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 $$\alpha\in F$$, The dimension of an algebra is the dimension of $$A$$ as a vector space.
 * 1) $$a(bc)=(ab)c$$,
 * 2) $$a(b+c)=ab+ac$$ and $$(a+b)c=ac+bc$$,
 * 3) $$\alpha(ab)=(\alpha a)b=a(\alpha b)$$.

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, $$Q=(Q_0,Q_1,s,t)$$, We will always assume that $$Q_0$$ is nonempty and that $$Q_0$$ and $$Q_1$$ are finite sets.
 * 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.

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 $$a=a_ma_{m-1}...a_1$$ 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: $$vu=v_n...v_1u_m...u_1$$. 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: $$e_ie_j=0$$ 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 $$\left(\sum_{i\in Q_0} e_i\right)a=\sum_{i\in Q_0} e_ia=e_ja=a$$. Similarily, $$a\left(\sum_{i\in Q_0} e_i\right)=a$$.

To be covered:

- General R-algebras