\documentclass[11pt,notitlepage]{article} \usepackage{amsmath,graphicx,amssymb,stmaryrd,datetime,txfonts,multicol} \usepackage[usenames,dvipsnames]{xcolor} % Following http://tex.stackexchange.com/a/847/22475: \usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks, linkcolor={green!50!black}, citecolor={green!50!black}, urlcolor=blue } \usepackage[all]{xy} % Following http://tex.stackexchange.com/questions/23521/tabular-vertical-alignment-to-top: \def\imagetop#1{\vtop{\null\hbox{#1}}} \def\red{\color{red}} \def\greenm#1{{\setlength{\fboxsep}{0pt}\colorbox{LimeGreen}{$#1$}}} \def\greent#1{{\setlength{\fboxsep}{0pt}\colorbox{LimeGreen}{#1}}} \def\pinkm#1{{\setlength{\fboxsep}{0pt}\colorbox{pink}{$#1$}}} \def\pinkt#1{{\setlength{\fboxsep}{0pt}\colorbox{pink}{#1}}} \def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}} \def\yellowt#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{#1}}} \def\ds{\displaystyle} \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\calA{{\mathcal A}} \def\calL{{\mathcal L}} \def\d{\downarrow} \def\dd{{\downarrow\downarrow}} \def\CW{\text{\it CW}} \def\FA{\text{\it FA}} \def\FL{\text{\it FL}} \def\Loneco{{\calL^{\text{1co}}}} \def\tbd{\text{\color{red} ?}} \def\tder{\operatorname{\mathfrak{tder}}} \def\TAut{\operatorname{TAut}} \paperwidth 8in \paperheight 10.5in \textwidth 8in \textheight 10.5in \oddsidemargin -0.75in \evensidemargin \oddsidemargin \topmargin -0.75in \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 OneCo}\hfill \parbox[b]{5in}{\tiny \null\hfill\url{http://drorbn.net/AcademicPensieve/2015-05/} \newline\null\hfill initiated 14/4/15; modified \today, \ampmtime; continues \href{http://drorbn.net/AcademicPensieve/2015-04/}{2015-04} } \vskip -3mm \rule{\textwidth}{1pt} \vspace{-8mm} \begin{multicols}{2}\raggedcolumns {\bf Models.} $\bullet$ In $[x,y]=\delta x$, $xf(y)=f(y+\delta)x$. If $\delta^2=0$, $[x,f(y)] = \delta f'(y)x$. $\bullet$ In $[x,y]=\delta x+z^2$, $xf(y)=f(y+\delta)x+\frac{z^2}{\delta}(f(y+\delta)-f(y))$. If $\delta^2=0$, $[x,f(y)] = \delta f'(y)x+z^2f'(y)$. $\bullet$ If $S_n\coloneqq\sum_{k=0}^{n-1}A^kCB^{n-1-k}$ then $AS_n-S_nB = A^nC-CB^n$ so $S_n = (L_A-R_B)^{-1}(A^nC-CB^n)$. $\bullet$ If $\psi(x)=\sum_{n\geq 0}a_nx^n$ then $\sum_{n\geq 0}a_n\sum_{k=0}^{n-1}b^n(-b)^{n-1-k} = {(\psi(b)-\psi(-b))/2b}$. {\bf Deriving Gassner.} $\yellowm{\calL^{2Dw}}$ is $\bbQ\llbracket\yellowm{b_i}\rrbracket\langle\yellowm{a_{ij}}\rangle$ modulo locality, $[a_{ij},a_{ik}] = 0$, $[a_{ik},a_{jk}] = -[a_{ij},a_{jk}] = b_ja_{ik}-b_ia_{jk}$, and $[a_{ij},a_{ji}] = b_ia_{ji}-b_ja_{ij}$. Acts on $\yellowm{V} = {\bbQ\llbracket b_i\rrbracket\langle\yellowm{x_i}=a_{i\infty}\rangle}$ by $[a_{ij},x_i]=0$, $[a_{ij},x_j] = b_ix_j-b_jx_i$. Hence $e^{\ad a_{ij}}x_i=x_i$, $e^{\ad a_{ij}}x_j = e^{b_i}x_j+\frac{b_j}{b_i}(1-e^{b_i})x_i$. Renaming $\yellowm{y_i}=x_i/b_i$, $\yellowm{t_i}=e^{b_i}$, get $[e^{\ad a_{ij}}]_{y_i,y_j} = \begin{pmatrix} 1 & 1-t_i \\ 0 & t_i \end{pmatrix}$. {\bf The $\calL^{2Dw}$ Adjoint representation.} $e^{\ad a_{ij}}$ acts by $\ds a_{kl} \mapsto a_{kl}$, \hfill $\ds a_{ik} \mapsto a_{ik}$, \hfill $\ds a_{kj} \mapsto e^{-b_i}a_{kj} + \frac{b_k}{b_i}(1-e^{-b_i})a_{ij}$, \hfill$\ds a_{ki} \mapsto a_{ki} + (1-e^{-b_i})a_{kj} + b_k\frac{e^{-b_i}-1}{b_i}a_{ij}$,\hfill\null \hfill$\ds a_{jk} \mapsto e^{b_i}a_{jk} + \frac{b_j}{b_i}(1-e^{b_i})a_{ik}$, \hfill$\ds a_{ji} \mapsto e^{b_i}a_{ji} + \frac{b_j}{b_i}(1-e^{b_i})a_{ij}$.\hfill\null {\bf Adjoint Gassner.} Renaming $\yellowm{\alpha_{ij}} = a_{ij}/b_i$ and $t_i=e^{b_i}$, get \[ \alpha_{kj} \mapsto t_i^{-1}\alpha_{kj} + (1-t_i^{-1})\alpha_{ij}, \] \[ \alpha_{ki} \mapsto a_{ki} + (1-t_i^{-1})\alpha_{kj} + (t_i^{-1}-1)\alpha_{ij} \] \[ \alpha_{jk} \mapsto t_i\alpha_{jk} + (1-t_i)\alpha_{ik}, \quad \alpha_{ji} \mapsto t_i\alpha_{ji} + (1-t_i)\alpha_{ij}. \] Implementation/verification: \href{http://drorbn.net/AcademicPensieve/2015-04/nb/ZeroCo.pdf}{pensieve://2015-04/nb/ZeroCo.pdf}. Interpretation: $\pi_T$-Artin? {\bf 2Dv.} $b$: bracket trace; $\yellowm{c}$: cobracket trace; ${\langle b,c\rangle = \yellowm{\delta}\in\{0,1\}}$; $\deg b_i = \deg c_j = \deg a_{ij} = \deg\delta = 1$. $\yellowm{\calA^{2Dv}}$ is $\bbQ\llbracket\delta\rrbracket\FA(b_i,c_j,a_{ij})$ (so $\calL^v = \{f+f^{ij}a_{ij}\}$) modulo locality, {\bf tt.}\hfill $[a_{ij},a_{ik}] = c_ka_{ij}-c_ja_{ik} \eqqcolon \yellowm{\gamma_{ijk}}$, \hfill (note $\gamma_{i\{jk\}}=0$) {\bf hh.} \hfill $[a_{ik},a_{jk}] = b_ja_{ik}-b_ia_{jk}$, {\bf ht.} \hfill $[a_{ij},a_{jk}] = b_ia_{jk}-b_ja_{ik}-\gamma_{ijk}$, {$\leftrightarrows$.} \hfill $[a_{ij},a_{ji}] = \pinkm{?}$, {\bf ab, ac.} \hfill $[a_{ij},b_i] = -[a_{ij},b_j] = -[a_{ij},c_i] = [a_{ij},c_j]$ \newline\null\hfill$ = \delta a_{ij}-b_ic_j \eqqcolon \yellowm{\gamma_{ij}}$, {\bf bc.} \hfill $[b_i,c_j]=0$. So\hfill$\ds a_{ij}f = f^\delta a_{ij} - \frac{b_ic_j}{\delta}(f^\delta-f), $\hfill$\ds [a_{ij},f] = (f^\delta-f)\left(a_{ij}-\frac{b_ic_j}{\delta}\right) $, with $\yellowm{f^\delta} \coloneqq f\act\left({b_i\to b_i+\delta\ b_j\to b_j-\delta \atop c_i\to c_i-\delta\ c_j\to c_j+\delta}\right)$. {\bf The Ascending Algebra $\yellowm{\calA^{2Dv}_+}$.} Same but with only $a_{ij},\ i