# 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[edit]

**Definition 1:** Let a field, and let be an -vector space on which we define the vector product . Then is called an *algebra over* provided that is a ring, where is the vector space addition, and if for all and ,

- ,
- and ,
- .

The *dimension* of an algebra is the dimension of 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 is a commutative ring, is called a *commutative algebra*. If it is a division ring, is called a *division algebra*. We reserve the terms *real* and *complex algebra* for algebras over and , respectively.

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

**Example 3:** Let be a field. The vector space forms a commutative -algebra under componentwise multiplication.

**Example 4:** The quaternions 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 , the vector space of polynomials is a commutative -algebra in a natural way.

**Example 6:** Let be a field. Then any matrix ring over , for example , gives rise to an -algebra in a natural way.

## Quivers and Path Algebras[edit]

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, ,

- is the set of
*vertices*of the quiver, - is the set of
*edges*, and - are functions associating with each edge a
*source vertex*and a*target vertex*, respectively.

We will always assume that is nonempty and that and are finite sets.

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

- The quiver with one point and no edges, represented by .
- The quiver with point and no edges, .
- The linear quiver with points, .
- The simplest quiver with a nontrivial loop, .

**Definition 9:** Let be a quiver. A *path* in is a sequence of edges where for all . We extend the domains of and and define and . We define the *length* of the path to be the number of edges it contains and write . With each vertex of a quiver we associate the *trivial path* with and . A nontrivial path with is called an *oriented loop* at .

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 be a quiver and a field. Let denote the free vector space generated by all the paths of . On this vector space, we define a vector porduct in the obvious way: if and are paths with , define their product by concatenation: . If , define their product to be . This product turns into an -algebra, called the *path algebra of* .

**Lemma 11:** Let be a quiver and field. If contains a path of length , then is infinite dimensional.

*Proof:* By a counting argument such a path must contain an oriented loop, , say. Evidently is a linearly independent set, such that is infinite dimensional.

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

*Proof:* Let be an oriented loop in . Then is infinite dimensional by the above argument. Conversely, assume 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 , there dimension of is bounded above by .

**Lemma 13:** Let be a quiver and a field. Then the trivial edges form an orthogonal idempotent set.

*Proof:* This is immediate from the definitions: if and .

**Corollary 14:** The element is the identity element in .

*Proof:* It sufficed to show this on the generators of . Let be a path in with and . Then . Similarily, .

*To be covered:*

- General R-algebras