cdots \otimes v_{n}\right)=f\left(v_{1} \otimes \cdots \otimes v_{n}\right)$. On the right-hand side, we have
\begin{aligned} (-1)^{k-1} \sum_{\pi \in \mathcal{M}{n}^{k}} Y^{\pi\left(\Lambda{n}\right)}\left(v_{1} \otimes \cdots \otimes v_{n}\right)=(-1)^{k-1} \sum_{\pi \in \mathcal{M}{n}^{k}}\left(Y^{\pi^{-1}}\right)^{\pi\left(\Lambda{n}\right)}\left(v_{1} \otimes \cdots \otimes v_{n}\right) \ &=(-1)^{k-1} \sum_{\pi \in \mathcal{M}{n}^{k}} \operatorname{sign}(\pi) Y^{\Lambda{n}}\left(v_{\pi(1)} \otimes \cdots \otimes v_{\pi(n)}\right) \ &=L_{k} f\left(v_{1} \otimes \cdots \otimes v_{n}\right), \end{aligned}
by the definition $(2.10)$ of $L_{k}$. Hence, $f$ satisfies Harrison’s conditions $(2.11)$ as claimed.

We next turn to the second claim of the lemma. Let $F \in C_{\partial, \mathrm{Har}}^{n}(V)$, i.e., $F: V^{\otimes n} \rightarrow V$ is an $\mathbb{F}[\partial]$-module homomorphism satisfying Harrison’s conditions (2.11). Then the corresponding $Y \in \mathrm{gr}^{n-1} W_{\mathrm{cl}}^{n-1}(\Pi V)$, such that $Y \Lambda_{n}=F$, is defined as follows. For $\Gamma \in R(n)$, or if $\Gamma \in \mathcal{L}(n)$ is not connected, we set
$$Y^{\Gamma}=0$$
For $\Gamma \in \mathcal{L}(n)$ connected, there exists a unique $\tau \in S_{n}$ such that $\tau(1)=1$ and $\Gamma=\tau\left(\Lambda_{n}\right)$. We then set
$$Y^{\Gamma}\left(v_{1} \otimes \cdots \otimes v_{n}\right)=\operatorname{sign}(\tau) F\left(v_{\tau(1)} \otimes \cdots \otimes v_{\tau(n)}\right)$$

(在Cl(圆周率在),广告⁡X)

(C∂,头发(在),d)

## 数学代写|表示论代写Representation theory代考|Lines

i_{1}^{l}=\min \left{i_{1}^{l}, \ldots, i_{k_{l}}^{l}\right} \quad \forall l=1, \ldots , si_{1}^{l}=\min \left{i_{1}^{l}, \ldots, i_{k_{l}}^{l}\right} \quad \forall l=1, \ldots , s

σ(Λn),σ∈小号n 这样 σ(1)=1

(i) 所有Γ∈G(n)∖G0(n)（即，包含循环的图）；(ii) 所有线性组合∑和∈CΓ∖和， 在哪里Γ∈G(n)和C⊂和(Γ)是一个定向循环。

## 数学代写|表示论代写Representation theory代考|Relation Between the Symmetry Property and Harrison’s Conditions

Grn−1在Cln−1(圆周率在)→∼C∂,H一个rn(在),是↦是Λn

(−1)ķ−1∑圆周率∈米nķ是圆周率(Λn)(在1⊗⋯⊗在n)=(−1)ķ−1∑圆周率∈米nķ(是圆周率−1)圆周率(Λn)(在1⊗⋯⊗在n) =(−1)ķ−1∑圆周率∈米nķ符号⁡(圆周率)是Λn(在圆周率(1)⊗⋯⊗在圆周率(n)) =大号ķF(在1⊗⋯⊗在n),

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