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## 数学代写|黎曼几何代写Riemannian geometry代考|Finite Lie Theory and Cohomology

In general very little is known about the structure of the bicovariant calculi $\Omega(G)$ for nonabelian finite groups, even the dimensions of the different degrees in most cases. Although one can define a simplicial complex (the Moore complex) associated to a Cayley graph, that has the same dimensions as the de Rham complex of a sphere of dimension $|\mathcal{C}|-1$, and that is very far from our case, as Example $1.60$ shows. For some general observations, we look first at the permutation group $S_{n}$ in general, with a focus on the left-invariant forms $\Lambda\left(S_{n}\right)$. We let $C_{n}(d)$ be the configuration space of ordered $n$-tuples in $\mathbb{R}^{d}$ with distinct entries, and $\mathrm{H}\left(C_{n}(d)\right)$ its cohomology with values in $\mathbb{Z}$ and hence in any field of characteristic zero.

Proposition $1.63$ The exterior algebra $\Omega\left(S_{n}\right)$ with $\mathcal{C}$ the 2-cycles has invariant 1forms $\left{e_{(i j)}\right}$ and at degree two the relations
$$\begin{gathered} e_{(i j)}^{2}=0, \quad e_{(i j)} \wedge e_{(m n)}+e_{(m n)} \wedge e_{(i j)}=0 \ e_{(i j)} \wedge e_{(j k)}+e_{(j k)} \wedge e_{(k i)}+e_{(k i)} \wedge e_{(i j)}=0 \end{gathered}$$
where $i, j, k, m, n$ are distinct. Moreover, if $\mathbb{k}$ has characteristic zero then
$$\Lambda\left(S_{n}\right) /\left\langle e_{(i j)} \wedge e_{(j k)}+e_{(j k)} \wedge e_{(i j)}\right\rangle \cong \mathrm{H}\left(C_{n}(2)\right)$$

## 数学代写|黎曼几何代写Riemannian geometry代考|Application to Naive Electromagnetism on Discrete

Given a quantum metric and our focus on differential forms, it is natural to ask for a Hodge operator $\circledast: \Omega^{i} \rightarrow \Omega^{n-i}$ in the case of volume dimension $n$. A general theory of this is missing but a naive approach, which we illustrate here, is to look for a bimodule map squaring to $\pm 1$, as happens classically. This could be of interest to physicists and we explain it in the context of an application to electromagnetism.
We recall that the modern approach to electromagnetism is to see the electric and magnetic fields as the components of a single curvature 2 -form $F$ on spacetime. Maxwell’s equations become
$$\mathrm{d} F=0, \quad \delta F=J,$$
where $J \in \Omega^{1}$ is the source. The first is solved usually by viewing $F$ as the curvature of a $U(1)$ bundle over spacetime and the second involves the divergence or Hodge codifferential $\delta=\circledast \mathrm{d} \circledast$. Noncommutative bundles are covered much later in the book and here we consider only the elementary layer of the theory which applies to trivial bundles, which we call ‘Maxwell theory’. The principal quantity of interest is the electromagnetic field $F$ but we write this as $F-\mathrm{d} \alpha$ in terms of a ‘gauge potential’ $\alpha \in S 2^{1}$ considered modulo exact forms, since adding $\mathrm{d}$ of something to $\alpha$ does not change $F$. The class of $\alpha$ in $\mathrm{H}_{\mathrm{dR}}^{1}$ is also of interest as a reflection of the nontrivial topology of spacetime. There is another ‘ $U(1)$-Yang-Mills’ elementary possibility which we mention at the end of the section. In the $$-algebra case we require \alpha^{}=-\alpha and J^{*}=-J. In order to have solutions it is often stated incorrectly in physics that we need \delta J=0, i.e. for the source to be conserved. What we need more precisely is that J is coexact, by which we mean J^{\circledast} is exact (using a super-script notation for the application of \circledast ). If the penultimate cohomology \mathrm{H}{\mathrm{dR}}^{n-1} is trivial then being conserved is equivalent to being coexact but otherwise coexact is stronger. Indeed, we can already do this theory in nice cases armed only with an exterior algebra over an algebra A and a quantum metric, but the calculus will typically be inner, \mathrm{H}{\mathrm{dR}}^{1}(A) will typically contain the nonclassical element \theta and the penultimate cohomology will typically contain \theta^{\circledast}. So \mathrm{H}{\mathrm{dR}}^{1}(A)=\mathbb{C} \theta and \mathrm{H}{\mathrm{dR}}^{n-1}(A)=\mathbb{C} \theta^{\oplus} will typically be the least amount of cohomology in these degrees that we will need to work with, even for elementary but noncommutative models. # 黎曼几何代考 ## 数学代写|黎曼几何代写Riemannian geometry代考|Finite Lie Theory and Cohomology 通常对双协变结石的结构知之其少 \Omega(G) 对于非阿贝尔有限群，在大多数情况下甚至是不同度数的维数。尽管可以定义与凯莱图相关的单纯复形（摩尔复形)，但 它与维球的德拉姆复形具有相同的维数 |\mathcal{C}|-1 ，这与我们的情况相去甚远，例如 1.60 显示。对于一些一般性的观察，我们首先看一下置换群 S_{n} 一般来说，重点是 左不变形式 \Lambda\left(S_{n}\right). 我们让 C_{n}(d) 是有序的配置空间 n – 元组 \mathbb{R}^{d} 具有不同的条目，并且 H\left(C_{n}(d)\right) 它与值的上同调 \mathbb{Z} 因此在任何特征为零的领域中。 主张1.63外代数 \Omega\left(S_{n}\right) 和 C 2 循环具有不变的 1 形式\left 的分隔符缺失或无法识别$$
e_{(i j)}^{2}=0, \quad e_{(i j)} \wedge e_{(m n)}+e_{(m n)} \wedge e_{(i j)}=0 e_{(i j)} \wedge e_{(j k)}+e_{(j k)} \wedge e_{(k i)}+e_{(k i)} \wedge e_{(i j)}=0
$$在哪里 i, j, k, m, n 是不同的。此外，如果纬么特征为零$$
\Lambda\left(S_{n}\right) /\left\langle e_{(i j)} \wedge e_{(j k)}+e_{(j k)} \wedge e_{(i j)}\right\rangle \cong \mathrm{H}\left(C_{n}(2)\right)


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