[<< 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

.
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