\documentclass[11pt,notitlepage]{article} \usepackage{amsmath,graphicx,amssymb,stmaryrd,datetime,txfonts,multicol,calc,import,amscd} \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} \usepackage{pstricks} \newcommand\entry[1]{{\bf\tiny (#1)}} % 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\purplem#1{{\setlength{\fboxsep}{0pt}\colorbox{Thistle}{$#1$}}} \def\purplet#1{{\setlength{\fboxsep}{0pt}\colorbox{Thistle}{#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\sder{\operatorname{\mathfrak{sder}}} \def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}} \def\bbQ{{\mathbb Q}} \def\calA{{\mathcal A}} \def\calL{{\mathcal L}} \def\calP{{\mathcal P}} \def\d{\downarrow} \def\dd{{\downarrow\downarrow}} \def\CW{\text{\it CW}} \def\FA{\text{\it FA}} \def\FL{\text{\it FL}} \def\frakt{{\mathfrak t}} \def\Loneco{{\calL^{\text{1co}}}} \def\PaT{\text{{\bf PaT}}} \def\remove{\!\setminus\!} \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/2015-10/} \newline\null\hfill Continued \href{http://drorbn.net/AcademicPensieve/2016-01}{2016-01}; Continues \href{http://drorbn.net/AcademicPensieve/2015-09}{2015-09}; modified \today } \vskip -3mm \rule{\textwidth}{1pt} \vspace{-8mm} \begin{multicols}{2}\raggedcolumns {\bf Road Map.} $\bullet$~Implement and verify scatter-level stitching. $\bullet$~Guess/deduce glow level. {\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 (mod~$\langle a_{ii}\rangle$) $[a_{ij},a_{ji}] = b_ia_{ji}-b_ja_{ij}$. 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 $\yellowm{\bar{x}_i}=x_i/b_i$, $\yellowm{t_i}=e^{b_i}$, get $[e^{\ad a_{ij}}]_{\bar{x}_i,\bar{x}_j} = \begin{pmatrix} 1 & 1-t_i \\ 0 & t_i \end{pmatrix}$. {\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 Implementation/verification: \href{http://drorbn.net/AcademicPensieve/2015-04/nb/ZeroCo.pdf}{pensieve://2015-04/nb/ZeroCo.pdf}. {\bf Adjoint Gassner.} Renaming $\yellowm{\bar{a}_{ij}} = a_{ij}/b_i$ and $t_i=e^{b_i}$, get $[{\bar a}_{ij},{\bar a}_{ik}] = 0$, $[\bar{a}_{ik},\bar{a}_{jk}] = -[\bar{a}_{ij},\bar{a}_{jk}] = \bar{a}_{ik}-\bar{a}_{jk}$, and (mod~$\langle \bar{a}_{ii}\rangle$) $[\bar{a}_{ij},\bar{a}_{ji}] = \bar{a}_{ji}-\bar{a}_{ij}$, so \hfill $\ds \bar{a}_{kj} \mapsto t_i^{-1}\bar{a}_{kj} + (1-t_i^{-1})\bar{a}_{ij}$,\hfill\null \hfill $\ds \bar{a}_{ki} \mapsto \bar{a}_{ki} + (1-t_i^{-1})\bar{a}_{kj} + (t_i^{-1}-1)\bar{a}_{ij}$,\hfill\null \hfill $\ds \bar{a}_{jk} \mapsto t_i\bar{a}_{jk} + (1-t_i)\bar{a}_{ik}, \quad \bar{a}_{ji} \mapsto t_i\bar{a}_{ji} + (1-t_i)\bar{a}_{ij}$.\hfill\null {\bf Questions.} $\bullet$ As Gassner is $\Gamma$ calculus, Adjoint Gassner must factor through Gassner. {\red How?} $\bullet$ Interpretation? $\pi_T$-Artin? \vskip 1mm %\includegraphics[width=\columnwidth]{figs/FoundationalRelations.pdf} %\def\svgwidth{\columnwidth} \subimport{../2015-07/}{FoundationalRelationsNT.pdf_tex} {\bf 2Dv.} $b$: bracket trace; $\yellowm{c}$: cobracket trace; ${\langle b,c\rangle = \yellowm{\delta}\in\{0,1\}}$; $\deg b_i = \deg c_j = \deg a_{ij} = \deg\delta = 1$. Implementation/verification: \href{http://drorbn.net/AcademicPensieve/2015-08/nb/abc.pdf}{pensieve://2015-08/nb/abc.pdf}. $\yellowm{\calA^{2Dv}}$ 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 $\yellowm{[a_{jk},a_{jl}] = c_la_{jk}-c_ka_{jl}}$, {\bf hh.} \hfill $\yellowm{[a_{jk},a_{ik}] = b_ia_{jk}-b_ja_{ik}}$, {\bf Swinging.} \hfill $\delta a_{ij}a_{kl} - \delta a_{il}a_{jk} = b_kc_la_{ij}-b_ic_la_{kj}-b_kc_ja_{il}+b_ic_ja_{kl}$ {\bf ht.} \hfill $\yellowm{[a_{jk},a_{kl}] = b_ja_{kl}-b_ka_{jl}-c_la_{jk}+c_ka_{jl}}$, {\bf ab,ac.} \hfill $\ad a_{jk}\colon b_j,-b_k,-c_j,c_k \mapsto \yellowm{\gamma_{jk}} \coloneqq \delta a_{jk}-b_jc_k$, {\bf Backie.} \hfill $[a_{jk},a_{kj}] = (b_j+c_k)a_{kj} - (b_k+c_j)a_{jk} + (b_j-c_j)a_{kk} - (b_k-c_k)a_{jj} + \gamma_{jk} - \gamma_{kj}$, \newline\null\hfill with $\yellowm{\gamma_{jk}} \coloneqq \delta a_{jk}-b_jc_k$, {\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) $, \hfill 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