10
INTRODUCTION
M(2,
A[1rn])
is injective. The representation Pn is studied in much greater detail
in Chapter 9, where we prove that
Pn :
H[7rn] ---
M(2,
A[1rn])
is injective and
identify its image. The argument outlined in the Addendum to Chapter 5 which
establishes a presentation of all of
H[7rn]
is an unusual argument, requiring lengthy
computations to show that a certain product is associative. We do not include
these computations in the paper. The argument in Chapter 6 which establishes a
presentation of
H+[7rn]
is more conceptual. In the Addendum to Chapter 6, we do
include the most difficult part of the associativity argument skipped in the Adden-
dum to Chapter 5, and obtain a second presentation of H+
[1r n],
as a finite module
over
Hci[7rn] =
k
[1~:1]
/(rank(S):; 3).
Also in the Addendum to Chapter 6, we show that the localizations of
H+[7rn]
defined by inverting either an element of the form s
=
lgp gq
I
or of the form
9q 9p
d =
l9r9s9tl
have very simple structures. For example,
il+[7rn][s-
1]
is isomorphic
to a simple localization of a polynomial ring on 3n- 3 generators, and H+
[1r n] [s-
1]
is a free module, iterated quadratic extension of
il+[7rn][s-
1].
The simplicity of
the localizations
H+[7rn][s-
1]
and
H+[7rn][d-
1]
is closely related to the study of
absolutely irreducible representations p :
7rn ---
SL(2, K), which we take up in
Chapter 7.
In Chapter 7, we first prove a number of results abou.t localizations of
H[1r]
for arbitrary groups
1r,
which are very similar to results in chapters 3 and 4 on
two-generator groups. The starting point is the result that if s
=
~~
;I
and d
=
ixyzi
E
H+[1r]
then both
H[1r][s-
1]
and
H[1r][d-
1]
are free ofrank 4 over
H+[7r][s-
1]
and
H+[7r][d-
1],
respectively, for any elements x,
y,
z E
H[1r].
The Tetrad Iden-
tity above shows that
H-[1r][d-
1]
is spanned by
{lxl, IYI, lzl}
over
H+[7r][d-
1],
and H-
[1r][s-
1]
turns out to be spanned by
{lxl, IYI, lxyl}
over
H+[7r][s-
1].
Of
course, these localizations give null rings unless the element s or d is not nilpo-
tent. However, once we do have a non-zero, free rank 4 algebra
R
0
H[1r]
for
H+["]
some H+
[1r] ---
R, it is easy to construct absolutely irreducible representations
p:
1r---
SL(2, K), where K is some field derived from R, by constructing a suitable
homomorphism I :
H[1r] ---
M(2, K). If
1r
is generated by elements
{g;}
then we
show that I corresponds to an absolutely irreducible representation if and only
if¢
(1~: ~:1)
=/= 0
E
K, some p,q, or
¢(19r9s9ti)
=/= 0
E
K, some
r,s,t,
where
¢ :
H+[1r] ---
K
is I
111
+["]'
or essentially the trace part of I. The condition
¢
(lgP gql)
=/= 0 is the same as
tr(p(gpgqg;; 1g;;
1
))
=/= 2
E
K. There are sometimes
9q 9p
exceptional irreducible representations of
1r
for which
tr(p(gpgqg;;
1
g;;
I))
=
2 for
every pair of generators
gp,gq,
but the condition
¢(i9r9s9tl)
=/= 0 for some
g,.,g"gt
captures these. Alternatively, we show that the first condition can be realized with
a new pair of generators
g,., 9.-9t 9.-:
I.
We insert here a general remark about the structure of
H[1r]
=
H+
[1r]
EB H-
[1r].
This decomposition is not a Z/2-graded algebra decomposition because products
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