\documentclass[12pt,notitlepage]{article} \usepackage{amsmath,graphicx,amssymb,xcolor,stmaryrd,datetime,txfonts} % Following http://tex.stackexchange.com/a/847/22475: \usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks, linkcolor={green!50!black}, citecolor={green!50!black}, urlcolor=blue } \def\sheeturl{{\url{http://drorbn.net/AcademicPensieve/2014-01/}}} \newcommand{\ad}{\operatorname{ad}} \newcommand{\bch}{\operatorname{bch}} \newcommand{\der}{\operatorname{der}} \newcommand{\diver}{\operatorname{div}} \def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}} \def\FL{\text{\it FL}} \def\tbd{\text{\color{red} ?}} \def\tder{\operatorname{\mathfrak{tder}}} \def\TAut{\operatorname{TAut}} \paperwidth 7.5in \paperheight 10in \textwidth 7.5in \textheight 10in \oddsidemargin -0.5in \evensidemargin \oddsidemargin \topmargin -0.5in \headheight 0in \headsep 0in \footskip 0in \parindent 0in \setlength{\topsep}{0pt} \pagestyle{empty} \dmyydate \newcounter{linecounter} \newcommand{\cheatline}{\vskip 1mm\noindent\refstepcounter{linecounter}\thelinecounter. } \begin{document} \setlength{\jot}{0ex} \setlength{\abovedisplayskip}{0.5ex} \setlength{\belowdisplayskip}{0.5ex} \setlength{\abovedisplayshortskip}{0ex} \setlength{\belowdisplayshortskip}{0ex} {\LARGE\bf Cheat Sheet $J$}\hfill \parbox[b]{5in}{\tiny \null\hfill\sheeturl \newline\null\hfill initiated 18/3/13; continues \href{http://drorbn.net/AcademicPensieve/2013-12/}{2013-12}; continued CheatSheetFreeLie @ \href{http://drorbn.net/AcademicPensieve/Projects/WKO4}{Projects/WKO4}; modified \today, \ampmtime } \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 $J_{uv}$ equation:\quad\hfill $\displaystyle J_u(\alpha) + J_v(\beta\sslash RC_u^\alpha)\sslash C_u^{-\alpha} = J_v(\beta) + J_u(\alpha\sslash RC_v^\beta)\sslash C_v^{-\beta} $ \cheatline \checkmark\ The $t$ equation:\quad\hfill $\displaystyle J_w(\gamma\sslash tm^{uv}_w) = \left( J_u(\gamma) + J_v(\gamma\sslash RC_u^\gamma)\sslash C_u^{-\gamma} \right) \sslash tm^{uv}_w $ \cheatline \checkmark\ The $h$ equation:\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 $\displaystyle C_u^\gamma\sslash RC_u^{-\gamma}=Id,$ \hfill $\displaystyle C_u^{\gamma\sslash RC_u^{\gamma}}=RC_u^\gamma$ \cheatline \checkmark\ $C_uC_v$ and $RC_uRC_v$:\quad\hfill $\displaystyle C_u^{\alpha\sslash RC_v^{-\beta}}\sslash C_v^\beta = C_v^{\beta\sslash RC_u^{-\alpha}}\sslash C_u^\alpha,$ \hfill $\displaystyle RC_u^\alpha\sslash RC_v^{\beta\sslash RC_u^\alpha} = RC_v^\beta\sslash RC_u^{\alpha\sslash RC_v^\beta}$ \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 $C$--$\diver$--$RC$ equations:\quad\hfill $\displaystyle \diver_u(\alpha\sslash RC_u^\gamma)\sslash C_u^\gamma = \tbd $ \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 $uv$: with $\ad_u^\gamma=\ad_u\{\gamma\}:=\der(u\to[\gamma, u])$, \newline\null\hfill $\displaystyle (\diver_u\alpha)\sslash\ad_v^\beta - (\diver_v\beta)\sslash\ad_u^\alpha = \diver_u(\alpha\sslash\ad_v^\beta) - \diver_v(\beta\sslash\ad_u^\alpha) $ \cheatline \checkmark\ $\diver$ property $uu$: \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) $ \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 $\gamma_s=\gamma\sslash RC^{s\gamma}_u$,\quad\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 \checkmark\ The growth map $\Gamma_u(\gamma)\coloneqq\beta(1)$. With $\beta(0)=0$ and $\beta'(s)=\gamma \act e^{\ad_u\{s\gamma\}} \act \frac{\ad\beta(s)}{e^{\ad\beta(s)}-1}$, \quad\hfill $\displaystyle e^{\ad_u\{\gamma\}} = C_u^{\beta(1)} $ \cheatline \checkmark\ Many-variable growth. With $\Gamma_{u\alpha\beta}(0)=\Gamma_{v\alpha\beta}(0)=0$, $\Gamma_{u\alpha\beta}'(s)=\alpha \act e^{\ad_u\{s\alpha\}+\ad_v\{s\beta\}} \act \frac{\ad\Gamma_{u\alpha\beta}(s)}{e^{\ad\Gamma_{u\alpha\beta}(s)}-1}$ \newline and $\Gamma_{v\alpha\beta}'(s)=\beta \act e^{\ad_u\{s\alpha\}+\ad_v\{s\beta\}} \act \frac{\ad\Gamma_{v\alpha\beta}(s)}{e^{\ad\Gamma_{v\alpha\beta}(s)}-1}$, \quad\hfill $\displaystyle e^{\ad_u\{s\alpha\}+\ad_v\{s\beta\}} = C_{u,v}^{\Gamma_{u\alpha\beta}(s),\Gamma_{v\alpha\beta}(s)} $ %\cheatline The relation with $\tder$, 2: \quad\hfill %$\displaystyle C_u^\gamma = e^{\ad_u\{\tbd\}}$ \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(\delta\gamma\sslash\frac{1-e^{-\ad\gamma}}{\ad\gamma}\right) = \left(\delta\gamma\sslash\frac{e^{\ad\gamma}-1}{\ad\gamma}\right)\cdot e^\gamma $ \cheatline \checkmark\ The differential of $\gamma=\bch(\alpha,\beta)$:\quad\hfill $\displaystyle \delta\gamma\sslash\frac{1-e^{-\ad\gamma}}{\ad\gamma} = \left(\delta\alpha\sslash\frac{1-e^{-\ad\alpha}}{\ad\alpha}\sslash e^{-\ad\beta}\right) + \left(\delta\beta\sslash\frac{1-e^{-\ad\beta}}{\ad\beta}\right) $ \cheatline \checkmark\ The differential of $C$:\quad\hfill $\displaystyle \delta C_u^\gamma = \ad_u\left\{ \delta\gamma\sslash\frac{e^{\ad \gamma}-1}{\ad\gamma}\sslash RC_u^{-\gamma} \right\} \sslash C_u^\gamma $ \cheatline \checkmark\ The differential of $C_{u,v,\ldots}$:\quad\hfill $\displaystyle \delta C_{u,v,\ldots}^{\alpha,\beta,\ldots} = \ad_{u,v\ldots}\left\{ \delta\alpha\act\frac{e^{\ad \alpha}-1}{\ad\alpha}\act RC_{u,v,\ldots}^{-\alpha,-\beta,\ldots}, \delta\beta\act\frac{e^{\ad \beta}-1}{\ad\beta}\act RC_{u,v,\ldots}^{-\alpha,-\beta,\ldots},\ldots \right\} \sslash C_{u,v,\ldots}^{\alpha,\beta,\ldots} $ \cheatline \checkmark\ The differential of $RC$:\quad\hfill $\displaystyle \delta RC_u^\gamma = RC_u^\gamma \sslash\ad_u\left\{ \delta\gamma\sslash\frac{1-e^{-\ad\gamma}}{\ad\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 \frac{1-e^{-\ad\gamma}}{\ad\gamma} \sslash RC_u^{\gamma} \sslash \diver_u \sslash C_u^{-\gamma} $ \end{document} \endinput