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\usepackage{amsmath,graphicx,amssymb,xcolor,stmaryrd,datetime,txfonts,multicol}
% Following http://tex.stackexchange.com/a/847/22475:
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}

\def\red{\color{red}}

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

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\def\bbQ{{\mathbb Q}}
\def\CW{\text{\it CW}}
\def\FL{\text{\it FL}}
\def\tbd{\text{\color{red} ?}}
\def\tder{\operatorname{\mathfrak{tder}}}
\def\TAut{\operatorname{TAut}}

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

\begin{document}
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{\LARGE\bf Cheat Sheet Free Lie}\hfill
\parbox[b]{5in}{\tiny
  \null\hfill\url{http://drorbn.net/AcademicPensieve/Projects/WKO4/}
  \newline\null\hfill initiated 18/3/13;
  continues Cheat Sheet $J$ @ \href{http://drorbn.net/AcademicPensieve/2014-01/}{2014-01};
  modified \today, \ampmtime
}

\vskip -3mm
\rule{\textwidth}{1pt}
\vspace{-8mm}

\begin{multicols}{2}

With alphabet $T$ and with $u,v,w\in T$, $\alpha,\beta,\gamma\in\FL(T)$, $\lambda\in\FL(T)^T$, $D\in\tder(T)$, $g,h\in\exp(\tder(T))=\TAut(T)$.

Checkmarks ($\checkmark$) as in {\tt CheatSheetFreeLie-Verification.nb}.

{\bf Definition.} $\ad_u^\gamma=\ad_u\{\gamma\}:=\der(u\mapsto[\gamma, u])$ and $\partial_\lambda = \partial\{\lambda\} \coloneqq -\sum_{s\in S}\ad_s^{\lambda_s}$.

{\bf Definition.} $C^\lambda \coloneqq \mor\left( u \mapsto (\Ad\lambda_u)(u) = e^{\lambda_u}ue^{-\lambda_u} = e^{\ad\lambda_u}u \right)$ and $C_u^\gamma\coloneqq C^{(u\to\gamma)}$.

\noindent\rule{\columnwidth}{1pt}

\cheatline \checkmark\ The meaning(s) of $RC$:\quad\hfill
$\displaystyle C_u^\gamma\act RC_u^{-\gamma}=1,$
\hfill
$\displaystyle C_u^{\gamma\act RC_u^{\gamma}}=RC_u^\gamma$

\cheatline \checkmark\ $C_uC_v$ and $RC_uRC_v$:\quad\hfill
$\displaystyle C_u^{\alpha\act RC_v^{-\beta}}\act C_v^\beta =   C_v^{\beta\act RC_u^{-\alpha}}\act C_u^\alpha,$
\newline\null\hfill
$\displaystyle RC_u^\alpha\act RC_v^{\beta\act RC_u^\alpha} = RC_v^\beta\act RC_u^{\alpha\act RC_v^\beta}$

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

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

\cheatline \checkmark\ $\Gamma$: With $\Gamma(t)\in\FL(T)^T$ solving $\Gamma(0)=0$, $\Gamma'(s)=\lambda \act e^{-\partial_{s\lambda}} \act \frac{\ad\Gamma(s)}{e^{\ad\Gamma(s)}-1}$,
\quad\hfill$\displaystyle e^{-\partial_\lambda} = C^{\Gamma(1)}$

\cheatline \checkmark\ $\Lambda$: With $\Lambda(t)\in\FL(T)^T$ solving $\Lambda(0)=0$, $\Lambda'(s)=\lambda \act e^{\partial_{\Lambda(s)}} \act \frac{\ad_{tb}\Lambda(s)}{e^{\ad_{tb}\Lambda(s)}-1}$,
\quad\hfill$\displaystyle e^{-\partial_{\Lambda(1)}} = C^{\lambda}$

\noindent\rule{\columnwidth}{1pt}

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

\cheatline \checkmark\ $\diver$ property $uv$:
\newline\null\hfill
$\displaystyle (\diver_u\alpha)\act\ad_v^\beta - (\diver_v\beta)\act\ad_u^\alpha
  = \diver_u(\alpha\act\ad_v^\beta) - \diver_v(\beta\act\ad_u^\alpha)
$

\cheatline \checkmark\ $\diver$ property $uu$:
\hfill $\displaystyle (\diver_u\alpha)\act\ad_u\{\beta\} - (\diver_u\beta)\act\ad_u\{\alpha\}$
\newline\null\hfill$  = \diver_u\left(
    [\alpha,\beta] + \alpha\act\ad_u\{\beta\} - \beta\act\ad_u\{\alpha\}
  \right)
$

\cheatline $C$--$\diver$--$RC$ eqns:\ \hfill
$\displaystyle \diver_u(\alpha\act RC_u^\gamma)\act C_u^\gamma
  = \tbd
$;
\hfill
$\displaystyle \diver_u(\alpha\act C_u^\gamma)\act RC_u^\gamma
  = \tbd
$

\noindent\rule{\columnwidth}{1pt}

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

\cheatline \checkmark\ $J_{uv}$ eqn:\ \hfill
$\displaystyle
  J_u(\alpha) + J_v(\beta\act RC_u^\alpha)\act C_u^{-\alpha}
  = J_v(\beta) + J_u(\alpha\act RC_v^\beta)\act C_v^{-\beta}
$

\cheatline \checkmark\ $t$ eqn:\ \hfill
$\displaystyle J_w(\gamma\act tm^{uv}_w)
  = \left(
    J_u(\gamma) + J_v(\gamma\act RC_u^\gamma)\act C_u^{-\gamma}
  \right) \act tm^{uv}_w
$

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

\noindent\rule{\columnwidth}{1pt}

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

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

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

\noindent\rule{\columnwidth}{1pt}

%\cheatline The relation with $\tder$, 2:  \quad\hfill
%$\displaystyle C_u^\gamma = e^{\ad_u\{\tbd\}}$

\cheatline $j$ following AT:\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{\columnwidth}{1pt}

\cheatline $d\exp$:\ \hfill
$\displaystyle \delta e^\gamma
  = e^\gamma\cdot\left(\delta\gamma\act\frac{1-e^{-\ad\gamma}}{\ad\gamma}\right)
  = \left(\delta\gamma\act\frac{e^{\ad\gamma}-1}{\ad\gamma}\right)\cdot e^\gamma
$

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

\cheatline \checkmark\ $dC$:\quad\hfill
$\displaystyle \delta C_u^\gamma
  = \ad_u\left\{
    \delta\gamma\act\frac{e^{\ad \gamma}-1}{\ad\gamma}\act RC_u^{-\gamma}
  \right\} \act C_u^\gamma
$

\cheatline \checkmark\ $dC^\lambda$:\quad\hfill
$\displaystyle \delta C^\lambda
  = -\partial\left\{
    \delta\lambda \act \frac{e^{\ad\lambda}-1}{\ad\lambda} \act RC^{-\lambda}
  \right\} \act C^\lambda
$

\cheatline \checkmark\ $dRC$:\quad\hfill
$\displaystyle \delta RC_u^\gamma
  = RC_u^\gamma \act\ad_u\left\{
    \delta\gamma\act\frac{1-e^{-\ad\gamma}}{\ad\gamma}\act RC_u^\gamma
  \right\}
$

\noindent\rule{\columnwidth}{1pt}

\cheatline \checkmark\ $dJ$:\quad\hfill
$\displaystyle \delta J_u(\gamma)
  = \delta\gamma \act \frac{1-e^{-\ad\gamma}}{\ad\gamma}
    \act RC_u^{\gamma} \act \diver_u \act C_u^{-\gamma}
$

\end{multicols}

\rule{\textwidth}{1pt}

\begin{multicols}{2}
{\bf\large The $\beta$ quotient $[u,v]=c_uv-c_vu$.} Let $R=R(T):=\bbQ\llbracket\{c_u\}_{u\in
T}\rrbracket$, $L=L(T):=R\otimes\bbQ T$. For $\gamma=\sum_u\gamma_uu\in L$ set $c_\gamma:=\sum_u\gamma_uc_u\in R$. With this,
\begin{eqnarray*}
  v \act C_u^{-\gamma} & = & v \act RC_u^\gamma\ =\ v
    \qquad\text{for $u\neq v\in T$}, \label{eq:betaonv} \\
  \rho \act C_u^{-\gamma} & = & \rho \act RC_u^\gamma\ =\ \rho
    \qquad\text{for $\rho\in R$}, \label{eq:betaonrho}
\end{eqnarray*}
\begin{eqnarray*}
  u\act C_u^{-\gamma} & = &
    e^{-c_\gamma}\left(
      u+c_u\frac{e^{c_\gamma}-1}{c_\gamma}\gamma
    \right) \label{eq:betaC0} \\
    &=& e^{-c_\gamma}\left(
      \left(1+c_u\gamma_u\frac{e^{c_\gamma}-1}{c_\gamma}\right)u
      +c_u\frac{e^{c_\gamma}-1}{c_\gamma}\sum_{v\neq u}\gamma_v v
    \right), \label{eq:betaC}
\end{eqnarray*}
\[
  u\act RC_u^\gamma =
    \left(1+c_u\gamma_u\frac{e^{c_\gamma}-1}{c_\gamma}\right)^{-1}
    \left(
      e^{c_\gamma}u
      -c_u\frac{e^{c_\gamma}-1}{c_\gamma}\sum_{v\neq u}\gamma_v v
    \right),
\]
\[
  \bch(\alpha,\beta) =
    \frac{c_\alpha+c_\beta}{e^{c_\alpha+c_\beta}-1}\left(
      \frac{e^{c_\alpha}-1}{c_\alpha}\alpha
      + e^{c_\alpha}\frac{e^{c_\beta}-1}{c_\beta}\beta
    \right)
\]
\[ \diver_u\gamma = c_u\gamma_u \]
\[ J_u(\gamma) = \log\left(1+\frac{e^{c_\gamma}-1}{c_\gamma}c_u\gamma_u\right). \]

\noindent{\red\rule{\columnwidth}{1pt}}

Further include: $j$, $C^\lambda$, $RC^\lambda$, $\Gamma$, $\Lambda$, $\partial$, $[\ ]_{tb}$.

Implement $\pi\colon\FL/\CW\to L/R$ (under {\tt LSeries}/{\tt RSeries}?), commutativity verifications.

Verifications of 1--24 in $\beta$.
\end{multicols}

\end{document}

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