0Introduction
III Algebras
0 Introduction
We start with the definition of an algebra. Throughout the course,
k
will be a
field.
Definition
(
k
-algebra)
.
A (unital) associative
k
-algebra is a
k
-vector space
A
together with a linear map
m
:
A ⊗ A → A
, called the product map, and linear
map u : k → A, called the unit map, such that
– The product induced by m is associative.
– u(1) is the identity of the multiplication.
In particular, we don’t require the product to be commutative. We usually
write m(x ⊗ y) as xy.
Example.
Let
K/k
be a finite field extension. Then
K
is a (commutative)
k-algebra.
Example.
The
n × n
matrices
M
n
(
k
) over
k
form a non-commutative
k
-algebra.
Example.
The quaternions
H
is an
R
-algebra, with an
R
-basis 1
, i, j, k
, and
multiplication given by
i
2
= j
2
= k
2
= 1, ij = k, ji = −k.
This is in fact a division algebra (or skew fields), i.e. the non-zero elements have
multiplicative inverse.
Example. Let G be a finite group. Then the group algebra
kG =
n
X
λ
g
g : g ∈ G, λ
g
∈ k
o
with the obvious multiplication induced by the group operation is a k-algebra.
These are the associative algebras underlying the representation theory of
finite groups.
Most of the time, we will just care about algebras that are finite-dimensional
as
k
-vector spaces. However, often what we need for the proofs to work is not
that the algebra is finite-dimensional, but just that it is Artinian. These algebras
are defined by some finiteness condition on the ideals.
Definition
(Ideal)
.
A left ideal of
A
is a
k
-subspace of
A
such that if
x ∈ A
and
y ∈ I
, then
xy ∈ I
. A right ideal is one where we require
yx ∈ I
instead.
An ideal is something that is both a left ideal and a right ideal.
Since the multiplication is not necessarily commutative, we have to make the
distinction between left and right things. Most of the time, we just talk about
the left case, as the other case is entirely analogous.
The definition we want is the following:
Definition
(Artinian algebra)
.
An algebra
A
is left Artinian if it satisfies the
descending chain condition (DCC ) on left ideals, i.e. if we have a descending
chain of left ideals
I
1
≥ I
2
≥ I
3
≥ · · · ,
then there is some N such that I
N+m
= I
N
for all m ≥ 0.
We say an algebra is Artinian if it is both left and right Artinian.
Example. Any finite-dimensional algebra is Artinian.
The main classification theorem for Artinian algebras we will prove is the
following result:
Theorem
(Artin–Wedderburn theorem)
.
Let
A
be a left-Artinian algebra such
that the intersection of the maximal left ideals is zero. Then
A
is the direct sum
of finitely many matrix algebras over division algebras.
When we actually get to the theorem, we will rewrite this in a way that
seems a bit more natural.
One familiar application of this theorem is in representation theory. The group
algebra of a finite group is finite-dimensional, and in particular Artinian. We
will later see that Maschke’s theorem is equivalent to saying that the hypothesis
of the theorem holds. So this puts a very strong constraint on how the group
algebra looks like.
After studying Artinian rings, we’ll talk about Noetherian algebras.
Definition
(Noetherian algebra)
.
An algebra is left Noetherian if it satisfies
the ascending chain condition (ACC ) on left ideals, i.e. if
I
1
≤ I
2
≤ I
3
≤ · · ·
is an ascending chain of left ideals, then there is some
N
such that
I
N+m
=
I
N
for all m ≥ 0.
Similarly, we can define right Noetherian algebras, and say an algebra is
Noetherian if it is both left and right Noetherian.
We can again look at some examples.
Example. Again all finite-dimensional algebras are Noetherian.
Example.
In the commutative case, Hilbert’s basis theorem tells us a polynomial
algebra
k
[
X
1
, · · · , X
k
] in finitely many variables is Noetherian. Similarly, the
power series rings k[[X
1
, · · · , X
n
]] are Noetherian.
Example.
The universal enveloping algebra of a finite-dimensional Lie algebra
are the (associative!) algebras that underpin the representation theory of these
Lie algebras.
Example.
Some differential operator algebras are Noetherian. We assume
char k
= 0. Consider the polynomial ring
k
[
X
]. We have operators “multipli-
cation by
X
” and “differentiate with respect to
X
” on
k
[
X
]. We can form the
algebra
k
[
X,
∂
∂x
] of differential operators on
k
[
X
], with multiplication given by
the composition of operators. This is called the Weyl algebra
A
1
. We will show
that this is a non-commutative Noetherian algebra.
Example.
Some group algebras are Noetherian. Clearly all group algebras of
finite groups are Noetherian, but the group algebras of certain infinite groups
are Noetherian. For example, we can take
G =
1 λ µ
0 1 ν
0 0 0
: λ, µ, ν ∈ Z
,
and this is Noetherian. However, we shall not prove this.
We will see that all left Artinian algebras are left Noetherian. While there is
a general theory of non-commutative Noetherian algebras, it is not as useful as
the theory of commutative Noetherian algebras.
In the commutative case, we often look at
Spec A
, the set of prime ideals of
A
. However, sometimes in the non-commutative there are few prime ideals, and
so Spec is not going to keep us busy.
Example. In the Weyl algebra A
1
, the only prime ideals are 0 and A
1
.
We will prove a theorem of Goldie:
Theorem
(Goldie’s theorem)
.
Let
A
be a right Noetherian algebra with no
non-zero ideals all of whose elements are nilpotent. Then
A
embeds in a finite
direct sum of matrix algebras over division algebras.
Some types of Noetherian algebras can be thought of as non-commutative
polynomial algebras and non-commutative power series, i.e. they are deformations
of the analogous commutative algebra. For example, we say
A
1
is a deformation
of the polynomial algebra
k
[
X, Y
], where instead of having
XY − Y X
= 0, we
have
XY − Y X
= 1. This also applies to enveloping algebras and Iwasawa
algebras. We will study when one can deform the multiplication so that it remains
associative, and this is bound up with the cohomology theory of associative
algebras — Hochschild cohomology. The Hochschild complex has rich algebraic
structure, and this will allow us to understand how we can deform the algebra.
At the end, we shall quickly talk about bialgebras and Hopf algebras. In a
bialgebra, one also has a comultiplication map
A → A⊗A
, which in representation
theory is crucial in saying how to regard a tensor product of two representations
as a representation.