2
INTRODUCTION
~ E
k.
We usually write H[n], rather than
Hk[n],
unless there is some particular
reason to emphasize the ground ring.
The group ring k[n] is an algebra with an involution,
L,
which is the klinear
antiautomorphism extending
L(g)
=
g 1
,
g E
Jr.
The involution
L
descends to
an involution on the quotient algebra H[n] of k[n]. The matrix ring M(2, A) is
also an algebra with an involution, which is an antiautomorphism,
L (
~ ~
)
=
(
d b ) . Our first result in Chapter 1 is the observation that group represen
c a
tations p:
1r

SL(2,
A) correspond bijectively to homomorphisms in the category
of kalgebras with involution
p:
H[n]  M(2, A). Now, arbitrary algebra homo
morphisms k[n] 
M(2,
A) correspond bijectively to representations
1r

GL(2,
A),
and involution preserving algebra homomorphisms k[n]  M(2, A) correspond bi
jectively to representations
1r
SL(2,
A). The point is, simply, if
1 E M(2,
A) then
L(l)
=
1
1
exactly when det(l)
=
1. So, our first result doesn't really have much
to do with H[n]. However, one of our last results, proved in Chapter 9, is that in
certain important cases there exists an injective involution preserving algebra map
P1r :
H[n]  M(2, A[n]), for a suitable kalgebra A[n]. Specifically, we prove this
assertion for arbitrary groups
1r
if the ground ring is a field of characteristic zero,
and we prove this assertion for arbitrary ground rings if the group
1r
is a free group.
Both results are essentially due to Procesi, in the context of his studies of invariant
theory of matrices. We will bring out this connection in graater detail later in this
Introduction. The point to be made here is that if an injective involution pre
serving algebra homomorphism H[n] 
M(2,
A[n]) exists, then the twosided ideal
((h(g
+
g
1
)
(g
+
g
1
)h))
c
k[n] contains
all
elements of k[n] which must map
to zero under all representations
p:
k[n] M(2, A) induced by p:
1r
SL(2,
A),
where
A
varies over commutative kalgebras. Thus, H[n] is a very natural quotient
algebra of k[n], in the context of studying
SL(2)
representations of
Jr.
The algebra H[n] is also an algebra with a trace
T(x)
=
x
+
L(x)
and a norm
N(x)
=
xL(x),
having certain useful properties which we establish in Chapter 1.
Note that for (
~ ~
)
=
1 E
M(2, A), we have tr(I)I
=
1
+
L(l) and det(I)I
=
IL(/), where L(l)
= (
d
b )
as above. Both
T(x)
and
N(x)
are central
c
a
elements in H[n] for all
x,
in fact, both belong to the central subring
TH[n]
C
H[n]
which is defined to be the ksubalgebra generated by all
T(g)
=
g+g
1
,
g
E
Jr.
The
CayleyHamilton Identity holds in H[n] in the form x
2
T(x)x+N(x)
=
0. Already
in the group ring k[n], one has the not too deep relation x
2

x(x+
L(x))
+xL(x)
=
0.
Nonetheless, the CayleyHamilton Identity in H[n] is very important. The real
subtlety is not the identity itself, but the fact that
T(x)
=
x+L(x)
and
N(x)
=
n(x)
are central elements in H[n].
There is a metatheorem in the combined framework of invariant theory of matri
ces and rings which satisfy Polynomial Identities, which asserts that all Polynomial
Identities which hold for matrices follow from the CayleyHamilton Identity, and
that any algebra which has a trace operator with certain properties and which sat
isfies CayleyHamilton Identities in a suitable sense, is a subring of a matrix ring.
Actually, this is a theorem, not just a metatheorem, again due to Procesi [20], but
it would not be productive to develop all the necessary hypotheses !Jere. Those