1Artinian algebras
III Algebras
1.3 Crossed products
Number theorists are often interested in representations of Galois groups and
kG
-modules where
k
is an algebraic number field, e.g.
Q
. In this case, the
D
i
’s
appearing in Artin–Wedderburn may be non-commutative.
We have already met one case of a non-commutative division ring, namely
the quaternions H. This is in fact an example of a general construction.
Definition
(Crossed product)
.
The crossed product of a
k
-algebra
B
and a
group G is specified by the following data:
– A group homomorphism φ : G → Aut
k
(B), written
φ
g
(λ) = λ
g
;
– A function
Ψ(g, h) : G × G → B.
The crossed product algebra has underlying set
X
λ
g
g : λ
g
∈ B.
with operation defined by
λg · µh = λµ
g
Ψ(g, h)(gh).
The function Ψ is required to be such that the resulting product is associative.
We should think of the
µ
g
as specifying what happens when we conjugate
g
pass µ, and then Ψ(g, h)(gh) is the product of g and h in the crossed product.
Usually, we take
B
=
K
, a Galois extension of
k
, and
G
=
Gal
(
K/k
). Then
the action
φ
g
is the natural action of
G
on the elements of
K
, and we restrict to
maps Ψ : G × G → K
×
only.
Example.
Consider
B
=
K
=
C
, and
k
=
R
. Then
G
=
Gal
(
C/R
)
∼
=
Z/
2
Z
=
{e, g}, where g is complex conjugation. The elements of H are of the form
λ
e
e + λ
g
g,
where λ
e
, λ
g
∈ C, and we will write
1 · g = g, i · g = k, 1 · e = 1, i · e = i.
Now we want to impose
−1 = j
2
= 1g · 1g = ψ(g, g)e.
So we set Ψ(
g, g
) =
−
1. We can similarly work out what we want the other
values of Ψ to be.
Note that in general, crossed products need not be division algebras.
The crossed product is not just a
k
-algebra. It has a natural structure of a
G-graded algebra, in the sense that we can write it as a direct sum
BG =
M
g∈G
Bg,
and we have Bg
1
· Bg
2
⊆ Bg
1
g
2
.
Focusing on the case where
K/k
is a Galois extension, we use the notation
(
K, G,
Ψ), where Ψ :
G × G → K
×
. Associativity of these crossed products is
equivalent to a 2-cocycle condition Ψ, which you will be asked to make precise
on the first example sheet.
Two crossed products (
K, G,
Ψ
1
) and (
K, G,
Ψ
2
) are isomorphic iff the map
G × G K
×
(g, h) Ψ
1
(g, h)(Ψ
2
(g, h))
−1
satisfies a 2-coboundary condition, which is again left for the first example sheet.
Therefore the second (group) cohomology
{2-cocycles : G × G → K
×
}
{2-coboundaries : G × G → K
×
}
determines the isomorphism classes of (associative) crossed products (K, G, Ψ).
Definition
(Central simple algebra)
.
A central simple
k
-algebra is a finite-
dimensional k-algebra which is a simple algebra, and with a center Z(A) = k.
Note that any simple algebra is a division algebra, say by Schur’s lemma.
So the center must be a field. Hence any simple
k
-algebra can be made into a
central simple algebra simply by enlarging the base field.
Example. M
n
(k) is a central simple algebra.
The point of talking about these is the following result:
Fact.
Any central simple
k
-algebra is of the form
M
n
(
D
) for some division
algebra
D
which is also a central simple
k
-algebra, and is a crossed product
(K, G, Ψ).
Note that when
K
=
C
and
k
=
R
, then the second cohomology group has
2 elements, and we get that the only central simple
R
-algebras are
M
n
(
R
) or
M
n
(H).
For amusement, we also note the following theorem:
Fact (Wedderburn). Every finite division algebra is a field.