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\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\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}}

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\begin{document}
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{\LARGE\bf Cheat Sheet OneCo}\hfill
\parbox[b]{5in}{\tiny
  \null\hfill\url{http://drorbn.net/AcademicPensieve/2015-07/}
  \newline\null\hfill initiated 14/4/15;
  modified \today, \ampmtime;
  continues \href{http://drorbn.net/AcademicPensieve/2015-06/}{2015-06}
}

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

\begin{multicols}{2}\raggedcolumns


{\bf Road Map.}
$\bullet$ Fully analyze and implement $\Loneco/2D$; verify Jacobi.
$\bullet$ Compute $\Ad$ and solve for $R$. $\bullet$ 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}

\import{../2015-06/figs/}{FoundationalRelations.pdf_tex}

{\bf 2Dv.} $b$: bracket trace; $\yellowm{c}$: cobracket trace; ${\langle b,c\rangle = \yellowm{\delta}\in\{0,1\}}$; $\deg b_i = \deg c_j = \deg a_{ij} = \deg\delta = 1$.

$\yellowm{\calA^{2Dv}}$ 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 (note $\gamma_{j\{kl\}}=0$) \hfill $\yellowm{[a_{jk},a_{jl}] = c_la_{jk}-c_ka_{jl} \eqqcolon \gamma_{jkl}}$,


{\bf hh.} \hfill $\yellowm{[a_{jk},a_{ik}] = b_ia_{jk}-b_ja_{ik}}$,

{\bf ht.} \hfill $\yellowm{[a_{jk},a_{kl}] = b_ja_{kl}-b_ka_{jl}-\gamma_{jkl}}$,

{\bf ab,ac.} \hfill $\ad a_{jk}\colon b_j,-b_k,-c_j,c_k \mapsto \delta a_{ij}-b_ic_j \eqqcolon \yellowm{\gamma_{ij}}$,

{$\leftrightarrows$.} \hfill $[a_{ij},a_{ji}] = \pinkm{?}$,

{\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 Repeating indices.} In $a_{ii}$, $\gamma_{ii}$, $\gamma_{iij}$, $\gamma_{ijj}$, and $\gamma_{iii}$, heads above tails.

{\bf The Ascending Algebra $\yellowm{\calA^{2Dv}_+}$.} Same but with only $a_{ij},\ i<j$.

{\bf The OneCo Sub-Quotient} is $\langle a_{ij}\rangle$ modulo $\delta^2=\delta c_i=c_jc_k=0$, so $\yellowm{\Loneco}$ is (coefficient functions non-central, in $\bbQ\llbracket b_i\rrbracket$)
\[ \left\{
    f^{ij}a_{ij} + f^{ijk}\gamma_{ijk} + f^{ijkl}\gamma_{ij}a_{kl}
  \right\}
  \left/\begin{array}{l}
    r_1\colon\ b_i\gamma_{ijk} = \gamma_{ij}a_{ik} - \gamma_{ik}a_{ij} \\
    r_2\colon\ b_i\gamma_{jkl} = \gamma_{jk}a_{il} - \gamma_{jl}a_{ik} \\
    \text{\footnotesize(is there a residual 4T/6T?)}
  \end{array}\right..
\]

{\bf Representations.} $\ad a_{-,-}$:

\vspace{-7mm}
\null\hfill
$\ds\xymatrix{
  \langle\beta\rangle \ar[r] &
  \langle\gamma_{ij}\rangle \ar@`{p+(16,16),p+(16,-16)}
}$
\hfill\hfill
\raisebox{7mm}{$\ds\xymatrix{
  \langle a_{ij}\rangle \ar@`{p+(-16,16),p+(-16,-16)} \ar[r] \ar[d] &
  \langle\gamma_{ijk}\rangle \ar@`{p+(16,16),p+(16,-16)} \ar[ld] \\
  \langle\gamma_{ij}a_{kl}\rangle \ar@`{p+(-16,16),p+(-16,-16)}
}$}
\hfill\null

$\bullet$ \pinkt{$\langle\gamma_{ij}\rangle$ is \ldots}

OneCo $\ad(a_{jk})$: \hfill (authorities: \href{http://drorbn.net/AcademicPensieve/2015-06/one/Cheat_Sheet_OneCo_Authorities.pdf}{pensieve://2015-06/})

\includegraphics[width=\columnwidth]{../2015-06/OneCoBrackets.pdf}

OneCo $\Ad(a_{jk})$:

\includegraphics[width=\columnwidth]{../2015-06/OneCoAd.pdf}

\vfill{\bf To do.}
$\bullet$ Perhaps I should find a way to highlight the fact that v is a perturbation of w.
$\bullet$ Position FiC.
$\bullet$ Position the 2D Lie bialgebras.
$\bullet$ Is there a meaningful $a_{ij}\to a_{ij}/b_i$ (etc) renormalization?
$\bullet$ Add: diagrammatic interpretations of $b_i$, $c_j$, $\gamma_{ij}$, $\gamma_{ijk}$.

\end{multicols}

\newpage\begin{multicols}{2}

{\bf Recycling.}

{\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 The primitivity condition.} $\ds\ker\left( f+f^{ij}a_{ij} \mapsto \delta f + f^{ij}b_ic_j\right)$. (Ignoring multiple arrows).

{\bf State Diagrams.} $\ad a_{ij}$ yields\hfill
{\small\greent{green:} roots. \pinkt{pink:} wrong.}

\vskip -5mm
\[
  \xymatrix{
    \greenm{fc_i} \ar[r]^{\hat{b_i}} \ar[d]_{-} &
    \greenm{fc_j} \ar[dl]^+ \ar@`{p+(16,16),p+(16,-16)}_{-\hat{b_i}} \\
    \delta fa_{ij} &
    \greenm{f} \ar[u]_(0.4){\hat{b_i}(\partial_j-\partial_i)} \ar[l]^-{\partial_i-\partial_j}
  }
  \qquad \xymatrix{
    \greenm{a_{jk}} \ar@`{p+(-16,16),p+(-16,-16)}^{\pinkm{\hat{b_i}}} \ar[r]^-+ \ar[d]_{-} \ar[rd]_{-\hat{b_j}} &
    c_ja_{ik} \ar@`{p+(0,22.6),p+(22.6,0)}_{-\hat{b_i}} \ar[rd]_+ & \\
    c_ka_{ij} &
    \greenm{a_{ik}} \ar[l]^-+ \ar[u]^{-} &
    \delta a_{ij}a_{ik}
  }
\]
\vskip -3mm
\[ \xymatrix{\greenm{a_{kj}} \ar@`{p+(-16,16),p+(-16,-16)}^{\pinkm{-\hat{b_i}}} \ar[r]^-{\hat{b_k}} & a_{ij}} \]

so with $\yellowm{\phi_0} \coloneqq \phi(0)$, $\yellowm{\phi_1} \coloneqq \phi'_0$, and $\yellowm{\phi_\d}(x) \coloneqq (\phi(x)-\phi_0)/x$, $\phi(\ad a_{ij})$ is

\vskip -8mm
\begin{align*}
  fc_i & \mapsto \phi_0fc_i+(b_i\phi_\dd(-b_i)-\phi_1)\delta fa_{ij}
    + b_i\phi_\d(-b_i)fc_j \\
  fc_j & \mapsto \phi(-b_i)fc_j+\phi_\d(-b_i)\delta fa_{ij} \\
  f & \mapsto \phi_0f + b_i\phi_\d(-b_i)(\partial_jf-\partial_if)c_j \\
    &\qquad + (b_i\phi_\dd(-b_i)-\phi_1)(\partial_jf-\partial_if)\delta a_{ij} \\
  \delta a_{\cdot\cdot} & \mapsto \text{as in Adjoint Gassner} \\
  a_{ik} & \mapsto \phi_0a_{ik} + \phi_1c_ka_{ij} - \phi_\d(-b_i)c_ja_{ik}
    - \phi_\dd(-b_i)\delta a_{ij}a_{ik} \\
  a_{jk} & \ \pinkm{\mapsto}\ \phi(b_i)a_{jk} - (\phi_\d(b_i)+b_j\phi_\dd(b_i))c_ka_{ij} - b_j\phi_\d(b_i)a_{ik} \\
    &\qquad + \frac{\phi(b_i)-\phi(-b_i) + b_j(\phi_\d(b_i)-\phi_\d(-b_i))}{2b_i}c_ja_{ik} \\
    &\qquad + \frac{\phi_\d(b_i)-\phi_\d(-b_i) + b_j(\phi_\dd(b_i)-\phi_\dd(-b_i))}{2b_i}\delta a_{ij}a_{ik} \\
  a_{kj} & \mapsto \\
  a_{ij} & \mapsto a_{ij}
\end{align*}

Then $[a_{ij},f] = (\partial_if-\partial_jf)\gamma_{ij}$ and

{\bf $\gamma$b.} \hfill $[\gamma_{ij},b_l]=0$ and $[\gamma_{ijk},b_l]=0$ \hfill incl.\ $l\in\{i,j,k\}$,

{\bf tt$\gamma$.} \hfill $[a_{jk}, \gamma_{jl}] = 0$,

{\bf hh$\gamma$.} \hfill $[a_{jk}, \gamma_{ik}] = -b_j\gamma_{ik}$,

{\bf th$\gamma$.} \hfill $[a_{jk}, \gamma_{ij}] = b_j\gamma_{ik}$,

{\bf ht$\gamma$.} \hfill $[a_{jk}, \gamma_{kl}] = b_j\gamma_{kl} - b_k\gamma_{jl}$,

{\bf tt$\gamma_3$.} \hfill $[a_{jk}, \gamma_{jlm}]=0$,

{\bf th$\gamma_3$.} \hfill $[a_{jk}, \gamma_{ijl}] = b_j\gamma_{ikl} + \gamma_{il}a_{jk}$,

{\bf ht$\gamma_3$.} \hfill $[a_{jk}, \gamma_{klm}] = b_k\gamma_{jkl} + b_j\gamma_{klm}$,

{\bf hh$\gamma_3$.} \hfill $[a_{jk}, \gamma_{nik}] = -b_j\gamma_{nik} + \gamma_{ni}a_{jk}$,

\hfill $[a_{jk}, \gamma_{jkl}] = -\gamma_{jk}a_{jl}$,

\hfill $[a_{jk}, \gamma_{ijk}] = - b_j\gamma_{ijk} + \gamma_{ij}a_{jk} + \gamma_{ik}a_{jk}$.

\end{multicols}

\end{document}

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