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\def\frakt{{\mathfrak t}}
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{\LARGE\bf Cheat Sheet OneCo-1604}\hfill
\parbox[b]{5in}{\tiny
  \null\hfill\url{http://drorbn.net/AcademicPensieve/Projects/OneCo-1604/}
  \newline\null\hfill
  Continues \href{http://drorbn.net/AcademicPensieve/2016-04}{2016-04};
  restarted \href{http://drorbn.net/AcademicPensieve/Projects/OneCo-1606/}{Projects/OneCo-1606};
  modified \today
}

\vskip -3mm
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\vspace{-8mm}

\begin{multicols}{2}\raggedcolumns

{\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 $[a_{ij},a_{ji}] = b_ia_{ji}-b_ja_{ij}+b_ia_{jj}-b_ja_{ii}$.
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 $[e^{\ad a_{ij}}]_{\bar{x}_i,\bar{x}_j} = \begin{pmatrix} 1 & 1-t_i \\ 0 & t_i \end{pmatrix}$. The radical contains $1$ and $a_{ii}$ (if included) and $\left\{\sum_i\frac{\alpha_i}{b_i}\sum_ja_{ij}\colon \sum\alpha_i=0\right\}$.

{\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}, \href{http://drorbn.net/AcademicPensieve/2016-04/nb/BurauAndAd.pdf}{pensieve://2016-04/nb/BurauAndAd.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 $e^{\ad a_{ij}}$ acts by

\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 Question.} 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
  $\delta a_{ij}a_{kl} - \delta a_{il}a_{kj} = b_kc_la_{ij}-b_ic_la_{kj}-b_kc_ja_{il}+b_ic_ja_{kl}$

{\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 $[b_i,c_j]=0$.

{\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 $\bbQ\llbracket b_i\rrbracket$)

{\bf The 1co Graphs.}

%\resizebox{\columnwidth}{!}{\subimport{../../2015-09/}{1coGraphs.pdf_t}}
\resizebox{\columnwidth}{!}{\subimport{./}{1coGraphs.pdf_t}}

In \href{http://drorbn.net/AcademicPensieve/Projects/OneCo-1604/nb/abc.pdf}{abc.nb}: $R^{jk}=e^{a_{jk}}\rho$ with $\rho \coloneqq$
\[
  \psi(b_j)\left(-c_k + \frac{c_ka_{jk}}{b_j} - \frac{\delta a_{jk}a_{jk}}{b_j^2}\right)
  + \frac{\phi(b_j)\psi(b_k)}{b_k\phi(b_k)}\left(c_ka_{kk} - \frac{\delta a_{jk}a_{kk}}{b_j}\right),
\]
and with $\phi(x)\coloneqq e^{-x}-1 = -x+x^2/2-\dots$, and $\psi(x)\coloneqq\left((x+2)e^{-x}-2+x\right)/(2x) = x^2/12-x^3/24+\dots$.

{\bf OneCo Monoblog.}

\entry{161621} 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 $\delta w=\e(c\wedge u)$
\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, $ad(-a_{12})=\{u_1\mapsto \e u_1c_2,\ u_2\mapsto -b_1u_2+b_2u_1-\e u_1c_2,\ b_1\mapsto -\e u_1w_2,\ b_2\mapsto \e u_1w_2,\ w_1\mapsto -b_1w_2-\e w_1c_2+\e c_1w_2,\ w_2\mapsto b_1w_2,\ c_1\mapsto u_1w_2,\ c_2\mapsto -u_1w_2\}$.

\entry{161620} I need concise descriptions of $\calP^{2,2}$ in $\calA^v$ terms \& in $\frakg$ terms.

\entry{160618b} A ``defining rep'' for the Euler ext.\ of Alexander-Gassner?

\entry{160618a} The ``Euler extension'' of a graded Lie algebra $L$ over $R\coloneqq\bbQ\llbracket b_i\rrbracket$, with $\deg b_i=1$, assuming as graded v.s.\ $L=R\otimes_\bbQ L_0$ with a graded f.d.\ $L_0$: Set $R_\epsilon\coloneqq R\otimes\bbQ[\epsilon]/(\epsilon^2=0)$ with $\deg\epsilon=1$ and $\iota\colon R\to R_\epsilon$ via $b_i\mapsto b_i+\epsilon$ (not $R$-linear!), set $L_E\coloneqq(R_\epsilon\otimes L_0)\rtimes R\langle E\rangle$ with $[E,b_i]=0$, $[E,\epsilon]=\epsilon$ and $[E,x]=(\deg x)x$ for $x\in L_0$, and with $\iota\colon L\to L_E$ in the obvious way. Then $L_E$ is finite rank over $R$ with a faithful adjoint rep, and it contains a $\bbQ$-linear copy of $L$.

\entry{160616} Consider ``degrons'' and their ``duals'', and non-linear embeddings of the Gassner Lie algebra.

\parpic[r]{\parbox{1.25in}{
$a_{12}=I\in\frakb_0^\ast\otimes\frakb_0$:
\newline\null\hfill$b_1c_2+u_1w_2$
\newline$[c,w]=w$\hfill$[b,\cdot]=0$
\newline$[u,c]=u$\hfill$[w,u]=b$
}}
\entry{160612} Let $\frakb_0\coloneqq\langle c,w\rangle$ with $[c,w]=w$, let $\frakb_0^\ast=\langle b,u\rangle$ with $c(b)=u(w)=1$ and $c(w)=u(b)=0$ be Abelian, let $\frakg_0 = I\frakb_0 = \frakb_0^\ast\rtimes\frakb_0$
so $[b,c]=[b,w]=[b,u]=0$ while $[c,u]=-u$ and $[w,u]=b$. Let $r=Id=b_1c_2+u_1w_2\in\frakb_0^\ast\otimes\frakb_0\subset \frakg_0\otimes \frakg_0$. Let $\calU = \calU(\frakg_0)$, degree-completed with respect to $\deg b,u=1$ and $\deg c,w=0$. Then $R=\exp(r)\in\calU\otimes\calU$ satisfies Yang-Baxter, $bc+uw$, $cb+wu$, and $b$ are central, and $(cb+wu)-(bc+uw)=b$. Also,
$ad(-r_{ij})=\{b_k\mapsto 0, u_i\mapsto 0, u_j\mapsto b_iu_j-b_ju_i, c_i\mapsto-u_iw_j, c_j\mapsto u_iw_j, w_i\mapsto b_iw_j, w_j\mapsto -b_iw_j\}$.

\entry{160611} I should re-write {\tt Local.nb} using $\{1, \delta, a_{ij}, \eta_k, \eta_ka_{ij}, u_i, w_j\}$, and aim to split the $a_{ij}$'s.

\entry{160608} Likely, $G^\ast\cong\Delta t\infty$.

\entry{160607} How is $G\otimes G^\ast$ related to ``Adjoint Gassner'' above?

\entry{160529} In \href{http://drorbn.net/AcademicPensieve/2016-06/}{2016-06/TurboGassner.nb}: \hfill (make presentable?)

\includegraphics[width=\linewidth]{GP.pdf}

\includegraphics[width=\linewidth]{FTG.pdf}

\includegraphics[width=\linewidth]{TG.pdf}

\entry{160529} Is there a use for $\delta^{-1}c_ic_j$ terms?

\entry{160527} Try rewriting $R$ as an exponential. Explains the $\phi_1$'s?

\entry{160505} A faithful representation for $\calA^{2,2}$? Ado suggests existence.

\entry{1504} If $S_n\coloneqq\sum_{k=0}^{n-1}A^kCB^{n-1-k}$ then $AS_n-S_nB = A^nC-CB^n$ so $S_n = (L_A-R_B)^{-1}(A^nC-CB^n)$.

\entry{151019a} To do: Find and implement the group-like condition.

\end{multicols}

\newpage

\begin{multicols}{2}

{\bf Recycling.}

\entry{160510} The next few steps: $\bullet$ $h\infty$ scattering in 1-co. $\bullet$ Solve again for $R$. $\bullet$ Find a manifestly polynomial formula for $R$. $\bullet$ Revisit stitching in 0-co. $\bullet$ Stitching for $\delta h\infty$ scattering. $\bullet$ Stitching for $h\infty$ scattering. $\bullet$ Full adjoint scattering in 1-co. $\bullet$ Stitching for full adjoint. $\bullet$ Glow. $\bullet$ Caps and cups.

\entry{151019c} Make the braid representation presentable?

\entry{160508} How would I present the TS stitching formula?

\entry{160317} To do: For 0-co $a$ and $b$, compute the 1-co part of $e^{-a}be^a$.

\entry{151019b} Switch to an EK basis?

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 $\yellowm{f^\delta} \coloneqq f\act\left({b_i\to b_i+\delta\ b_j\to b_j-\delta \atop c_i\to c_i-\delta\ c_j\to c_j+\delta}\right)$.

\vskip -2mm
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{\bf The Abstract Context.} (From LesDiablerets-1508)

{\red Definition.} A meta-monoid is a functor $M\colon$(finite sets,
injections)$\to$(sets) along with natural operations $\ast\colon M(S_1)\times M(S_2)\to M(S_1\sqcup
S_2)$ whenever $S_1\cap S_2=\emptyset$ and $m^{ab}_c\colon
M(S)\to M((S\remove\{a,b\})\sqcup\{c\})$ whenever $a\neq b\in S$ and
$c\notin S\remove\{a,b\}$, such that
\[ \text{meta-associativity:}\quad
  m^{ab}_y\act m^{yc}_x = m^{bc}_y\act m^{ay}_x \]
\[ \text{meta-locality:}\quad
  m^{ab}_x\act m^{de}_y = m^{de}_y\act m^{ab}_x \]
and, with $\epsilon_b=M(S\hookrightarrow S\sqcup\{b\})$,
\[ \text{meta-unit:}\quad
  \epsilon_b\act m^{ab}_a = Id = \epsilon_b\act m^{ba}_a.
\]

{\red Theorem.} $S\mapsto\Gamma_0(S)$ is a meta-monoid and
$z_0\colon\PvT\to\Gamma_0$ is a morphism of meta-monoids.

{\red Theorem.} There exists an extension of $\Gamma_0$ to a
bigger meta-monoid $\Gamma_{01}(S) = \Gamma_0(S)\times\Gamma_1(S)$, along with
an extension of $z_0$ to $z_{01}\colon\PvT\to\Gamma_{01}$, with
\[ \Gamma_1(S) = R_S \oplus
  V\oplus V^{\otimes 2}\oplus V^{\otimes 3}\oplus \calS^2(V)^{\otimes 2}
  \qquad(\text{with }V\coloneqq R_S\langle S\rangle).
\]
{\red Furthermore,} upon reducing to a single variable everything is
polynomial size and polynomial time.

{\red Furthermore,} $\Gamma_{01}$ is given using a
``meta-2-cocycle $\rho^{ab}_c$ over $\Gamma_0$'': In
addition to $m^{ab}_c\to m^{ab}_{0c}$, there are $R_S$-linear
$m^{ab}_{1c}\colon\Gamma_1(S\sqcup\{a,b\})\to\Gamma_1(S\sqcup\{c\})$,
a meta-right-action
$\alpha^{ab}\colon\Gamma_1(S)\times\Gamma_0(S)\to\Gamma_1(S)$
$R_S$-linear in the first variable, and a
first order differential operator (over $R_S$)
$\rho^{ab}_c\colon\Gamma_0(S\sqcup\{a,b\})\to\Gamma_1(S\sqcup\{c\})$
such that
\[ (\zeta_0,\zeta_1)\act m^{ab}_c
  = \left(
    \zeta_0\act m^{ab}_{0c},
    (\zeta_1,\zeta_0)\act\alpha^{ab}\act m^{ab}_{1c}
      + \zeta_0\act\rho^{ab}_c
  \right)
\]

In \href{http://drorbn.net/AcademicPensieve/Projects/OneCo-1604/nb/MostGeneralR.pdf}{MostGeneralR.nb}:

\vskip 1mm\includegraphics[width=\columnwidth]{rule2.png}

\entry{151019d} Perhaps I should switch to a circuit algebra perspective, plus meta-monoid ops.

\entry{160612} Let $\frakg=\FL(x^1,x^2)/[x^1,x^2]=x^2$,
let $\frakg^\ast=\langle\phi_1,\phi_2\rangle$ with
$\phi_i(x^j)=\delta_i^j$, let $I\frakg=\frakg^\ast\rtimes\frakg$
so $[\phi_i,\phi_j]=[\phi_1,x^i]=0$ while $[x^1,\phi_2]=-\phi_2$ and
$[x^2,\phi_2]=\phi_1$.
Let $r=Id=\phi_1\otimes x^1+\phi_2\otimes
x^2\in\frakg^\ast\otimes\frakg\subset I\frakg\otimes I\frakg$.
Let $\calU=\calU(I\frakg)$,
degree-completed with respect to $\deg\phi_i=1$ and $\deg x^i=0$ (so
$\calU\equiv(\text{power series is 4 variables})$).
Then $R=\exp(r)\in\calU\otimes\calU$ satisfies CYBE, $\phi_ix^i$, $x^i\phi_i$, $\phi_1$
are central, $x^i\phi_i-\phi_ix^i=\phi_1$,
and $[x^j,\phi_i]=\delta_i^j\phi_1-\delta_1^j\phi_i$. $\phi_1$ is ``bracket-trace''.

\end{multicols}

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