Homework Assignment 6

Question 1. Let $\Theta:{\mathcal A}\to{\mathcal A}$ be the multiplication operator by the 1-chord diagram $\theta$, and let $\partial_\theta=\frac{d}{d\theta}$ be the adjoint of multiplication by $W_\theta$ on ${\mathcal A}^\star$, where $W_\theta$ is the obvious dual of $\theta$ in ${\mathcal A}^\star$. Let $P:{\mathcal A}\to{\mathcal A}$ be defined by

1. $\left[\partial_\theta,\Theta\right]=1$, where $1:{\mathcal A}\to{\mathcal A}$ is the identity map and where $[A,B]:=AB-BA$ for any two operators.
2. $P$ is a degree $0$ operator; that is, $\deg Pa=\deg a$ for all $a\in{\mathcal A}$.
3. $\partial_\theta$ satisfies Leibnitz' law: $\partial_\theta(ab)=(\partial_\theta a)b+a(\partial_\theta b)$ for any $a,b\in{\mathcal A}$.
4. $P$ is an algebra morphism: $P1=1$ and $P(ab)=(Pa)(Pb)$.
5. $\Theta$ satisfies the co-Leibnitz law: $\Box\circ\Theta=(\Theta\otimes 1+1\otimes\Theta)\circ\Box$ (why does this deserve the name "the co-Leibnitz law"?).
6. $P$ is a co-algebra morphism: $\eta\circ P=\eta$ (where $\eta$ is the co-unit of ${\mathcal A}$) and $\Box\circ P=(P\otimes P)\circ\Box$.
7. $P\theta=0$ and hence $P\langle\theta\rangle=0$, where $\langle\theta\rangle$ is the ideal generated by $\theta$ in the algebra ${\mathcal A}$.
8. If $Q:{\mathcal A}\to{\mathcal A}$ is defined by $Q = \sum_{n=0}^\infty \frac{(-\Theta)^n}{(n+1)!}\partial_\theta^{(n+1)}$ then $a=\theta Qa+Pa$ for all $a\in{\mathcal A}$.
9. $\ker P=\langle\theta\rangle$.
10. $P$ descends to a Hopf algebra morphism ${\mathcal A}^r\to{\mathcal A}$, and if $\pi:{\mathcal A}\to{\mathcal A}^r$ is the obvious projection, then $\pi\circ P$ is the identity of ${\mathcal A}^r$. (Recall that ${\mathcal A}^r={\mathcal A}/\langle\theta\rangle$).
11. $P^2=P$.

Question 2. Note that the definition of $W_{\mathfrak g}$ extends to diagrams that do not have a skeleton (the circle around the diagram). In fact, in that case, no representation of ${\mathfrak g}$ is needed. Let $D$ be a connected "vacuum diagram" - a connected diagram that has no skeleton. Let $v$ be the number of vertices in $D$. Prove that $W_{gl_N}(D)$ is a polynomial in $N$ of degree at most $\frac{v}{2}+2$, and that if $D$ is 2-connected (it remains connected even if any one edge is removed from it) the coefficiant of $N^{v/2+2}$ in $W_{gl_N}(D)$ is non-zero iff $D$ can be embedded in the plane without self intersections.

(Note that "2-connectedness" is precisely the condition necessary to make the complement of a diagram a map with no self-bordering countries, so the application to the 4CT remains standing).

This assignment is due on Crowdmark by the end of Friday October 28, 2020.