In Preparation
The information below is preliminary and cannot be trusted! (v)
A HOMFLY Braidor
The Algebra
Let
be the vector-space tensor product of the group ring
of the permutation group
(with coefficients in
, polynomials in the variable
) with the free associative algebra
on (non-commuting) generators
(that is,
is the ring of non-commutative polynomials in the variables
). We put an algebra structure on
as follows:
Let
be the free associative (but non-commutative) algebra generated by the elements of the symmetric group
on
and by formal variables
and
, and let
be the quotient of
by the following relations:
commutes with everything else.
- The product of permutations is as in the symmetric group
.
- If
is a permutation then
.
, where
is the transposition of
and
.
Finally, declare that
while
for every
and every
, and let
be the graded completion of
.
The Equations
The Equations in Functional Terms
A Solution