[<< wikibooks] 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
      
    
    {\displaystyle F}
   be a field, and let 
  
    
      
        A
      
    
    {\displaystyle A}
   be an 
  
    
      
        F
      
    
    {\displaystyle F}
  -vector space on which we define the vector product 
  
    
      
        ⋅
        
        :
        
        A
        ×
        A
        →
        A
      
    
    {\displaystyle \cdot \,:\,A\times A\rightarrow A}
  . Then 
  
    
      
        A
      
    
    {\displaystyle A}
   is called an algebra over 
  
    
      
        F
      
    
    {\displaystyle F}
   provided that 
  
    
      
        (
        A
        ,
        +
        ,
        ⋅
        )
      
    
    {\displaystyle (A,+,\cdot )}
   is a ring, where 
  
    
      
        +
      
    
    {\displaystyle +}
   is the vector space addition, and if for all 
  
    
      
        a
        ,
        b
        ,
        c
        ∈
        A
      
    
    {\displaystyle a,b,c\in A}
   and 
  
    
      
        α
        ∈
        F
      
    
    {\displaystyle \alpha \in F}
  ,

  
    
      
        a
        (
        b
        c
        )
        =
        (
        a
        b
        )
        c
      
    
    {\displaystyle a(bc)=(ab)c}
  ,

  
    
      
        a
        (
        b
        +
        c
        )
        =
        a
        b
        +
        a
        c
      
    
    {\displaystyle a(b+c)=ab+ac}
   and 
  
    
      
        (
        a
        +
        b
        )
        c
        =
        a
        c
        +
        b
        c
      
    
    {\displaystyle (a+b)c=ac+bc}
  ,

  
    
      
        α
        (
        a
        b
        )
        =
        (
        α
        a
        )
        b
        =
        a
        (
        α
        b
        )
      
    
    {\displaystyle \alpha (ab)=(\alpha a)b=a(\alpha b)}
  .The dimension of an algebra is the dimension of 
  
    
      
        A
      
    
    {\displaystyle 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
        ,
        +
        ,
        ⋅
        )
      
    
    {\displaystyle (A,+,\cdot )}
   is a commutative ring, 
  
    
      
        A
      
    
    {\displaystyle A}
   is called a commutative algebra. If it is a division ring, 
  
    
      
        A
      
    
    {\displaystyle A}
   is called a division algebra. We reserve the terms real and complex algebra for algebras over 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
   and 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  , respectively.
The reader is invited to check that the following examples really are examples of algebras.
Example 3: Let 
  
    
      
        F
      
    
    {\displaystyle F}
   be a field. The vector space 
  
    
      
        
          F
          
            n
          
        
      
    
    {\displaystyle F^{n}}
   forms a commutative 
  
    
      
        F
      
    
    {\displaystyle F}
  -algebra under componentwise multiplication.
Example 4: The quaternions 
  
    
      
        
          H
        
      
    
    {\displaystyle \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
      
    
    {\displaystyle F}
  , the vector space of polynomials 
  
    
      
        F
        [
        x
        ]
      
    
    {\displaystyle F[x]}
   is a commutative 
  
    
      
        F
      
    
    {\displaystyle F}
  -algebra in a natural way.
Example 6: Let 
  
    
      
        F
      
    
    {\displaystyle F}
   be a field. Then any matrix ring over 
  
    
      
        F
      
    
    {\displaystyle F}
  , for example 
  
    
      
        
          (
          
            
              
                
                  F
                
                
                  0
                
              
              
                
                  F
                
                
                  F
                
              
            
          
          )
        
      
    
    {\displaystyle \left({\begin{array}{cc}F&0\\F&F\end{array}}\right)}
  , gives rise to an 
  
    
      
        F
      
    
    {\displaystyle 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
        )
      
    
    {\displaystyle Q=(Q_{0},Q_{1},s,t)}
  ,

  
    
      
        
          Q
          
            0
          
        
      
    
    {\displaystyle Q_{0}}
   is the set of vertices of the quiver,

  
    
      
        
          Q
          
            1
          
        
      
    
    {\displaystyle Q_{1}}
   is the set of edges, and

  
    
      
        s
        ,
        t
        
        :
        
        
          Q
          
            1
          
        
        →
        
          Q
          
            0
          
        
      
    
    {\displaystyle 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
          
        
      
    
    {\displaystyle Q_{0}}
   is nonempty and that 
  
    
      
        
          Q
          
            0
          
        
      
    
    {\displaystyle Q_{0}}
   and 
  
    
      
        
          Q
          
            1
          
        
      
    
    {\displaystyle Q_{1}}
   are finite sets.
Example 8: The following are the simplest examples of quivers:

The quiver with one point and no edges, represented by 
  
    
      
        1
      
    
    {\displaystyle 1}
  .
The quiver with 
  
    
      
        n
      
    
    {\displaystyle n}
   point and no edges, 
  
    
      
        1
        
        2
        
        .
        .
        .
        
        n
      
    
    {\displaystyle 1\quad 2\quad ...\quad n}
  .
The linear quiver with 
  
    
      
        n
      
    
    {\displaystyle n}
   points, 
  
    
      
        1
        
        
          
            
              
                ⟶
              
              
                
                  a
                  
                    1
                  
                
              
            
          
        
        
        2
        
        
          
            
              
                ⟶
              
              
                
                  a
                  
                    2
                  
                
              
            
          
        
        
        .
        .
        .
        
        
          
            →
            
              
                a
                
                  n
                  −
                  1
                
              
            
          
        
        
        n
      
    
    {\displaystyle 1\,{\stackrel {a_{1}}{\longrightarrow }}\,2\,{\stackrel {a_{2}}{\longrightarrow }}\,...\,{\xrightarrow {a_{n-1}}}\,n}
  .
The simplest quiver with a nontrivial loop, 
  
    
      
        1
        
          
            
              
                
                  ⇆
                
                
                  b
                
              
            
            a
          
        
        2
      
    
    {\displaystyle 1{\underset {a}{\stackrel {b}{\leftrightarrows }}}2}
  .Definition 9: Let 
  
    
      
        Q
      
    
    {\displaystyle Q}
   be a quiver. A path in 
  
    
      
        Q
      
    
    {\displaystyle Q}
   is a sequence of edges 
  
    
      
        a
        =
        
          a
          
            m
          
        
        
          a
          
            m
            −
            1
          
        
        .
        .
        .
        
          a
          
            1
          
        
      
    
    {\displaystyle a=a_{m}a_{m-1}...a_{1}}
   where 
  
    
      
        s
        (
        
          a
          
            i
          
        
        )
        =
        t
        (
        
          a
          
            i
            −
            1
          
        
        )
      
    
    {\displaystyle s(a_{i})=t(a_{i-1})}
   for all 
  
    
      
        i
        =
        2
        ,
        .
        .
        .
        ,
        m
      
    
    {\displaystyle i=2,...,m}
  . We extend the domains of 
  
    
      
        s
      
    
    {\displaystyle s}
   and 
  
    
      
        t
      
    
    {\displaystyle t}
   and define 
  
    
      
        s
        (
        a
        )
        ≡
        s
        (
        
          a
          
            0
          
        
        )
      
    
    {\displaystyle s(a)\equiv s(a_{0})}
   and 
  
    
      
        t
        (
        a
        )
        ≡
        t
        (
        
          a
          
            m
          
        
        )
      
    
    {\displaystyle 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
      
    
    {\displaystyle l(a)=m}
  . With each vertex 
  
    
      
        i
      
    
    {\displaystyle i}
   of a quiver we associate the trivial path 
  
    
      
        
          e
          
            i
          
        
      
    
    {\displaystyle e_{i}}
   with 
  
    
      
        s
        (
        
          e
          
            i
          
        
        )
        =
        t
        (
        
          e
          
            i
          
        
        )
        =
        i
      
    
    {\displaystyle s(e_{i})=t(e_{i})=i}
   and 
  
    
      
        l
        (
        
          e
          
            i
          
        
        )
        =
        0
      
    
    {\displaystyle l(e_{i})=0}
  . A nontrivial path 
  
    
      
        a
      
    
    {\displaystyle a}
   with 
  
    
      
        s
        (
        a
        )
        =
        t
        (
        a
        )
        =
        i
      
    
    {\displaystyle s(a)=t(a)=i}
   is called an oriented loop at 
  
    
      
        i
      
    
    {\displaystyle 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
      
    
    {\displaystyle Q}
   be a quiver and 
  
    
      
        F
      
    
    {\displaystyle F}
   a field. Let 
  
    
      
        F
        Q
      
    
    {\displaystyle FQ}
   denote the free vector space generated by all the paths of 
  
    
      
        Q
      
    
    {\displaystyle Q}
  . On this vector space, we define a vector product in the obvious way: if 
  
    
      
        u
        =
        
          u
          
            m
          
        
        .
        .
        .
        
          u
          
            1
          
        
      
    
    {\displaystyle u=u_{m}...u_{1}}
   and 
  
    
      
        v
        =
        
          v
          
            n
          
        
        .
        .
        .
        
          v
          
            1
          
        
      
    
    {\displaystyle v=v_{n}...v_{1}}
   are paths with 
  
    
      
        s
        (
        v
        )
        =
        t
        (
        u
        )
      
    
    {\displaystyle s(v)=t(u)}
  , define their product 
  
    
      
        v
        u
      
    
    {\displaystyle vu}
   by concatenation: 
  
    
      
        v
        u
        =
        
          v
          
            n
          
        
        .
        .
        .
        
          v
          
            1
          
        
        
          u
          
            m
          
        
        .
        .
        .
        
          u
          
            1
          
        
      
    
    {\displaystyle vu=v_{n}...v_{1}u_{m}...u_{1}}
  . If 
  
    
      
        s
        (
        v
        )
        ≠
        t
        (
        u
        )
      
    
    {\displaystyle s(v)\neq t(u)}
  , define their product to be 
  
    
      
        v
        u
        =
        0
      
    
    {\displaystyle vu=0}
  . This product turns 
  
    
      
        F
        Q
      
    
    {\displaystyle FQ}
   into an 
  
    
      
        F
      
    
    {\displaystyle F}
  -algebra, called the path algebra of 
  
    
      
        Q
      
    
    {\displaystyle Q}
  .
Lemma 11: Let 
  
    
      
        Q
      
    
    {\displaystyle Q}
   be a quiver and 
  
    
      
        F
      
    
    {\displaystyle F}
   field. If 
  
    
      
        Q
      
    
    {\displaystyle Q}
   contains a path of length 
  
    
      
        
          |
        
        
          Q
          
            0
          
        
        
          |
        
      
    
    {\displaystyle |Q_{0}|}
  , then 
  
    
      
        F
        Q
      
    
    {\displaystyle FQ}
   is infinite dimensional.
Proof: By a counting argument such a path must contain an oriented loop, 
  
    
      
        a
      
    
    {\displaystyle a}
  , say. Evidently 
  
    
      
        {
        
          a
          
            n
          
        
        
          }
          
            n
            ∈
            
              N
            
          
        
      
    
    {\displaystyle \{a^{n}\}_{n\in \mathbb {N} }}
   is a linearly independent set, such that 
  
    
      
        F
        Q
      
    
    {\displaystyle FQ}
   is infinite dimensional.
Lemma 12: Let 
  
    
      
        Q
      
    
    {\displaystyle Q}
   be a quiver and 
  
    
      
        F
      
    
    {\displaystyle F}
   a field. Then 
  
    
      
        F
        Q
      
    
    {\displaystyle FQ}
   is infinite dimensional if and only if 
  
    
      
        Q
      
    
    {\displaystyle Q}
   contains an oriented loop.
Proof: Let 
  
    
      
        a
      
    
    {\displaystyle a}
   be an oriented loop in 
  
    
      
        Q
      
    
    {\displaystyle Q}
  . Then 
  
    
      
        F
        Q
      
    
    {\displaystyle FQ}
   is infinite dimensional by the above argument. Conversely, assume 
  
    
      
        Q
      
    
    {\displaystyle 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
      
    
    {\displaystyle |Q_{0}|-1}
  , there dimension of 
  
    
      
        F
        Q
      
    
    {\displaystyle FQ}
   is bounded above by 
  
    
      
        
          d
          i
          m
        
        
        F
        Q
        ≤
        
          |
        
        
          Q
          
            0
          
        
        
          
            |
          
          
            2
          
        
        −
        
          |
        
        
          Q
          
            0
          
        
        
          |
        
        <
        ∞
      
    
    {\displaystyle \mathrm {dim} \,FQ\leq |Q_{0}|^{2}-|Q_{0}|<\infty }
  .
Lemma 13: Let 
  
    
      
        Q
      
    
    {\displaystyle Q}
   be a quiver and 
  
    
      
        F
      
    
    {\displaystyle F}
   a field. Then the trivial edges 
  
    
      
        
          e
          
            i
          
        
      
    
    {\displaystyle e_{i}}
   form an orthogonal idempotent set.
Proof: This is immediate from the definitions: 
  
    
      
        
          e
          
            i
          
        
        
          e
          
            j
          
        
        =
        0
      
    
    {\displaystyle e_{i}e_{j}=0}
   if 
  
    
      
        i
        ≠
        j
      
    
    {\displaystyle i\neq j}
   and 
  
    
      
        
          e
          
            i
          
          
            2
          
        
        =
        
          e
          
            i
          
        
      
    
    {\displaystyle e_{i}^{2}=e_{i}}
  .
Corollary 14: The element 
  
    
      
        
          ∑
          
            i
            ∈
            
              Q
              
                0
              
            
          
        
        
          e
          
            i
          
        
      
    
    {\displaystyle \sum _{i\in Q_{0}}e_{i}}
   is the identity element in 
  
    
      
        F
        Q
      
    
    {\displaystyle FQ}
  .
Proof: It sufficed to show this on the generators of 
  
    
      
        F
        Q
      
    
    {\displaystyle FQ}
  . Let 
  
    
      
        a
      
    
    {\displaystyle a}
   be a path in 
  
    
      
        Q
      
    
    {\displaystyle Q}
   with 
  
    
      
        s
        (
        a
        )
        =
        j
      
    
    {\displaystyle s(a)=j}
   and 
  
    
      
        t
        (
        a
        )
        =
        k
      
    
    {\displaystyle t(a)=k}
  . Then 
  
    
      
        
          (
          
            
              ∑
              
                i
                ∈
                
                  Q
                  
                    0
                  
                
              
            
            
              e
              
                i
              
            
          
          )
        
        a
        =
        
          ∑
          
            i
            ∈
            
              Q
              
                0
              
            
          
        
        
          e
          
            i
          
        
        a
        =
        
          e
          
            j
          
        
        a
        =
        a
      
    
    {\displaystyle \left(\sum _{i\in Q_{0}}e_{i}\right)a=\sum _{i\in Q_{0}}e_{i}a=e_{j}a=a}
  . Similarily, 
  
    
      
        a
        
          (
          
            
              ∑
              
                i
                ∈
                
                  Q
                  
                    0
                  
                
              
            
            
              e
              
                i
              
            
          
          )
        
        =
        a
      
    
    {\displaystyle a\left(\sum _{i\in Q_{0}}e_{i}\right)=a}
  .
To be covered:
- General R-algebras