A Super Technical Lemma
| Speaker: | Cameron Marcott |
|---|---|
| Affiliation: | University of Waterloo |
| Room: | Mathematics and Computer Building (MC) 5426 |
Abstract:
I
will
present
a
technical
lemma
about
extending
matrix
algebras
by
an
idempotent.
There
will
be
no
movie
references,
fun,
or
combinatorics.
Specifically,
I
will
be
presenting
Jones'
Basic
Construction.
Given
a
matrix
algebra
A
equipped
with
a
trace
function
and
a
subalgebra
B,
there
is
a
unique
idempotent
which
projects
A
onto
B
in
a
way
that
plays
nicely
with
the
trace
function.
Jones'
Basic
Construction
is
the
algebra
obtained
by
adjoining
this
idempotent
to
A.
Its
structure
is
completely
determined
by
B
and
knowledge
about
the
restriction
from
A
to
B.
Jones'
Basic
Construction
has
applications
anywhere
you
might
expect
the
words
"subfactors
of
type
$II_1$
von
Neumann
algebas"
to
pop
up,
including:
knot
theory
(where
it
is
related
to
the
Jones'
polynomial),
statistical
mechanics
(where
it
is
related
to
certain
transfer
matrix
algebras),
and
representation
theory
(where
it
is
used
to
study
certain
towers
of
algebras).
I
will
not
be
discussing
any
of
these
applications;
I
will
only
be
giving
technical
results
about
the
basic
construction
itself.