The Jones–Witten theory gives rise to representations of
the (extended) mapping class group of any closed surface
indexed by a
semi-simple Lie group
and a level . In the case
these representations
(denoted )
have a particularly simple description in terms of the Kauffman skein modules with parameter
a primitive
th root of
unity ().
In each of these representations (as well as the general
case), Dehn
twists act as transformations of finite order, so none represents the mapping class group
faithfully. However, taken
together, the quantum
representations are faithful on non-central elements of
. (Note that
has non-trivial
center only if is
a sphere with
or punctures, a
torus with or
punctures, or the
closed surface of genus .)
Specifically, for a non-central
there is an such
that if and
is a primitive
th root of unity then
acts projectively nontrivially on
. Jones’ original representation
of the braid groups
, sometimes called the generic
–analog––representation,
is not known to be faithful. However, we show that any braid
admits a
cabling
so that ,
.
Keywords
quantum invariants, Jones–Witten theory, mapping class
groups