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Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
4
votes
0
answers
170
views
Does a dual basis for $U_h(\mathfrak{sl}_2(\mathbb{C}))$ exist?
Let $\mathcal{F}_h(\operatorname{SL}_2(\mathbb{C}))$ be the $\mathbb{C}[[h]]$-algebra generated by $a, b, c, d$ subject to the following relations:
\begin{align*}
& ac = e^{-h}ca, \quad bd = e^{-h}db, …
5
votes
1
answer
151
views
Explicit correspondence between classical double and quantum double
Proposition 12.3 of Etingof and Schiffmann's "Lectures on Quantum Groups" states the following claim.
Proposition 12.3. Let $H$ be a quantized enveloping algebra and let $\mathfrak{g}$ be the
quasi-cl …
2
votes
0
answers
119
views
Is there an explicit description of a gauge transformation $F$ such that $U_{\hbar}(\mathfra...
Let $\mathfrak{g}$ be a semisimple Lie algebra, let $t$ be its canonical 2-tensor, and let $\Phi_{KZ}$ be a Drinfeld associator.When $R_{KZ}=e^{\hbar t/2}$, $(U(\mathfrak{g})[[\hbar]],\Phi_{KZ},R_{KZ} …
2
votes
0
answers
80
views
The dual of elements $E$, $F$, and $H$ of $U_h(\mathfrak{sl}_2)$ corresponds to which elemen...
$\newcommand{\sl}{\mathfrak{sl}}\DeclareMathOperator\SL{SL}$Let $U_h(\sl_2)$ be the quantized universal enveloping algebra of $\sl_2(\mathbb{C})$ and $F_h(\SL_2)$ be the quantized function algebra of …
4
votes
0
answers
111
views
Explicit construction of the Drinfeld double for quasi-bialgebras
Let $A$ be a quasi-bialgebra over a field $k$ and $A$-$\mathrm{mod}$ be the category of finite dimensional left $A$-modules. Since $A$-$\mathrm{mod}$ is a monoidal category, its Drinfel’d center $\ma …
3
votes
1
answer
312
views
Is there a definition of Heisenberg double for quasi-Hopf algebras?
$\newcommand{\dmod}{\text{-}\mathrm{mod}}$Let $A$ be a finite-dimensional quasi-Hopf algebra over a field $k$ and $A\dmod$ be the category of finite dimensional left $A$-modules.
Since $A\dmod$ is a r …