INTRODUCTION 11
of elements of H-[n] have components in both summands. For
x,y
E
H-[n], we
have
xy
=
~~~
+
ixyi
E H+[n] EB H-[n], where
~~~
=
~(xy
+
yx):
H-[n] x H-[n]-+ H+[n]
is a symmetric H+[n]-bilinear form on H-[n] and where
1
ixyi
=
2(xy- yx):
H-[n] x H-[n]-+ H-[n]
is a skew-symmetric Lie algebra product on H- [n], over H+ [n]. It is relatively easy
to see that whenever R
®
H[n] is a free rank 4 algebra over R then the form
lXI
on
H+~
y
the 3-dimensional Lie algebra
R
®
H-[n] coincides with the (slightly modified)
H+[7r]
Killing form ~tr(ad(x)ad(y)), where
ad(x)(z)
=
ixzi.
Quaternion algebras again play a major role in Chapter 7 and the Addendum
to Chapter 7. Free module quaternion algebras
H(r, s; R)
appear in two ways. For
any group
1r
and any elements x, y
E
H[n], let r
=
~~~,
s
=
~~
;I
E
H+[n]. Then
H[n][r-
1
,s-
1]
=
H(r,s;H+[n][r-l,s-
1]).
The quaternion basis is
{1,i,j,k} = {1,
lxl,
ixyi,
lxllxy!}.
Again, it is necessary to
find a pair of elements
x,
y so that
~~~~~
;I
is not nilpotent in H+[n], to get a non-
zero localization. For a free group
1r
n,
one can take x and y to be two generators.
Secondly, if
x, y
E
H[n] are two elements with
T(x)
=
T(y)
and
N(x)
=
N(y),
for
example two meridian generators of a knot group, then, again with s
=
~~
;I,
H[n][s-
1]
=
H(I,
J;
H+[n][s-
1]).
Here, the quaternion basis is {1, i, j, k}
=
{1,
lxl - IYI, lxl + IYI, 2lxyl}
and I
=
i
2
=
2
(~~~-~~~),
J
=
P
=
2
(1~1 +~~I)·
The assumption
T(x)
=
T(y), N(x)
=
N(y)
implies
lxl
2
=
~~~
=
~~~
=
IYI
2
,
so i
=
lxi-IYI
and j
=
lxl + IYI
anticommute.
As in Chapter 4, once we have quaternion algebra localizations of H[n], it
is easy to construct irreducible representations of
1r.
Also, in the Addendum to
Chapter 7, we show how to write down very efficient presentations of H+ [
1r ][
s -
1]
and H+[n][d-
1],
where s
=
lgP gql
and d
=
lgrgsgtl,
using quaternion algebras and
gq gp
our knowledge of the localizations H+[nn][s-
1]
and H+[nn][d-
1]
when
1fn
is free.
We obtain some algorithms for describing, say, all conjugate classes of absolutely
irreducible representations p :
1r
-+ SL(2, K), where
1r
is a knot group and K is a
field, but in practice one would face serious computational problems carrying out
these algorithms and making use of the results.
Following
the rather formidable algebraic developments in chapters 5,6, and
7, we make a completely fresh, elementary start in Chapter 8. The idea, roughly,
is to develop the classical view of representations of
1r
as points of an algebraic
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