# 数学代写|表示论代写Representation theory代考|Quivers of Type Have Infinite Representation Type

## 数学代写|表示论代写Representation theory代考|Quivers of Type Have Infinite Representation Type

According to Definition 11.30 we have to show that every endomorphism of $\mathcal{M}$ is a scalar multiple of the identity. So let $\varphi: \mathcal{M} \rightarrow \mathcal{M}$ be a homomorphism of representations. Then as before each linear map $\varphi_i: M(i) \rightarrow M(i)$ is the restriction of $\varphi_4$ to $M(i)$. We again use that each subspace $M(i)$ is invariant under $\varphi_4$ to get restrictions for $\varphi_4$.

First, $\varphi_4(g)=c_1 g$ for some $c_1 \in K$ since $\varphi_4(g) \in M(2) \cap M(7)=\operatorname{span}{g}$. Similarly, $\varphi_4(e) \in M(1) \cap M(5)=\operatorname{span}{e}$, so $\varphi_4(e)=c_2 e$ for some $c_2 \in K$. Moreover, we have $\varphi_4(k) \in M(8)$, thus $\varphi_4(k)=c k+z h$ for $c, z \in K$. With this, we have by linearity
$$\varphi_4(e+g+k)=c_2 e+c_1 g+(c k+z h) .$$
But this must lie in $M(3)$. Since $l$ and $f$ do not occur in the above expression, it follows from the definition of $M(3)$ that $\varphi_4(e+g+k)$ must be a scalar multiple of $e+g+k$ and hence $z=0$ and $c_1=c_2=c$. In particular, $\varphi_4(k)=c k$.

We may write $\varphi_4(l)=u e+v l$ with $u, v \in K$ since it is in $M(1)$, and $\varphi_4(h)=r h+s k$ with $r, s \in K$ since it is in $M(8)$. We have $\varphi_4(h+l) \in M(3)$. In $\varphi_4(h)+\varphi_4(l)$, basis vectors $g$ and $f$ do not occur, and it follows that $\varphi_4(h+l)$ is a scalar multiple of $h+l$. Hence
$$\varphi_4(l)=v l, \varphi_4(h)=r h, \text { and } v=r .$$
We can write $\varphi_4(f)=a f+b g$ with $a, b \in K$ since it is in $M(2) \cap M(6)=\operatorname{span}{f, g}$. Now we have by linearity that
$$\varphi_4(e+f+h)=c_2 e+(a f+b g)+r h \in M(3) .$$
Since the basis vectors $l$ and $k$ do not occur in this expression, it follows that $\varphi_4(e+f+h)$ is a scalar multiple of $e+f+h$. So $b=0$ and $a=c_2=r$; in particular, we have $\varphi_4(f)=a f$.

In total we have now seen that $c=c_1=c_2=a=r=v$, so all six basis vectors of $M(4)$ are mapped to the same scalar multiple of themselves. This proves that $\varphi_4$ is a scalar multiple of the identity, and then so is $\varphi$.

## 数学代写|表示论代写Representation theory代考|Morita Equivalence

A precise definition of Morita equivalence needs the setting of categories and functors (and it can be found in more advanced texts). Here, we will give an informal account, for finite-dimensional algebras over a field $K$, and we give some examples as illustrations. Roughly speaking, two algebras $A$ and $B$ over the same field $K$ are Morita equivalent ‘if they have the same representation theory’. That is, there should be a canonical correspondence between finite-dimensional $A$-modules and finitedimensional $B$-modules, say $M \rightarrow F(M)$, and between module homomorphisms: If $M$ and $N$ are $A$-modules and $f: M \rightarrow N$ is an $A$-module homomorphism then $F(f)$ is a $B$-module homomorphism $F(M) \rightarrow F(N)$, such that
(i) it preserves (finite) direct sums, and takes indecomposable $A$-modules to indecomposable $B$-modules, and simple $A$-modules to simple $B$-modules,
(ii) every $B$-module is isomorphic to $F(M)$ for some $A$-module $M$, where $M$ is unique up to isomorphism,
(iii) the correspondence between homomorphisms gives a vector space isomorphism
$$\left.\operatorname{Hom}_A(M, N)\right) \cong \operatorname{Hom}_B(F(M), F(N))$$
for arbitrary $A$-modules $M, N$.
Clearly, isomorphic algebras are Morita equivalent. But there are many nonisomorphic algebras which are Morita equivalent, and we will illustrate this by considering some examples which appeared in this book.

# 表示论代考

## 数学代写|表示论代写Representation theory代考|Quivers of Type Have Infinite Representation Type

$\varphi: \mathcal{M} \rightarrow \mathcal{M}$ 是表示的同态。然后像以前一样每个线性映射 $\varphi_i: M(i) \rightarrow M(i)$ 是限制 $\varphi_4$ 到 $M(i)$. 我们再次使用每个子空间 $M(i)$ 在以下是不变的 $\varphi_4$ 获得限制 $\varphi_4$.

$$\varphi_4(e+g+k)=c_2 e+c_1 g+(c k+z h) .$$

$$\varphi_4(l)=v l, \varphi_4(h)=r h, \text { and } v=r .$$

$$\varphi_4(e+f+h)=c_2 e+(a f+b g)+r h \in M(3) .$$

## 数学代写|表示论代写Representation theory代考|Morita Equivalence

(i) 它保留 (有限的) 直和，并采用不可分解的 $A$-不可分解的模块 $B$-模块，简单 $A$ – 简单 的模块 $B$-模块，
(ii) 每个 $B$-模块同构于 $F(M)$ 对于一些 $A$-模块 $M$ ，在哪里 $M$ 直到同构是唯一的，
(iii) 同态之间的对应给出了向量空间同构
$$\left.\operatorname{Hom}_A(M, N)\right) \cong \operatorname{Hom}_B(F(M), F(N))$$

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