The purple box is one polytope colored two different ways. In general a polytope is colored according to the following scheme:
- Light blue dots are from the previous polytope.
- The red dot is the “moving” dot.
- Dark Blue dots come from a reflection (connected by a blue arc to its preimage).
- Green dots are added so the shape will be “Pseudo-Weyl”.
Where “Pseudo-Weyl”, in this case, means that non-vertical edges have integer slope and any point in the closed polytope minus its vertical lines must be included.
The process for constructing a new polytope goes in that order:
- Include all previous points
- Move the “moving point” in the desired direction
- Reflect any corners on the appropriate side of the reflection line and include those, and
- Include any additional points necessary to make it Pseudo-Weyl
Where the reflection line is placed so that the first corner before the red dot (in the direction it moved) would be reflected onto the red dot. Only corners on the side opposite the red dot get reflected.
Note: I use “reflect” loosley. In this case all reflection lines will be vertical, but when moving diagonally the reflection will be shear-reflection.
and 
. In a sense its as though the weight in in the middle of that line is sometimes included in the polytope and sometimes not. (Though, how this plays out for more complicated partitions I don’t yet know.)
with highest weight 
we define new polytopes 
with highest weight 
(where 
is a fundamental coroot.) 
. If you test, you will see that only two of them will fit into the parabola for the basic representation.
acts by -2 , (c by 1 and d by zero). I calculated that:
, none of these are zero.
, so the outline looks like \_/, a truncated cone, not a parabola.
![=\mathbb{C}[t,t^{-1}]](http://l.wordpress.com/latex.php?latex=%3D%5Cmathbb%7BC%7D%5Bt%2Ct%5E%7B-1%7D%5D&bg=ffffff&fg=000000&s=0 "=\mathbb{C}[t,t^{-1}]")
![,[d,t^k\otimes x]=kt^k\otimes x](http://l.wordpress.com/latex.php?latex=+%2C%5Bd%2Ct%5Ek%5Cotimes+x%5D%3Dkt%5Ek%5Cotimes+x&bg=ffffff&fg=000000&s=0 ",[d,t^k\otimes x]=kt^k\otimes x")
![,[t^{-k_1}\otimes x,t^{k_2}\otimes y]=t^{k_2-k_1}[x,y]+k_1\delta^{k_1}_{k_2}(x,y)c](http://l.wordpress.com/latex.php?latex=+%2C%5Bt%5E%7B-k_1%7D%5Cotimes+x%2Ct%5E%7Bk_2%7D%5Cotimes+y%5D%3Dt%5E%7Bk_2-k_1%7D%5Bx%2Cy%5D%2Bk_1%5Cdelta%5E%7Bk_1%7D_%7Bk_2%7D%28x%2Cy%29c&bg=ffffff&fg=000000&s=0 ",[t^{-k_1}\otimes x,t^{k_2}\otimes y]=t^{k_2-k_1}[x,y]+k_1\delta^{k_1}_{k_2}(x,y)c")
is the killing form:
, 
and 
.
(similarly for 
). (Don’t confuse it with similar notation used for higher rank Kac-Moody Algebras.)
where 
(the large green dot) is generated by 
, to define a Verma module 
, where the product is taken over 
(
acts trivially on 
and 
).
where
.
(by left multiplication) of lowest weight 
is itself a 
.
be the Verma module with a highest weight 
), with a similar decomposition 
. With a choice of basis for each 
and 
we define a pairing 
such that 
for basis elements 
. The rank of 
-weight space of the quotient 
:
) is 
and will be generated by the following maps.
) is 
and will be generated by
on 
breaks down like this:
.
.
by solving the equations:
and
is orthogonal to 
)
be a Lie group. We are interested in the infinite dimensional Lie group 
where composition is done pointwise. One way to understand a group is by understanding its representations. In this particular case our interest is quickly narrowed to smooth, projective, positive energy representations. It turns out that a better way object of study is the semidirect product 
where 
is a particular one dimensional central extension and 
acts on 
by rotating loops (that is precomposing 
with a rotation).
. My charge, by this Sunday, is to calculate the Lie bracket. Suffice we will consider the (dense?) subalgebra ![\mathbb{C}_\text{rot} \oplus \mathfrak{g}[t^{-1},t] \oplus \mathbb{C}_\text{cent}](http://l.wordpress.com/latex.php?latex=%5Cmathbb%7BC%7D_%5Ctext%7Brot%7D+%5Coplus+%5Cmathfrak%7Bg%7D%5Bt%5E%7B-1%7D%2Ct%5D+%5Coplus+%5Cmathbb%7BC%7D_%5Ctext%7Bcent%7D&bg=ffffff&fg=000000&s=0 "\mathbb{C}_\text{rot} \oplus \mathfrak{g}[t^{-1},t] \oplus \mathbb{C}_\text{cent}")
![\left[(z_1,0,0),(z_2,0,0)\right]=(0,0,0)](http://l.wordpress.com/latex.php?latex=%5Cleft%5B%28z_1%2C0%2C0%29%2C%28z_2%2C0%2C0%29%5Cright%5D%3D%280%2C0%2C0%29&bg=ffffff&fg=000000&s=0 "\left[(z_1,0,0),(z_2,0,0)\right]=(0,0,0)")
![\left[(z,t^k\alpha,w_1),(0,0,w_2)\right]=(0,0,0)](http://l.wordpress.com/latex.php?latex=%5Cleft%5B%28z%2Ct%5Ek%5Calpha%2Cw_1%29%2C%280%2C0%2Cw_2%29%5Cright%5D%3D%280%2C0%2C0%29&bg=ffffff&fg=000000&s=0 "\left[(z,t^k\alpha,w_1),(0,0,w_2)\right]=(0,0,0)")
![\left[(z,0,0),(0,t^l\beta,0)\right]](http://l.wordpress.com/latex.php?latex=%5Cleft%5B%28z%2C0%2C0%29%2C%280%2Ct%5El%5Cbeta%2C0%29%5Cright%5D&bg=ffffff&fg=000000&s=0 "\left[(z,0,0),(0,t^l\beta,0)\right]")
![\left[(0,t^k\alpha,0),(0,t^l\beta,0)\right]](http://l.wordpress.com/latex.php?latex=%5Cleft%5B%280%2Ct%5Ek%5Calpha%2C0%29%2C%280%2Ct%5El%5Cbeta%2C0%29%5Cright%5D&bg=ffffff&fg=000000&s=0 "\left[(0,t^k\alpha,0),(0,t^l\beta,0)\right]")
![\left[(z,0),(0,t^l\beta)\right]](http://l.wordpress.com/latex.php?latex=%5Cleft%5B%28z%2C0%29%2C%280%2Ct%5El%5Cbeta%29%5Cright%5D&bg=ffffff&fg=000000&s=0 "\left[(z,0),(0,t^l\beta)\right]")
such that
is closed,
is a manifold of dimension k, and
latex X_{n-2}$.
where 
.
is covered by open sets in 
such that each open set 
is of the form 
where L (called the “link”) is a stratified space depending only on the strata (or possibly on the component of the strata) and 
indicates the open cone ![( (0,1]\times L) / (1\times L)](http://l.wordpress.com/latex.php?latex=%28+%280%2C1%5D%5Ctimes+L%29+%2F+%281%5Ctimes+L%29&bg=ffffff&fg=000000&s=0 "( (0,1]\times L) / (1\times L)")
.
is called admissible. A chain, 
, that intersects 
will be called admissible if it can be written as the product 
where 
is an admissible chain (defined inductively) for the space 
, the next strata consists of three copies of 
at their intersections, and the final strata consists of two points, the origin and point of compactification.
the link is two circles, one for each hyperplane. Once again, 1-chains cannot cross the singular stratum. Also no 2-chain can touch unless it wraps around the singular part.
on 
comes from an action of LG given by
=
.
on 
.
with the bracket: ![\left[(\xi,\lambda),(\eta,\mu)\right]](http://l.wordpress.com/latex.php?latex=%5Cleft%5B%28%5Cxi%2C%5Clambda%29%2C%28%5Ceta%2C%5Cmu%29%5Cright%5D&bg=ffffff&fg=000000&s=0 "\left[(\xi,\lambda),(\eta,\mu)\right]")
![=([\xi,\eta ],\omega (\xi,\eta ))](http://l.wordpress.com/latex.php?latex=%3D%28%5B%5Cxi%2C%5Ceta+%5D%2C%5Comega+%28%5Cxi%2C%5Ceta+%29%29&bg=ffffff&fg=000000&s=0 "=([\xi,\eta ],\omega (\xi,\eta ))")
(notice that 
is central in the sense of not contributing to the bracket.)
is given by (WLOG in some sense):
on 
.)
given by pulling back the evaluation map 
. (
is based loops.) We define 
to be a left invariant 3-form on G whose value at the identity is given by ![\sigma_{id}(\xi,\eta,\zeta)=\langle[\xi,\eta],\zeta\rangle](http://l.wordpress.com/latex.php?latex=%5Csigma_%7Bid%7D%28%5Cxi%2C%5Ceta%2C%5Czeta%29%3D%5Clangle%5B%5Cxi%2C%5Ceta%5D%2C%5Czeta%5Crangle&bg=ffffff&fg=000000&s=0 "\sigma_{id}(\xi,\eta,\zeta)=\langle[\xi,\eta],\zeta\rangle")
. When we pullback this form and evaluate it at 
on the vector 
we get, according to the book,![\frac{1}{4\pi}\langle\gamma(\theta)^{-1}\gamma'(\theta),[\gamma(\theta)^{-1}\delta_1\gamma(\theta),\gamma(\theta)^{-1}\delta_2\gamma(\theta)]\rangle\delta\theta](http://l.wordpress.com/latex.php?latex=%5Cfrac%7B1%7D%7B4%5Cpi%7D%5Clangle%5Cgamma%28%5Ctheta%29%5E%7B-1%7D%5Cgamma%27%28%5Ctheta%29%2C%5B%5Cgamma%28%5Ctheta%29%5E%7B-1%7D%5Cdelta_1%5Cgamma%28%5Ctheta%29%2C%5Cgamma%28%5Ctheta%29%5E%7B-1%7D%5Cdelta_2%5Cgamma%28%5Ctheta%29%5D%5Crangle%5Cdelta%5Ctheta&bg=ffffff&fg=000000&s=0 "\frac{1}{4\pi}\langle\gamma(\theta)^{-1}\gamma'(\theta),[\gamma(\theta)^{-1}\delta_1\gamma(\theta),\gamma(\theta)^{-1}\delta_2\gamma(\theta)]\rangle\delta\theta")
a bit. Let 
where w is an element of the Weyl group and 
is a fundamental weight. Before calculating 
such that 
where 
indicates the lift of w.
.
.
when 
.)
. The lifts are much the same as for the finite case only there will be t’s in places. What I don’t have nailed down yet is the 
vectors: I don’t even know where they live.
there’s only one set of two. We can still get two towers, but these will both be described by the same set which is self-dual.
we start getting more intermediate data. We still have the “level 1″ set in the data that describes an outer tower and a a level 
set in the data that describes an inner tower. But now we have additional data. I’d like to understand the additional restrictions this set (and further middle sets for higher n) will put on towers.
.
denote row vectors and 
denote column vectors of a representative in 
of an element in 
. The 
will be unchanged despite the elimination of this row. Its because of this that I say the middle data arising in 
