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\begin{document}
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\parbox[b]{4in}{
  {\small Dror Bar-Natan: Academic Pensieve: 2013-12:}
  \newline
  {\Large\bf Cheat Sheet \v{S}evera Quantization}
}
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  \null\hfill\sheeturl
  \newline\null\hfill initiated 3 December 2013; modified \today
}

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\begin{multicols}{2}

{\bf \v{S}evera's construction.} (maintained at monoblog)

Given a Braided Monoidal Category (BMC) $\calD$ {\magenta (with Manin $(\partial, \frakg, \frakg^\star)$, set $\calD\coloneqq\calU(\partial)-\Mod^\Phi$)}, given a co-braided co-algebra $(M,\Delta\colon M\to M^2,\epsilon\colon M\to 1_\calD)$ {\magenta ($M\coloneqq\calU(\frakg)=\calU(\partial)/\calU(\partial)\frakg^*$)}, given a second BMC $\calC$ {\magenta (Vect)}, a functor $F\colon\calD\to\calC$  {\magenta ($F(X)\coloneqq X/\frakg X$)} and a comonoidal structure $c$ (namely a natural  $c_{X,Y}\colon F(XY)\to F(X)F(Y)$ and $c_1\colon F(1_\calD)\to 1_\calC$ respecting the braiding and associativity) such that
\[ \begin{CD}F(XMY)@>F(1\Delta 1)>>F(XMMY)@>c_{XM,MY}>>F(XM)F(MY)\end{CD} \]
\[ \text{and}\quad\begin{CD}F(M)@>F(\epsilon)>>F(1_\calD)@>c_1>>1_\calC\end{CD} \]
\vskip 2mm
are isomorphisms {\magenta(the clear $c_{X,Y}\colon XY/\frakg(XY)\to(X/\frakg X)(Y/\frakg Y)$)}, construct a Hopf algebra structure on $H\coloneqq F(M^2)$:
\[ \Delta_H\colon\
  \begin{CD}
    F(M^2)@>F(\Delta\Delta)>>F(M^4)@>F(1R1)>>F(M^4)@>c_{M,M}>>F(M^2)^2,
  \end{CD}
\]
\[ m_H\colon\qquad
  \begin{CD}
    F(M^2)^2 @<c_{M^2,M^2}\circ F(1\Delta 1)<\raisebox{2mm}{\Large $\sim$}< F(M^3) @>F(1\epsilon 1)>> F(M^2),
  \end{CD}
\]
\[ S_H\colon\qquad\qquad
  \begin{CD}
    F(M^2) @>F(R)>> F(M^2).
  \end{CD}
\]

Set also $G\colon X\mapsto F(MX)$ {\magenta($G\colon X\mapsto\frac{\calU(\frakg)X}{\frakg(\calU(\frakg)X)}$)}, ``The Twist''.

{\bf Questions. }
$\bullet$ Is $H$ the symmetry algebra of something?

$\bullet$ In the non-quasi case, can we reconstruct $\calU(\frakg)$ from the category of $\partial$-modules?

$\bullet$ In the abstract context, what is the relation between $H$ and $M$?

$\bullet$ How does this restrict to AT/AET in the commutative case?

\end{multicols}


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{\bf Tannakian reconstruction.} (maintained at Confessions)

--- Given an algebra $A$ let $\calD\coloneqq A-\text{Mod}$ (projective (?) left $A$-modules), let $\calC\coloneqq{\text{Vect}}$ and $G\colon\calD\to\calC$ be the forGetful functor. Then $A\simeq\End(G)$ by
\[ a\in A\mapsto(\text{the action of $a$ on any $X\in\calD$}), \]
\[ \{a_X\colon G(X)\to G(X)\}_{X\in\calD} \mapsto a_A(1)\in A. \]

--- Given a monoidal $\calD$ and an exact $G\colon\calD\to\calC\eqqcolon\text{Vect}$ with a natural isomorphism $\alpha_{X,Y}\colon G(X)G(Y)\to G(XY)$, there is a Hopf algebra structure on $H\coloneqq\End(G)$: product is composition, coproduct $\Delta\colon H\to H^2 = \End(G^2\colon\calD\times\calD\to\calC)$ by
\[ (h_X)_{X\in\calD}\mapsto\left(
  (X,Y)\mapsto \alpha_{X,Y}\act h_{XY}\act \alpha_{X,Y}^{-1} \in \End(G(X)G(Y))
\right). \]

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