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 k-linear
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 k-algebras 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 k-algebra 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 two-sided 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 k-algebras. 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 k-subalgebra generated by all
T(g)
=
g+g-
1
,
g
E
Jr.
The
Cayley-Hamilton 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 Cayley-Hamilton 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 Cayley-Hamilton Identity, and
that any algebra which has a trace operator with certain properties and which sat-
isfies Cayley-Hamilton 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
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