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