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{\LARGE\bf Cheat Sheet Unitarity}
\hfill$b_i=\log(T_i)$\hfill
\parbox[b]{2in}{\tiny
  \null\hfill\url{http://drorbn.net/AcademicPensieve/2015-09/}
  \newline\null\hfill
  split from \href{http://drorbn.net/AcademicPensieve/2015-09/OneCo.pdf}{Cheat Sheet OneCo};
  modified \today
}

\vskip -3mm
\rule{\textwidth}{1pt}
\vspace{-5mm}

\textbf{$\sigma$ calculus.}
\hfill $\sigma_1\ast\sigma_2=\sigma_1\cup\sigma_2$,
\hfill $tm^{uv}_w=({T_{u,v}\to T_w})$,
\hfill $hm^{xy}_z(\sigma)=(\sigma\remove\{x,y\})\cup(z\to\sigma_x\sigma_y)$,
\hfill $tha^{ux}=I$,
\hfill $R^\pm_{ux}\mapsto T_u^{\pm 1}$

\rule{\textwidth}{1pt}

\textbf{Gassner calculus $\Gamma$.}
\hfill Preserves $C_1\coloneqq[\text{col sum}=1]$ ($\Leftrightarrow$OC) and $\green\checkmark$ $C_2\coloneqq[\forall a,b,\, (T_a-1)\mid(A_{ab}-\delta_{ab}\sigma_b)]$
\newline\null\hfill$\bullet$ At $T_\ast=1$, $\omega=1$ and $A=I$.

\vskip -3mm
$\displaystyle
  \begin{CD}
    \begin{array}{c|ccc}
      \omega & a & b & S \\
      \hline
      a & \alpha & \beta & \theta \\
      b & \gamma & \delta & \epsilon \\
      S & \phi & \psi & \Xi
    \end{array}
    @>{m^{ab}_{c}}>{\mu\coloneqq 1-\beta \atop T_a,T_b\to T_c}>
    \begin{array}{c|cc}
      \mu\omega & c & S \\
      \hline
      c & \gamma+\alpha\delta/\mu & \epsilon+\delta\theta/\mu \\
      S & \phi+\alpha\psi/\mu & \Xi+\psi\theta/\mu
    \end{array}
  \end{CD}
$
\hfill$\Theta_{ij} \underset{\Gamma}{=} \begin{array}{c|cc}
  1 & i & j \\
  \hline
  i & \frac{b_j e^{(b_i+b_j)/2}+b_i}{b_i+b_j} & \frac{b_i \left(1-e^{(b_i+b_j)/2}\right)}{b_i+b_j} \\
  j & \frac{b_j \left(1-e^{(b_i+b_j)/2}\right)}{b_i+b_j} & \frac{b_i e^{(b_i+b_j)/2}+b_j}{b_i+b_j}
\end{array}$
\hfill$R^{\pm}_{ab} \underset{\Gamma}{=} \begin{array}{c|cc}
  1 & a & b \\
  \hline
  a & 1 & 1-T_a^{\pm 1} \\
  b & 0 & T_a^{\pm 1}
\end{array}$

The map (tangle $T$ $\mapsto$ matrix $A$) is anti-multiplicative.

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{\bf Unitarity} (rough). With
$\Omega(\tau)\coloneqq
  \begin{pmatrix}
    (1-t_{\tau 1})^{-1} & 0 & \cdots & 0 \\
    1 & (1-t_{\tau 2})^{-1} & \cdots & 0 \\
    \vdots & \vdots & \ddots & \vdots \\
    1 & 1 & \ldots & (1-t_{\tau n})^{-1}
  \end{pmatrix}
$,
have
$\Omega(\tau)\gamma^{-1}=\bar{\gamma}^T\Omega(\iota)$,
or
$\gamma^{-1}=\Omega(\tau)^{-1}\bar{\gamma}^T\Omega(\iota)$.

\rule{\textwidth}{1pt}\par\vspace{0mm}

KV in $\Gamma$: \hfill (Represent!)

\vskip 2mm

\includegraphics[width=\textwidth]{../Projects/MetaCalculi/KVinGamma.png}

\vspace{-2mm}\par\rule{\textwidth}{1pt}\par\vspace{-5mm}

\begin{multicols}{2}\raggedcolumns

{\bf Unitarity Monoblog.}

\entry{150922} Is the unitarity pairing hidden in Kawazumi-Kuno?

\entry{150921b} Is there unitarity in Cimasoni-Turaev?

\entry{150921a} I can develop a ``unitary Burau (Gassner) calculus'', a functor on $\PaT$ (or on parenthesized bottom tangles). May follow Cimasoni-Turaev notation.

%\columnbreak

\entry{150920} \$75 KaL question: Let $R_n=\bbQ[b_1,\ldots,b_n]$. Consider the adjoint action of the primitives $\calA^w_{\text{prim}}(\uparrow^n)$ on $\FL(n)\coloneqq\FL(x_1,\ldots x_n)$ via the inclusion $\FL(n)\hookrightarrow\calA^w(\uparrow^n\uparrow_\infty)$ given by $x_i\mapsto a_{i\infty}$. Reducing mod $\beta\colon[a_{ik},a_{jk}] = b_ja_{ik}-b_ia_{jk}$ makes $R_n^n$ an $\calA^w_{\text{prim}}(\uparrow^n)$-module. Restricting to $\calA^u_{\text{prim}}(\uparrow^n)$ and regarding the $b_i$ as imaginary, this module is formally Hermitian relative to $\langle x_i,x_j\rangle=b_i\delta_{ij}$. E.g.\ on $t^{jk}\in\frakt_n$ is the matrix $A=\begin{pmatrix} b_k & -b_k \\ -b_j & b_j \end{pmatrix}$ relative to the basis $\{x_j,x_k\}$, and with $D = \begin{pmatrix} b_j & 0 \\ 0 & b_k \end{pmatrix}$, we have $\bar{A}^TD+DA=0$. Whence cometh? Verification: \href{http://drorbn.net/AcademicPensieve/2015-09/nb/RhoHermitian.pdf}{pensieve://2015-09/nb/RhoHermitian.pdf}.

\entry{150916} The composition $S\times\FL(S)\to\sder_S\to\FL(S)^S$ does not descend mod $\beta$.

\entry{140609} Determine the image of $\calP^u\to\calP^w\to\calP^w/\beta\subset M_{S\times S}(R_S)$.

\entry{150915} A unitarity / hermitian property in $\calA^u_{\exp}$ / $\calA^u_{\operatorname{prim}}$?

\entry{150906b} A $\Gamma$ meaning for the $u$ full- and half-twist belt trick?

\entry{150906a} In $u$, (lazy mirror)$=$(radical mirror). $\Gamma$ meaning?

\columnbreak

{\footnotesize In \url{http://drorbn.net/AP/2012-05/beta5.1/}:}

\includegraphics[width=\columnwidth]{../2012-05/beta5.1/SimplifyingTheExact.pdf}

\end{multicols}

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