# Abstract Algebra/Algebras

In this section we will talk about structures with three operations. These are called algebras. We will start by defining an algebra over a field, which is a vector space with a bilinear vector product. After giving some examples, we will then move to a discussion of quivers and their path algebras.

## Algebras over a Field

Definition 1: Let $F$  be a field, and let $A$  be an $F$ -vector space on which we define the vector product $\cdot \,:\,A\times A\rightarrow A$ . Then $A$  is called an algebra over $F$  provided that $(A,+,\cdot )$  is a ring, where $+$  is the vector space addition, and if for all $a,b,c\in A$  and $\alpha \in F$ ,

1. $a(bc)=(ab)c$ ,
2. $a(b+c)=ab+ac$  and $(a+b)c=ac+bc$ ,
3. $\alpha (ab)=(\alpha a)b=a(\alpha b)$ .

The dimension of an algebra is the dimension of $A$  as a vector space.

Remark 2: The appropriate definition of a subalgebra is clear from Definition 1. We leave its formal statement to the reader.

Definition 2: If $(A,+,\cdot )$  is a commutative ring, $A$  is called a commutative algebra. If it is a division ring, $A$  is called a division algebra. We reserve the terms real and complex algebra for algebras over $\mathbb {R}$  and $\mathbb {C}$ , respectively.

The reader is invited to check that the following examples really are examples of algebras.

Example 3: Let $F$  be a field. The vector space $F^{n}$  forms a commutative $F$ -algebra under componentwise multiplication.

Example 4: The quaternions $\mathbb {H}$  is a 4-dimensional real algebra. We leave it to the reader to show that it is not a 2-dimensional complex algebra.

Example 5: Given a field $F$ , the vector space of polynomials $F[x]$  is a commutative $F$ -algebra in a natural way.

Example 6: Let $F$  be a field. Then any matrix ring over $F$ , for example $\left({\begin{array}{cc}F&0\\F&F\end{array}}\right)$ , gives rise to an $F$ -algebra in a natural way.

## Quivers and Path Algebras

Naively, a quiver can be understood as a directed graph where we allow loops and parallell edges. Formally, we have the following.

Definition 7: A quiver is a collection of four pieces of data, $Q=(Q_{0},Q_{1},s,t)$ ,

1. $Q_{0}$  is the set of vertices of the quiver,
2. $Q_{1}$  is the set of edges, and
3. $s,t\,:\,Q_{1}\rightarrow Q_{0}$  are functions associating with each edge a source vertex and a target vertex, respectively.

We will always assume that $Q_{0}$  is nonempty and that $Q_{0}$  and $Q_{1}$  are finite sets.

Example 8: The following are the simplest examples of quivers:

1. The quiver with one point and no edges, represented by $1$ .
2. The quiver with $n$  point and no edges, $1\quad 2\quad ...\quad n$ .
3. The linear quiver with $n$  points, $1\,{\stackrel {a_{1}}{\longrightarrow }}\,2\,{\stackrel {a_{2}}{\longrightarrow }}\,...\,{\xrightarrow {a_{n-1}}}\,n$ .
4. The simplest quiver with a nontrivial loop, $1{\underset {a}{\stackrel {b}{\leftrightarrows }}}2$ .

Definition 9: Let $Q$  be a quiver. A path in $Q$  is a sequence of edges $a=a_{m}a_{m-1}...a_{1}$  where $s(a_{i})=t(a_{i-1})$  for all $i=2,...,m$ . We extend the domains of $s$  and $t$  and define $s(a)\equiv s(a_{0})$  and $t(a)\equiv t(a_{m})$ . We define the length of the path to be the number of edges it contains and write $l(a)=m$ . With each vertex $i$  of a quiver we associate the trivial path $e_{i}$  with $s(e_{i})=t(e_{i})=i$  and $l(e_{i})=0$ . A nontrivial path $a$  with $s(a)=t(a)=i$  is called an oriented loop at $i$ .

The reason quivers are interesting for us is that they provide a concrete way of constructing a certain family of algebras, called path algebras.

Definition 10: Let $Q$  be a quiver and $F$  a field. Let $FQ$  denote the free vector space generated by all the paths of $Q$ . On this vector space, we define a vector product in the obvious way: if $u=u_{m}...u_{1}$  and $v=v_{n}...v_{1}$  are paths with $s(v)=t(u)$ , define their product $vu$  by concatenation: $vu=v_{n}...v_{1}u_{m}...u_{1}$ . If $s(v)\neq t(u)$ , define their product to be $vu=0$ . This product turns $FQ$  into an $F$ -algebra, called the path algebra of $Q$ .

Lemma 11: Let $Q$  be a quiver and $F$  field. If $Q$  contains a path of length $|Q_{0}|$ , then $FQ$  is infinite dimensional.

Proof: By a counting argument such a path must contain an oriented loop, $a$ , say. Evidently $\{a^{n}\}_{n\in \mathbb {N} }$  is a linearly independent set, such that $FQ$  is infinite dimensional.

Lemma 12: Let $Q$  be a quiver and $F$  a field. Then $FQ$  is infinite dimensional if and only if $Q$  contains an oriented loop.

Proof: Let $a$  be an oriented loop in $Q$ . Then $FQ$  is infinite dimensional by the above argument. Conversely, assume $Q$  has no loops. Then the vertices of the quiver can be ordered such that edges always go from a lower to a higher vertex, and since the length of any given path is bounded above by $|Q_{0}|-1$ , there dimension of $FQ$  is bounded above by $\mathrm {dim} \,FQ\leq |Q_{0}|^{2}-|Q_{0}|<\infty$ .

Lemma 13: Let $Q$  be a quiver and $F$  a field. Then the trivial edges $e_{i}$  form an orthogonal idempotent set.

Proof: This is immediate from the definitions: $e_{i}e_{j}=0$  if $i\neq j$  and $e_{i}^{2}=e_{i}$ .

Corollary 14: The element $\sum _{i\in Q_{0}}e_{i}$  is the identity element in $FQ$ .

Proof: It sufficed to show this on the generators of $FQ$ . Let $a$  be a path in $Q$  with $s(a)=j$  and $t(a)=k$ . Then $\left(\sum _{i\in Q_{0}}e_{i}\right)a=\sum _{i\in Q_{0}}e_{i}a=e_{j}a=a$ . Similarily, $a\left(\sum _{i\in Q_{0}}e_{i}\right)=a$ .

To be covered:

- General R-algebras