\documentclass[11pt,notitlepage]{article} \usepackage{amsmath, graphicx, amssymb, stmaryrd, datetime, txfonts, multicol, calc, import, amscd, picins, enumitem, wasysym} \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} \usepackage[greek,english]{babel} \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\bbE{{\mathbb E}} \def\bbO{{\mathbb O}} \def\bbQ{{\mathbb Q}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calL{{\mathcal L}} \def\calP{{\mathcal P}} \def\calS{{\mathcal S}} \def\calU{{\mathcal U}} \def\d{\downarrow} \def\dd{{\downarrow\downarrow}} \def\e{\epsilon} \def\CW{\text{\it CW}} \def\FA{\text{\it FA}} \def\FL{\text{\it FL}} \def\frakb{{\mathfrak b}} \def\frakg{{\mathfrak g}} \def\frakt{{\mathfrak t}} \def\Loneco{{\calL^{\text{1co}}}} \def\PaT{\text{{\bf PaT}}} \def\PvT{{\mathit P\!v\!T}} \def\remove{\!\setminus\!} \def\tbd{\text{\color{red} ?}} \def\tder{\operatorname{\mathfrak{tder}}} \def\TAut{\operatorname{TAut}} \def\bbs#1#2#3{{\href{http://drorbn.net/bbs/show?shot=#1-#2-#3.jpg}{BBS:\linebreak[0]#1-\linebreak[0]#2}}} \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-1606}\hfill \parbox[b]{5in}{\tiny \null\hfill\url{http://drorbn.net/AcademicPensieve/Projects/OneCo-1606/} \newline\null\hfill Restarts \href{http://drorbn.net/AcademicPensieve/Projects/OneCo-1604/}{Projects/OneCo-1604}; modified \today; continued \href{http://drorbn.net/AcademicPensieve/2016-11/}{2016-11} } \vskip -3mm \rule{\textwidth}{1pt} \vspace{-8mm} \begin{multicols*}{2}%\raggedcolumns {\bf $\frakg_0$ bi-local exponentiation relations.} In $\frakg_0 = \calD\gemini = \calD\mathfrak{ii} = \calD\text{\scshape{ii}} = \calD\textsc{ii} \coloneqq \langle b,c,u,w\rangle/([b,\cdot]=0,\,[c,u]=u,\,[c,w]=-w,\,[u,w]=b)$ with $\deg(b,c,u,w)=(1,0,1,0)$ and $a_{12}=b_1c_2+u_1w_2$:\hfill\text{\footnotesize(verifications in {\tt G0.nb})} \begin{enumerate}[leftmargin=*,labelindent=0pt] \item The Yang-Baxter element, \newline\null\hfill$\exp_\calU(a_{12}) = \exp\left(b_1c_2+\frac{e^{b_1}-1}{b_1}u_1w_2\right)\act m^{u_1}_1\act m^{c_2w_2}_2$ \item $\green\checkmark$\hfill $[c,uw]=0$ \newline$\green\checkmark$\hfill$e^{\beta u}e^{\alpha c}=e^{\alpha c}e^{e^{-\alpha}\beta u}$ and $e^{\beta w}e^{\alpha c}=e^{\alpha c}e^{e^{\alpha}\beta w}$ \item $\green\checkmark$\hfill$[w,e^{\gamma u}] = -b\gamma e^{\gamma u}$ and $[u,e^{\gamma w}] = b\gamma e^{\gamma w}$ \item With $M_{uw}=M_{uw}(\gamma)\coloneqq e^{\gamma uw}\act m^{uw}=\sum_{k\geq 0}\frac{\gamma^k}{k!}u^kw^k$, \newline$\green\checkmark$\hfill $[u,M_{uw}]=b\gamma uM_{uw}$ \quad and\quad $[w,M_{uw}]=-b\gamma M_{uw}w$ \newline$\green\checkmark$\hfill$M_{uw}^{-1}(\gamma(\alpha))\partial_\alpha M_{uw}(\gamma(\alpha)) = \frac{\partial_\alpha\gamma(\alpha)}{1-b\gamma(\alpha)}uw$ \item With $M_{wu} = M_{wu}(\delta) \coloneqq e^{\delta uw}\act m^{wu}=\sum_{k\geq 0}\frac{\delta^k}{k!}w^ku^k$, \newline$\green\checkmark$\hfill $[u,M_{wu}]=b\delta M_{wu}u$ \quad and\quad $[w,M_{wu}]=-b\delta w M_{wu}$ \newline$\green\checkmark$\hfill$M_{wu}^{-1}(\alpha\delta)\partial_\alpha M_{wu}(\alpha\delta) = \frac{\delta}{1+b\alpha\delta}wu = \frac{\delta}{1+b\alpha\delta}(uw-b)$ \item $\green\checkmark$\hfill $M_{wu}(\delta) = \frac{1}{1+b\delta}M_{uw}\left(\frac{\delta}{1+b\delta}\right)$ \item $\green\checkmark$ The hard core $uw$ relation.\hfill $e^{\alpha w}e^{\beta u} = e^{-b\alpha\beta}e^{\beta u}e^{\alpha w}$ \newline with $\nu=(1+b\delta)^{-1}$, \hfill$e^{\alpha w}M_{wu}(\delta)e^{\beta u} = \nu e^{-b\nu\alpha\beta}e^{\nu\beta u}M_{uw}(\nu\delta)e^{\nu\alpha w}$ \end{enumerate} \rule{\linewidth}{1pt} $\green\checkmark$ {\bf 1-Smidgen $sl_2$ / $\frakg_1$ bi-local exponentiation relations.} With $\epsilon^2=0$, in $\frakg_1 \coloneqq \bbQ[\epsilon]\langle b,c,u,w\rangle/([b,\cdot]=0,\,[c,u]=u,\,[c,w]=-w,\, [u,w]=b-2\epsilon c)$ with $\deg(b,c,u,w,\epsilon)=(1,0,1,0,1)$ and $a_{12}=(b_1-\epsilon c_1)c_2+u_1w_2$: \hfill\text{\footnotesize(verifications in {\tt G1.nb})} \begin{enumerate}[leftmargin=*,labelindent=0pt] \item $c$ relations: \hfill$e^{\beta u}e^{\alpha c}=e^{\alpha c}e^{e^{-\alpha}\beta u}$ and $e^{\beta w}e^{\alpha c}=e^{\alpha c}e^{e^{\alpha}\beta w}$ \item With $\nu=(1+b\delta)^{-1}$ and $\Lambda$ as below, \newline\null\hfill $\bbO\left(e^{\alpha w+\beta u+\delta uw}\mid wu\right) = \bbO\left(\nu (1+\epsilon\nu\Lambda) e^{\nu(-b\alpha\beta+\alpha w+\beta u+\delta uw)}\mid cuw\right)$ \end{enumerate} \rule{\linewidth}{1pt} \entry{160805} $\Lambda$ for {\greektext L'ogos}, ``a principle of order and knowledge'': $\Lambda = % Copy-pasted from G1.nb: -\frac{1}{2} b \nu \left(\alpha ^2 \beta ^2 \nu ^2+4 \alpha \beta \delta \nu +2 \delta ^2\right)-\frac{1}{2} \beta ^2 \delta \nu ^3 u^2 (3 b \delta +2)-\frac{1}{2} b \delta ^4 \nu ^3 u^2 w^2-\beta \delta ^2 \nu ^3 u^2 w (2 b \delta +1)-\beta \nu ^2 u (2 b \delta +1) (\alpha \beta \nu +2 \delta )-2 b \delta ^2 \nu ^2 u w (\alpha \beta \nu +\delta )+\frac{1}{2} \alpha ^2 \delta \nu ^3 w^2 (b \delta +2)+2 c (\alpha \beta \nu +\delta )+2 \beta c \delta \nu u+2 c \delta ^2 \nu u w+2 \alpha c \delta \nu w+\alpha \delta ^2 \nu ^3 u w^2+\alpha \nu ^2 w (\alpha \beta \nu +2 \delta )$. \entry{160801a} Roland's $sm_qsl(2)$ formulas, \bbs{VanDerVeen}{160731}{180741}, with $q=e^\epsilon$, $t=e^b$: $b$ central, $[w,c]=w$, $[c,u]=u$, $wu-quw=1-te^{2\epsilon c}$ (at $\epsilon^2=0$: $[w,u]=\epsilon uw+1-t-2\epsilon tc$), $R=\sum_{m,n}\frac{u^n(b+\epsilon c)^m\otimes c^mw^n}{m![n]_q!}\to \sum_{m,n}\frac{u^n(b+\epsilon c)^m\otimes c^mw^n}{m!n!}\left(1-\frac{\epsilon}{2}\binom{n}{2}\right)$. Also, $\Delta(b,c,u,w)=(b_1+b_2,c_1+c_2,t_2e^{\epsilon c_2}u_1+u_2, e^{\epsilon c_2}w_1+w_2)$ and $S(b,c,u,w)=(-b,-c,-t^{-1}ue^{-\epsilon c},-we^{-\epsilon c})$. Verified {\tt VdVAlgebraAt1-Testing.nb}. \entry{160801b} $sm_0sl(2)$ formulas, $t=e^b$: $b$ central, $[w,c]=w$, $[c,u]=u$, $wu-uw=1-t$, $R=\sum_{m,n}\frac{u^nb^m\otimes c^mw^n}{m!n!}$ (verified {\tt VdVAlgebraAt0.nb}). Also, $\Delta(b,c,u,w)=(b_1+b_2,c_1+c_2,t_2u_1+u_2, w_1+w_2)$ and $S(b,c,u,w)=(-b,-c,-t^{-1}u,-w)$ (unverified). \entry{160730} Lessons from Roland: $\bullet$ There is an additional grading, with $ht(b,c,u,w)=(0,0,1,-1)$. $\bullet$ Rescale $u\to\frac{b}{e^b-1}u$. $\bullet$ A simple $R$-matrix for 1-co. \entry{160725} Challenge: In $\frakg_0$, understand $e^{\beta uw+\alpha u+\gamma w}\act m^{uw}_x$ and $e^{\beta uw+\alpha u+\gamma w}\act m^{wu}_x$. \entry{160628} Figure out duality in $\frakg_1$! \entry{160622b} The $(b-\e c)$-scapegoated 1-co low algebra $\frakg_1$ ``1-smidgen $sl_2$'' ($\e^2=0$, $b$ central) with \hfill $a_{12}=(b_1-\e c_1)c_2+u_1w_2$, \newline $[w,c]=w$ \hfill $[c,u]=u$ \hfill $[u,w]=b-2\e c$. Also $ad(-a_{12})=\{c_1\mapsto u_1w_2$, \hfill $c_2\mapsto -u_1w_2$,\newline \null\quad$u_1\mapsto \e u_1c_2$, \hfill ${u_2\mapsto -(b_1-\e c_1)u_2+(b_2-2\e c_2)u_1}$, \newline \null\quad$w_1\mapsto -b_1w_2-\e w_1c_2+2\e c_1w_2$, \hfill $w_2\mapsto (b_1-\e c_1)w_2\}$. {\bf Claim.} Over $\bbQ\llbracket \e, b_i\rrbracket$ the following generate a sub-Lie algebra, sub-meta-monoid, and contains the $a_{ij}$'s: \[ \{ 1, c_i, u_i, w_i, u_iw_j\} \quad \text{and} \] \[ \e\{c_ic_j, c_iu_j, c_iw_j, c_iu_jw_k, u_iu_jw_k, u_iw_jw_k, u_iu_jw_kw_l\} \] \entry{160621} 1-co low algebra ($\e^2=0$): \hfill $a_{12}=I=b_1c_2+u_1w_2\in\frakb_\e^\ast\otimes\frakb_\e$ \newline $[w,c]=w$ \hfill $[b,u]=-\e u$ \hfill $\delta c=0$ \hfill $\delta w=\e(c\wedge w)$ \newline $[b,c]=0$ \hfill $[b,w]=\e w$ \hfill $[c,u]=u$ \hfill $[u,w]=b-\e c$ \newline\null\hfill(verification in \href{http://drorbn.net/AcademicPensieve/2016-06}{pensieve://2016-06}) Also, $ad(-a_{12})=\{u_1\mapsto \e u_1c_2,\ u_2\mapsto -b_1u_2+b_2u_1-\e u_1c_2,\ b_1\mapsto -\e u_1w_2,\ b_2\mapsto \e u_1w_2,\ w_1\mapsto -b_1w_2-\e w_1c_2+\e c_1w_2,\ w_2\mapsto b_1w_2,\ c_1\mapsto u_1w_2,\ c_2\mapsto -u_1w_2\}$. \rule{\linewidth}{1pt} {\bf Recycling.} \entry{160622a} Scapegoated 1-co low algebra ($\e^2 \!=\! s^2 \!=\! \e s \!=\! 0$, $b$ central): \[ a_{12}=I=(b_1+s_1)c_2+u_1w_2\in\frakb_\e^\ast\otimes\frakb_\e \] $[w,c]=w$ \hfill $[s,u]=-\e u$ \hfill $\delta c=0$ \hfill $\delta w=\e(c\wedge w)$ \newline $[s,c]=0$ \hfill $[s,w]=\e w$ \hfill $[c,u]=u$ \hfill $[u,w]=b+s-\e c$ Also, $ad(-a_{12})=\{$\newline \null\quad$u_1\mapsto \e u_1c_2$, \hfill ${u_2\mapsto -(b_1+s_1)u_2+(b_2+s_2)u_1-\e u_1c_2}$, \newline \null\quad$s_1\mapsto -\e u_1w_2$, \hfill $s_2\mapsto \e u_1w_2$,\hfill $c_1\mapsto u_1w_2$, \hfill $c_2\mapsto -u_1w_2$,\newline \null\quad$w_1\mapsto -(b_1+s_1)w_2-\e w_1c_2+\e c_1w_2$, \hfill $w_2\mapsto (b_1+s_1)w_2$,\newline $\}$. \entry{160629} Let $A=(A_0=\langle 1\rangle)\oplus A_{>0}$ be a graded unital algebra over an augmented ring $\eta\colon R\to\bbQ$, and let $T\colon A^{\otimes n}\to A^{\otimes n}$ be such that ???. Then there is a unique \end{multicols*} \end{document} \endinput