<s>
In	O
mathematics	O
the	O
Lawrence	B-Algorithm
–	I-Algorithm
Krammer	I-Algorithm
representation	I-Algorithm
is	O
a	O
representation	O
of	O
the	O
braid	B-Application
groups	I-Application
.	O
</s>
<s>
The	O
first	O
Lawrence	O
representation	O
is	O
the	O
Burau	B-Algorithm
representation	I-Algorithm
and	O
the	O
second	O
is	O
the	O
Lawrence	B-Algorithm
–	I-Algorithm
Krammer	I-Algorithm
representation	I-Algorithm
.	O
</s>
<s>
The	O
Lawrence	B-Algorithm
–	I-Algorithm
Krammer	I-Algorithm
representation	I-Algorithm
is	O
named	O
after	O
Ruth	O
Lawrence	O
and	O
Daan	O
Krammer	O
.	O
</s>
<s>
Consider	O
the	O
braid	B-Application
group	I-Application
to	O
be	O
the	O
mapping	O
class	O
group	O
of	O
a	O
disc	O
with	O
n	O
marked	O
points	O
,	O
.	O
</s>
<s>
The	O
Lawrence	B-Algorithm
–	I-Algorithm
Krammer	I-Algorithm
representation	I-Algorithm
is	O
defined	O
as	O
the	O
action	O
of	O
on	O
the	O
homology	O
of	O
a	O
certain	O
covering	O
space	O
of	O
the	O
configuration	O
space	O
.	O
</s>
<s>
is	O
the	O
Lawrence	B-Algorithm
–	I-Algorithm
Krammer	I-Algorithm
representation	I-Algorithm
.	O
</s>
<s>
Using	O
Bigelow	O
's	O
conventions	O
for	O
the	O
Lawrence	B-Algorithm
–	I-Algorithm
Krammer	I-Algorithm
representation	I-Algorithm
,	O
generators	O
for	O
the	O
group	O
are	O
denoted	O
for	O
.	O
</s>
<s>
Letting	O
denote	O
the	O
standard	O
Artin	O
generators	O
of	O
the	O
braid	B-Application
group	I-Application
,	O
we	O
obtain	O
the	O
expression	O
:	O
</s>
<s>
Stephen	O
Bigelow	O
and	O
Daan	O
Krammer	O
have	O
given	O
independent	O
proofs	O
that	O
the	O
Lawrence	B-Algorithm
–	I-Algorithm
Krammer	I-Algorithm
representation	I-Algorithm
is	O
faithful	O
.	O
</s>
<s>
The	O
Lawrence	B-Algorithm
–	I-Algorithm
Krammer	I-Algorithm
representation	I-Algorithm
preserves	O
a	O
non-degenerate	O
sesquilinear	B-Algorithm
form	I-Algorithm
which	O
is	O
known	O
to	O
be	O
negative-definite	O
Hermitian	O
provided	O
are	O
specialized	O
to	O
suitable	O
unit	O
complex	O
numbers	O
(	O
q	O
near	O
1	O
and	O
t	O
near	O
i	O
)	O
.	O
</s>
<s>
Thus	O
the	O
braid	B-Application
group	I-Application
is	O
a	O
subgroup	O
of	O
the	O
unitary	O
group	O
of	O
square	O
matrices	O
of	O
size	O
.	O
</s>
<s>
Recently	O
it	O
has	O
been	O
shown	O
that	O
the	O
image	O
of	O
the	O
Lawrence	B-Algorithm
–	I-Algorithm
Krammer	I-Algorithm
representation	I-Algorithm
is	O
a	O
dense	O
subgroup	O
of	O
the	O
unitary	O
group	O
in	O
this	O
case	O
.	O
</s>
<s>
The	O
sesquilinear	B-Algorithm
form	I-Algorithm
has	O
the	O
explicit	O
description	O
:	O
</s>
