# 数学代写|表示论代写Representation theory代考|Group Algebras

## 数学代写|表示论代写Representation theory代考|Group Algebras

Let $G$ be a group and $K$ a field. We define a vector space over $K$ which has basis the set ${g \mid g \in G}$, and we call this vector space $K G$. This space becomes a $K$-algebra if one defines the product on the basis by taking the group multiplication, and extends it to linear combinations. We call this algebra $K G$ the group algebra.
Thus an arbitrary element of $K G$ is a finite linear combination of the form $\sum_{g \in G} \alpha_g g$ with $\alpha_g \in K$. We can write down a formula for the product of two elements, following the recipe in Remark 1.4. Let $\alpha=\sum_{g \in G} \alpha_g g$ and $\beta=\sum_{h \in G} \beta_h h$ be two elements in $K G$; then their product has the form
$$\alpha \beta=\sum_{x \in G}\left(\sum_{g h=x} \alpha_g \beta_h\right) x .$$
Since the multiplication in the group is associative, it follows that the multiplication in $K G$ is associative. Furthermore, one checks that the multiplication in $K G$ is distributive. The identity element of the group algebra $K G$ is given by the identity element of $G$.

Note that the group algebra $K G$ is finite-dimensional if and only if the group $G$ is finite, in which case the dimension of $K G$ is equal to the order of the group $G$. The group algebra $K G$ is commutative if and only if the group $G$ is abelian.

Example 1.10. Let $G$ be the cyclic group of order 3 , generated by $y$, so that $G=\left{1_G, y, y^2\right}$ and $y^3=1_G$. Then we have
$$\left(a_0 1_G+a_1 y+a_2 y^2\right)\left(b_0 1_G+b_1 y+b_2 y^2\right)=c_0 1_G+c_1 y+c_2 y^2,$$
with
$$c_0=a_0 b_0+a_1 b_2+a_2 b_1, c_1=a_0 b_1+a_1 b_0+a_2 b_2, c_2=a_0 b_2+a_1 b_1+a_2 b_0$$

## 数学代写|表示论代写Representation theory代考|Path Algebras of Quivers

Path algebras of quivers are a class of algebras with an easy multiplication formula, and they are extremely useful for calculating examples. They also have connections to other parts of mathematics. The underlying basis of a path algebra is the set of paths in a finite directed graph. It is customary in representation theory to call such a graph a quiver. We assume throughout that a quiver has finitely many vertices and finitely many arrows.

Definition 1.11. A quiver $Q$ is a finite directed graph. We sometimes write $Q=\left(Q_0, Q_1\right)$, where $Q_0$ is the set of vertices and $Q_1$ is the set of arrows.

We assume that $Q_0$ and $Q_1$ are finite sets. For any arrow $\alpha \in Q_1$ we denote by $s(\alpha) \in Q_0$ its starting point and by $t(\alpha) \in Q_0$ its end point.

A non-trivial path in $Q$ is a sequence $p=\alpha_r \ldots \alpha_2 \alpha_1$ of arrows $\alpha_i \in Q_1$ such that $t\left(\alpha_i\right)=s\left(\alpha_{i+1}\right)$ for all $i=1, \ldots, r-1$. Note that our convention is to read paths from right to left. The number $r$ of arrows is called the length of $p$, and we denote by $s(p)=s\left(\alpha_1\right)$ the starting point, and by $t(p)=t\left(\alpha_r\right)$ the end point of $p$.
For each vertex $i \in Q_0$ we also need to have a trivial path of length 0 , which we call $e_i$, and we $\operatorname{set} s\left(e_i\right)=i=t\left(e_i\right)$.

We call a path $p$ in $Q$ an oriented cycle if $p$ has positive length and $s(p)=t(p)$.
Definition 1.12. Let $K$ be a field and $Q$ a quiver. The path algebra $K Q$ of the quiver $Q$ over the field $K$ has underlying vector space with basis given by all paths in $Q$.

The multiplication in $K Q$ is defined on the basis by concatenation of paths (if possible), and extended linearly to linear combinations. More precisely, for two paths $p=\alpha_r \ldots \alpha_1$ and $q=\beta_s \ldots \beta_1$ in $Q$ we set
$$p \cdot q=\left{\begin{array}{cl} \alpha_r \ldots \alpha_1 \beta_s \ldots \beta_1 & \text { if } t\left(\beta_s\right)=s\left(\alpha_1\right) \ 0 & \text { otherwise } \end{array}\right.$$

# 表示论代考

## 数学代写|表示论代写Representation theory代考|Group Algebras

$$\alpha \beta=\sum_{x \in G}\left(\sum_{g h=x} \alpha_g \beta_h\right) x .$$

$$\left(a_0 1_G+a_1 y+a_2 y^2\right)\left(b_0 1_G+b_1 y+b_2 y^2\right)=c_0 1_G+c_1 y+c_2 y^2$$

$$c_0=a_0 b_0+a_1 b_2+a_2 b_1, c_1=a_0 b_1+a_1 b_0+a_2 b_2, c_2=a_0 b_2+a_1 b_1+a_2 b_0$$

## 数学代写|表示论代写Representation theory代考|Path Algebras of Quivers

\alpha_r \ldots \alpha_1 \beta_s \ldots \beta_1 \quad \text { if } t\left(\beta_s\right)=s\left(\alpha_1\right) 0 \quad \text { otherwise }


myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部:

myassignments-help服务请添加我们官网的客服或者微信/QQ，我们的服务覆盖：Assignment代写、Business商科代写、CS代考、Economics经济学代写、Essay代写、Finance金融代写、Math数学代写、report代写、R语言代考、Statistics统计学代写、物理代考、作业代写、加拿大代考、加拿大统计代写、北美代写、北美作业代写、北美统计代考、商科Essay代写、商科代考、数学代考、数学代写、数学作业代写、physics作业代写、物理代写、数据分析代写、新西兰代写、澳洲Essay代写、澳洲代写、澳洲作业代写、澳洲统计代写、澳洲金融代写、留学生课业指导、经济代写、统计代写、统计作业代写、美国Essay代写、美国代考、美国数学代写、美国统计代写、英国Essay代写、英国代考、英国作业代写、英国数学代写、英国统计代写、英国金融代写、论文代写、金融代考、金融作业代写。

Scroll to Top