\documentclass[12pt,notitlepage]{article}
\usepackage{amsmath,graphicx,amssymb,color,stmaryrd,datetime}
\usepackage[setpagesize=false]{hyperref}

\def\sheeturl{{\url{http://drorbn.net/AcademicPensieve/2013-03/}}}

\newcommand{\ad}{\operatorname{ad}}
\newcommand{\bch}{\operatorname{bch}}
\newcommand{\der}{\operatorname{der}}
\newcommand{\diver}{\operatorname{div}}

\def\FL{\text{\it FL}}
\def\tbd{\text{\color{red} ?}}
\def\tder{\operatorname{\frak{tder}}}
\def\TAut{\operatorname{TAut}}

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\newcounter{linecounter}
\newcommand{\cheatline}{\vskip 1mm\noindent\refstepcounter{linecounter}\thelinecounter. }

\begin{document}
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{\LARGE\bf Cheat Sheet $J$}\hfill
\parbox[b]{4.5in}{\small
  \null\hfill\sheeturl
  \newline\null\hfill initiated 18/3/13; modified \today, \ampmtime; continued \href{http://drorbn.net/AcademicPensieve/2013-04/}{2013-04}
}

\vskip 1mm

With alphabet $T$ and with $u,v,w\in T$, $\alpha,\beta,\gamma\in\FL(T)$, $D\in\tder(T)$, $g,h\in\exp(\tder(T))=\TAut(T)$. Checkmarks ($\checkmark$) as in {\tt CheatSheetJ-Verification.nb}.

\noindent\rule{\textwidth}{1pt}

\cheatline The definition of $J$:\quad\hfill
$\displaystyle J_u(\gamma) := \int_0^1ds\,\diver_u\!\left(
    \gamma \sslash RC_u^{s\gamma}
  \right) \sslash C_u^{-s\gamma}
$

\cheatline \checkmark\ The $t$ equation (desired):
\newline\null\hfill
$\displaystyle J_w(\gamma\sslash tm^{uv}_w)\sslash RC_w^{\gamma\sslash tm^{uv}_w}
  = J_u(\gamma)\sslash tm^{uv}_w \sslash RC_w^{\gamma\sslash tm^{uv}_w}
    + J_v(\gamma\sslash RC_u^{\gamma})
      \sslash RC_v^{\gamma\sslash RC_u^{\gamma}}\sslash tm^{uv}_w
$

\cheatline \checkmark\ The $h$ equation (desired):\quad\hfill
$\displaystyle J_u(\bch(\alpha,\beta))
  = J_u(\alpha)
  + J_u(\beta\sslash RC_u^{\alpha}) \sslash C_u^{-\alpha}
$

\noindent\rule{\textwidth}{1pt}

\cheatline \checkmark\ The meaning(s) of $RC$:\quad\hfill
$\displaystyle CC_u^\gamma\sslash RC_u^{-\gamma}=Id,$
\hfill
$\displaystyle CC_u^{\gamma\sslash RC_u^{\gamma}}=RC_u^\gamma$

\cheatline RC equation $t$:\quad\hfill
$\displaystyle tm^{uv}_w \sslash RC_w^{\gamma\sslash tm^{uv}_w}
  = RC_u^{\gamma}\sslash RC_v^{\gamma\sslash RC_u^{\gamma}}\sslash tm^{uv}_w
$

\cheatline RC equation $h$:\quad\hfill
$\displaystyle RC^{\bch(\alpha,\beta)}_u
  = RC^{\alpha}_u \sslash RC^{\beta\sslash RC^{\alpha}_u}_u
$

\cheatline RCC equation $\diver$:\quad\hfill
$\displaystyle \diver_u(\alpha\sslash RC_u^\gamma)\sslash C_u^\gamma
  = \tbd
$

\cheatline CRC equation $\diver$:\quad\hfill
$\displaystyle \diver_u(\alpha\sslash C_u^\gamma)\sslash RC_u^\gamma
  = \tbd
$

\noindent\rule{\textwidth}{1pt}

\cheatline $\diver$ property $t$:\quad\hfill
$\displaystyle \diver_w(\gamma\sslash tm^{uv}_w)
  = \left(\diver_u(\gamma)+\diver_v(\gamma)\right)\sslash tm^{uv}_w
$

\cheatline \checkmark\ $\diver$ property $h$ --- the ``cocycle condition'': with $\ad_u\{\gamma\}:=\der(u\to[\gamma, u])$,
\newline\null\hfill
$\displaystyle (\diver_u\alpha)\sslash\ad_u\{\beta\} - (\diver_u\beta)\sslash\ad_u\{\alpha\}
  = \diver_u\left(
    [\alpha,\beta] + \alpha\sslash\ad_u\{\beta\} - \beta\sslash\ad_u\{\alpha\}
  \right)
$

\cheatline $\diver$ of $\bch$:\quad\hfill
$\displaystyle \diver_u(\bch(\alpha,\beta))
  = \tbd
$

\noindent\rule{\textwidth}{1pt}

\def\JA{{\text{\it JA}}}

\cheatline The definition of $\JA$:\quad\hfill
$\displaystyle \JA_u(\gamma) := J_u(\gamma)\sslash RC_u^\gamma$

\cheatline The ODE for $JA$: with $\gamma_s=\gamma\sslash RC^{s\gamma}_u$,
\newline\null\hfill
$\displaystyle \JA(0)=0,
  \quad\frac{d\JA(s)}{ds} = \JA(s)\sslash\ad_u\{\gamma_s\} + \diver_u\gamma_s,
  \quad \JA(1)=\JA_u(\gamma)
$

\noindent\rule{\textwidth}{1pt}

\cheatline The relation with $\tder$:\quad\hfill
$\displaystyle e^{\ad_u\{\gamma\}}
  = C_u^{\tbd}
$
and
$\displaystyle C_u^\gamma
  = e^{\ad_u\{\tbd\}}
$

\cheatline The definition of $j$ (following A-T):\quad\hfill
$\displaystyle j(e^D)
  = \int_0^1ds\,e^{sD}(\diver D)
  = \frac{e^D-1}{D}(\diver D)
$

\cheatline $j$'s cocycle property:\quad\hfill
$\displaystyle j(gh)=j(g)+g\cdot j(h)$

\noindent\rule{\textwidth}{1pt}

\cheatline The differential of $\exp$:\quad\hfill
$\displaystyle \delta e^\gamma
  = e^\gamma\cdot\left(
    \frac{1-e^{-\ad\gamma}}{\ad\gamma}
  \right)(\delta\gamma)
  = \left(
    \frac{e^{\ad\gamma}-1}{\ad\gamma}
  \right)(\delta\gamma)\cdot e^\gamma
$

\cheatline \checkmark\ The differential of $\gamma=\bch(\alpha,\beta)$:
\newline\null\hfill
$\displaystyle \left(\frac{1-e^{-\ad\gamma}}{\ad\gamma}\right)(\delta\gamma)
  = \left(e^{-\ad\beta}\frac{1-e^{-\ad\alpha}}{\ad\alpha}\right)(\delta\alpha)
  + \left(\frac{1-e^{-\ad\beta}}{\ad\beta}\right)(\delta\beta)
$

\cheatline \checkmark\ The differential of $C$:\quad\hfill
$\displaystyle \delta C_u^\gamma
  = \ad_u\left\{
    \left(\frac{e^{\ad \gamma}-1}{\ad\gamma}\right)(\delta\gamma)
    \sslash RC_u^{-\gamma}
  \right\}
    \sslash C_u^\gamma
$

\cheatline \checkmark\ The differential of $RC$:\quad\hfill
$\displaystyle \delta RC_u^\gamma
  = RC_u^\gamma \sslash\ad_u\left\{
    \left(\frac{1-e^{-\ad\gamma}}{\ad\gamma}\right)(\delta\gamma) \sslash RC_u^\gamma
  \right\}
$

\noindent\rule{\textwidth}{1pt}

\cheatline \checkmark\ The differential of $J$:\quad\hfill
$\displaystyle \delta J_u(\gamma)
  = \delta\gamma \sslash \left(\frac{1-e^{-\ad\gamma}}{\ad\gamma}\right)
    \sslash RC_u^{\gamma} \sslash \diver_u \sslash C_u^{-\gamma}
$

\newpage\noindent{\bf Recycling.}

\noindent\checkmark\ The differential of $J$:\quad\hfill
$\displaystyle \delta J_u(\gamma)
  = \int_0^1 ds\, \diver_u\left(\delta\gamma\sslash e^{-\ad s\gamma}\sslash
    RC_u^{s\gamma}\right) \sslash C_u^{-s\gamma}
$
\newline\null\hfill$\displaystyle
  + \int_0^1 ds\, \diver_u\left(
    \left(\frac{1-e^{-\ad s\gamma}}{\ad\gamma}\right)(\delta\gamma) \sslash RC_u^{s\gamma}
      \sslash \ad_u\{\gamma\sslash RC_u^{s\gamma}\}
    \right) \sslash C_u^{-s\gamma}
$
\newline\null\hfill$\displaystyle
  - \int_0^1 ds\, \diver_u\left(
    \left(\frac{1-e^{-\ad s\gamma}}{\ad\gamma}\right)(\delta\gamma) \sslash RC_u^{s\gamma}
    \right) \sslash \ad_u\{\gamma\sslash RC_u^{s\gamma}\} \sslash C_u^{-s\gamma}
$

\end{document}

\endinput