Abstract Algebra/Algebras

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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 FieldEdit

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  ,

  1.  ,
  2.   and  ,
  3.  .

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 AlgebrasEdit

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,  ,

  1.   is the set of vertices of the quiver,
  2.   is the set of edges, and
  3.   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:

  1. The quiver with one point and no edges, represented by  .
  2. The quiver with   point and no edges,  .
  3. The linear quiver with   points,  .
  4. 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