### 数学代写|表示论代写Representation theory代考|MATH4031

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## 数学代写|表示论代写Representation theory代考|Hochschild Cohomology Complex

First, we review the Hochschild cohomology complex, of which Harrison’s is a subcomplex, see [Hoc45] and [Har62]. We use the original Harrison’s definition. For other definitions see [GS87, Lod13].

Let $A$ be an associative algebra over the base field $\mathbb{F}$, and $M$ be an $A$-bimodule. We will write $A^{\otimes n}$ for the $n$-fold tensor product $A \otimes \cdots \otimes A$. The Hochschild cohomology complex is defined as follows. The space of $n$-cochains is
$$\operatorname{Hom}\left(A^{\otimes n}, M\right),$$
and the differential $d: \operatorname{Hom}\left(A^{8 n}, M\right) \rightarrow \operatorname{Hom}\left(A^{8 n+1}, M\right)$ is defined by
\begin{aligned} (d f)\left(a_{1} \otimes\right.&\left.\cdots \otimes a_{n+1}\right)=a_{1} f\left(a_{2} \otimes \cdots \otimes a_{n+1}\right) \ &+\sum_{i=1}^{n}(-1)^{i} f\left(a_{1} \otimes \cdots \otimes a_{i-1} \otimes a_{i} a_{i+1} \otimes a_{i+2} \otimes \cdots \otimes a_{n+1}\right) \ &+(-1)^{n+1} f\left(a_{1} \otimes \cdots \otimes a_{n}\right) a_{n+1} \end{aligned}
Then $d^{2}=0$, and we get the Hochschild cohomology complex

$$0 \longrightarrow M \stackrel{d}{\longrightarrow} \operatorname{Hom}(A, M) \stackrel{d}{\longrightarrow} \operatorname{Hom}\left(A^{\otimes 2}, M\right) \stackrel{d}{\longrightarrow} \cdots$$
If $A$ is an associative algebra with a derivation $\partial: A \rightarrow A$, and $M$ is a differential bimodule over $A$ (i.e., the action of $\partial$ is compatible with the bimodule structure), we may consider the differential Hochschild cohomology complex by taking the subspace of $n$-cochains
$$\operatorname{Hom}{2}[2]\left(A^{3 n}, M\right)$$ It is clear by the definition $(2.2)$ that the differential $d$ maps $\operatorname{Hom}{F[\partial]}\left(A^{\otimes n}, M\right)$ to $\operatorname{Hom}_{F[2]}\left(A^{\otimes n+1}, M\right)$. Hence, we have a cohomology subcomplex.

Remark $2.1$ It is straightforward, using the Koszul-Quillen rule, to extend the definition of the Hochschild complex to the case when $A$ is an associative superalgebra, as well as all other definitions and results of the paper. We restricted here to the purely even case for the simplicity of the exposition.

## 数学代写|表示论代写Representation theory代考|Monotone Permutations

Consider the symmetric group $S_{n}$. Using Harrison’s notation in [Har62] (see also [GS87]), we have the following definition:

Definition $2.2$ A permutation $\pi \in S_{n}$ is called monotone if, for each $i=1, \ldots, n$, one of the following two conditions holds:
(a) $\pi(j)<\pi$ (i) for all $j\pi(i)$ for all $j<i$.
(Not necessarily the same condition (a) or (b) holds for every i.) When (b) holds, we call $i$ a drop of $\pi$. Also, $\pi(1)=k$ is called the start of $\pi$ (and we say that $\pi$ starts at $k$ ).

We denote by $\mathcal{M}{n} \subset S{n}$ the set of monotone permutations, and by $\mathcal{M}{n}^{k} \subset \mathcal{M}{n}$ the set of monotone permutations starting at $k$.

Here is a simple description of all monotone permutations starting at $k$. Let us identify the permutation $\pi \in S_{n}$ with the $n$-tuple $[\pi(1), \ldots, \pi(n)]$. To construct all $\pi \in \mathcal{M}_{n}^{k}$, we let $\pi(1)=k$. Then, for every choice of $k-1$ positions in ${2, \ldots, n}$ we get a monotone permutation $\pi$ as follows. In the selected positions we put the numbers 1 to $k-1$ in decreasing order from left to right; in the remaining positions we write the numbers $k+1$ to $n$ in increasing order from left to right. (The selected positions are the drops of $\pi$.)
According to the above description, we have a bijective correspondence

associating the monotone permutation $\pi \in \mathcal{M}_{n}^{k}$ to the set $D(\pi)$ of drops of $\pi$, which are
$$\pi^{-1}(k-1)<\pi^{-1}(k-2)<\cdots<\pi^{-1}(1) \in{2, \ldots, n}$$

## 数学代写|表示论代写Representation theory代考|Differential Harrison Cohomology Complex

Let us now recall Harrison’s original definition of his cohomology complex [Har62]. Let $A$ be a commutative associative algebra, and $M$ be a symmetric $A$-bimodule, i.e., such that $a m=m a$, for all $a \in A$ and $m \in M$. For every $1<k \leq n$ define the following endomorphism on the space $\operatorname{Hom}\left(A^{8 n}, M\right)$ :
$$\left(L_{k} F\right)\left(a_{1} \otimes \cdots \otimes a_{n}\right):=\sum_{\pi \in \mathcal{M}{k}^{k}}(-1)^{\operatorname{dr}(\pi)} F\left(a{\pi(1)} \otimes \cdots \otimes a_{\pi(n)}\right) .$$
A Harrison $n$-cochain is defined as a Hochschild $n$-cochain $F \in \operatorname{Hom}\left(A^{\otimes n}, M\right)$ fixed by all operators $L_{k}$ :
$$L_{k} F=F, \text { for every } 2 \leq k \leq n .$$
We will denote by
$$C_{\text {Har }}^{n}(A, M) \subset \operatorname{Hom}\left(A^{\otimes n}, M\right)$$
the space of Harrison $n$-cochains.

## 数学代写|表示论代写Representation theory代考|Hochschild Cohomology Complex

(dF)(一个1⊗⋯⊗一个n+1)=一个1F(一个2⊗⋯⊗一个n+1) +∑一世=1n(−1)一世F(一个1⊗⋯⊗一个一世−1⊗一个一世一个一世+1⊗一个一世+2⊗⋯⊗一个n+1) +(−1)n+1F(一个1⊗⋯⊗一个n)一个n+1

0⟶米⟶d他⁡(一个,米)⟶d他⁡(一个⊗2,米)⟶d⋯

## 数学代写|表示论代写Representation theory代考|Monotone Permutations

(a)圆周率(j)<圆周率(一) 为所有人j圆周率(一世)对所有人j<一世.
（不一定相同的条件 (a) 或 (b) 对每个 i 都成立。）当 (b) 成立时，我们称一世一滴圆周率. 还，圆周率(1)=ķ被称为开始圆周率（我们说圆周率开始于ķ ).

## 数学代写|表示论代写Representation theory代考|Differential Harrison Cohomology Complex

(大号ķF)(一个1⊗⋯⊗一个n):=∑圆周率∈米ķķ(−1)博士⁡(圆周率)F(一个圆周率(1)⊗⋯⊗一个圆周率(n)).

C头发 n(一个,米)⊂他⁡(一个⊗n,米)

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