A generalization to H2 (Part 2)

in which I layout some foundation for a process of translating from one half of a 2D polytope to the other.

For some simplification in the formulas, I will index the vertices ( $\mu_i$ ) of a representative of half of a 2D polytope (in the affine or hyperbolic case) by the following set of surreal numbers: $\mathbb N/2\cup \{\omega+\mathbb Z\}\cup\{2\omega-\mathbb N/2\}$ with the following conventions:

$\mu_0=0$ ,
for $i\in\mathbb N$ ,

$\mu_{i+\frac 1 2}-\mu_i = a_iR^i\alpha_1$ where $\alpha_1$ is a simple root and $R=r_2r_1$ as in the previous post,
$\mu_{i+1}-\mu_{i+\frac 1 2}=a_{i+\frac 1 2}R^ir_2\alpha_2$ ,
$\mu_{2\omega-i}-\mu_{2\omega-i-\frac 1 2}=a_{2\omega-i}(r_1r_2)^i\alpha_2$ , and
$\mu_{2\omega-i-\frac 1 2}-\mu_{2\omega-i-1}=a_{2\omega-i-\frac 1 2}(r_1r_2)^ir_1\alpha_1$ , and

for $i\in\mathbb Z$ , $\lambda_i=\mu_{\omega+i+1}-\mu_{\omega+i}$ is an imaginary root or zero.

Note that for a sequence $\bar\lambda$ comprised of imaginary roots and zeros the equivalence defined in the last post will always freely shift the zeroes around, so we may take a number of conventions regarding its support (which must be finite). I will leave it to the future to determine whether it is more convenient in different contexts to have, e.g,. $supp\bar\lambda=[0,l]$ or $[-l,0]$ or $2[0,l]$ , etc.

With this notation I define:

$G_k=\langle \alpha_1,\mu_{k+\frac 1 2}\rangle+\langle (C-I)\alpha_1,\mu_{k-\frac 1 2}\rangle$

where $C$ is the Cartan matrix and matrix multiplication and $\langle,\rangle$ are understood with the convention that $\alpha_i$ is the $i$ th basis element.

If $max(G_k)>0$ let $k$ be the smallest to achieve this maximum. Then we set $\bar a_0=max(G_k)$ , $\bar a_i=a_{i-\frac 1 2}$ for $0<i<k$ , $i\in \mathbb N\cup\{2\omega-\mathbb N\}$ and $\eta_i=r_1\lambda_i$ for $i\in\mathbb Z$ , $\omega+i<k$ then inductively consider the smaller polytope.

In a fashion similar to that described by Tingley for 2D affine polytopes we may handle the cases when $max(G_k)\le0$ but at this point I think I may need to delay particulars to a third installment.

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A generalization to H2 (Part 2)

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