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 Field[edit]

Definition 1: Let F 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[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, 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_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 porduct 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