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{\LARGE\bf Cheat Sheet OneCo}\hfill
\parbox[b]{5in}{\tiny
  \null\hfill\url{http://drorbn.net/AcademicPensieve/2015-04/}
  \newline\null\hfill initiated 14/4/15;
  modified \today, \ampmtime
}

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\begin{multicols}{2}

{\bf Background.}

$d\exp$:\ \hfill
$\displaystyle \delta e^\gamma
  = e^\gamma\cdot\left(\delta\gamma\act\frac{1-e^{-\ad\gamma}}{\ad\gamma}\right)
  = \left(\delta\gamma\act\frac{e^{\ad\gamma}-1}{\ad\gamma}\right)\cdot e^\gamma
$

The differential of $\gamma=\bch(\alpha,\beta)$:
\newline\null\hfill
$\displaystyle \delta\gamma\act\frac{1-e^{-\ad\gamma}}{\ad\gamma}
  = \left(\delta\alpha\act\frac{1-e^{-\ad\alpha}}{\ad\alpha}\act e^{-\ad\beta}\right)
  + \left(\delta\beta\act\frac{1-e^{-\ad\beta}}{\ad\beta}\right)
$

{\bf Models.} $\bullet$ In $[x,y]=\delta x$, $xf(y)=f(y+\delta)x$. If $\delta^2=0$, $[x,f(y)] = \delta f'(y)$.

{\bf Deriving Gassner.} Locality,
$[a_{ij},a_{ik}] = 0$,
$[a_{ik},a_{jk}] = -[a_{ij},a_{jk}] = b_ja_{ik}-b_ia_{jk}$,
$[a_{ij},a_{ji}] = b_ia_{ji}-b_ja_{ij}$,
$b_i$ central.
Acts on $V = {\bbQ\llbracket b_i\rrbracket\langle x_i=a_{i\infty}\rangle}$ 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 $y_i=x_i/b_i$, $t_i=e^{b_i}$, get $[e^{\ad a_{ij}}]_{y_i,y_j} = \begin{pmatrix} 1 & 1-t_i \\ 0 & t_i \end{pmatrix}$.

\columnbreak

{\bf The $\calA^w/2D$ Adjoint representation.} $e^{\ad a_{ij}}$ acts by
\[ a_{kl} \mapsto a_{kl},\qquad a_{ik} \mapsto a_{ik}, \qquad a_{kj} \mapsto e^{-b_i}a_{kj} + \frac{b_k}{b_i}(1-e^{-b_i})a_{ij}, \]
\[ \text{and}\quad a_{jk} \mapsto e^{b_i}a_{jk} + \frac{b_j}{b_i}(1-e^{b_i})a_{ik}\qquad\text{also for $k=i$}. \]

An interpretation?

{\bf 2Dv.} $b$: bracket trace; $c$: cobracket trace; $\langle b,c\rangle=\delta\in\{0,1\}$.
Relations: locality,
$[a_{ij},a_{ik}] = c_ka_{ij}-c_ja_{ik}$,
$[a_{ik},a_{jk}]=b_ja_{ik}-b_ia_{jk}$,
$[a_{ij},a_{jk}] = (c_j-b_j)a_{ik}+b_ia_{jk}-c_ka_{ij}$,
$[a_{ij},a_{ji}] = ?$,
$[a_{ij},b_i] = -[a_{ij},b_j] = -[a_{ij},c_i] = [a_{ij},c_j] = \delta a_{ij}-b_ic_j$,
$[b_i,c_i]=0$.

$\displaystyle
  a_{ij}f = \left(f^\delta - \frac{b_ic_j}{\delta}(f^\delta-f)\right)a_{ij},
$\hfill$\displaystyle
  [a_{ij},f] = \left(1-\frac{b_ic_j}{\delta}\right)(f^\delta-f)a_{ij},
$

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

{\bf The primitivity condition.} $\displaystyle\ker\left( f+f^{ij}a_{ij} \mapsto \delta f + f^{ij}b_ic_j\right)$.

{\bf The OneCo Quotient} is $\delta c_i=c_jc_k=0$. Then $[a_{ij},f] = (\delta-b_ic_j) \left[(\partial_{b_i}-\partial_{b_j}-\partial_{c_i}+\partial_{c_j})f\right]_{c_i=0} \cdot a_{ij}$.

\end{multicols}

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