Introduction
The purpose of this work is to develop a somewhat new approach to the algebraic
theory of
S£(2)
representations of groups. The initial idea is very simple. If
A is a commutative ring and if p : 1r
---+
S£(2,
A) is a homomorphism from a
group 1r to the group of 2 x 2 matrices of determinant one over A, then there is
a unique algebra homomorphism
p :
Z[1r]
---+
M(2, A)
extending p, where Z[1r] is
the integral group ring of 1r and
M(2,
A) is the ring of
2
x
2
matrices over A.
Certain elements of Z[1r] are always in the kernel of
p.
For example, first observe
that if 'Y
E S£(2, A)
then 'Y
+
1-
1 =
tr('Y
)I E M(2, A),
where
I
= (
~ ~
)
and
tr('Y)
E
A is the trace of 'Y· This identity, 'Y
+
1-1
=
tr('Y)I for 'Y
E
S£(2,
A),
is equivalent to the Cayley-Hamilton Identity for a 2 x 2 matrix of determinant
one, 1
2
-
tr('Y)'Y
+I
=
0, which can be rewritten 'Y(tr('Y)I- 'Y)
=
I.
It follows
that for
g
E 1r,
p(g
+
g- 1
)
=
p(g)
+
p(g)-
1
belongs to
AI=
center
M(2, A),
hence
h(g
+
g- 1
) -
(g
+
g- 1
)h
E
kernel(p) for all
g, h
E
1r. Thus,
p
factors through the
quotient ring of Z[1r] obtained by dividing by the two-sided ideal generated by all
elements
h(g
+
g-
1
)-
(g
+
g-
1
)h.
Slightly more generally, if k is any commutative ground ring and if A is a
commutative k-algebra then a representation p : 1r
---+
S£(2, A)
extends k-linearly
top:
k[1r]
---+
M(2, A),
with
p(h(g
+
g- 1
)-
(g
+
g- 1
)h)=
0. We thus define a
k-
algebra, which is a functorial invariant of 1r, as follows.
Hk[1r]
=
k[1r]/((h(g
+
g- 1
)-
(g
+
g- 1
)h)),
all
g,
hE 1r.
Our approach to the study of
S£(2)
representations will be to unravel the structure
of the algebra
Hk
[1r], and relate that structure to a description of the
S£(2)
rep-
resentations of 1r and the conjugate classes of
S£(2)
representations of 1r. In this
Introduction, we will outline our main results, indicate the connections of our ap-
proach with other approaches to
S£(2)
representations, discuss some applications
and potential applications of our algebras Hk[1r] to the study of knot groups and
knot invariants, and explain a little bit of the philosophy behind our approach.
We are not really interested in arbitrary ground rings. For example, Hk[1r]
=
k
0
Hz[1r], so if we understood Hz[1r], we would understand all Hk[1r]. It turns
out that we don't know much about Hz[7r] in general, but we do have extensive
results about
Hz 1 ~j[1r],
and thus also about Hk[1r]
=
k 0
Hz 1 ~j[1r]
whenever
~ E
k.
At times it is convenient to assume that k is a field, of characteristic zero or of
finite characteristic
=f.
2, which basically means k =
Q
or Zjp, since, again, one
can easily extend scalers to larger fields. Nonetheless, it is relatively harmless, and
more natural, to work over an arbitrary commutative ground ring k, or some k with
http://dx.doi.org/10.1090/conm/187
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