Drinfeld double and Heisenberg double of Super Hopf algebras
This note aimed to get explicit multiplication formulas for the Drinfeld double and the Heisenberg double.
I draw string diagram and use GPT, but there may be some mistakes. If you noticed, please let me know!
Conventions
Let $U=(U,M,1,\Delta,\varepsilon,S)$ be a Super Hopf algebra ($\mathbb{Z}_2$-graded) with bijective antipode $S$.
- Sweedler notation is used with implicit summation.
- $\Delta^{(2)}=(\Delta\otimes\mathrm{id})\Delta$.
- On $U^*$, $\Delta_{\mathrm{cop}}$ denotes the opposite coproduct.
- The super swap is$ \tau(x\otimes y)=(-1)^{|x||y|}\,y\otimes x . $
- Pairing $\langle\cdot,\cdot\rangle:U^*\otimes U\to\Bbb k$
Evaluation and Coevaluation
Left evaluation:$
\overleftarrow{\mathrm{ev}}:U^*\otimes U\to\Bbb k:f\otimes x\mapsto\langle f,x\rangle .
$
Right evaluation:$
\overrightarrow{\mathrm{ev}}:U\otimes U^*\to\Bbb k:x\otimes f\mapsto(-1)^{|x||f|}\langle f,x\rangle .
$
Left coevaluation:$
\overleftarrow{\mathrm{coev}}:\Bbb k\to U\otimes U^*:1\mapsto\sum_i e_i\otimes e^i .
$
Right coevaluation:$
\overrightarrow{\mathrm{coev}}:\Bbb k\to U^*\otimes U:1\mapsto\sum_i (-1)^{|e_i|} e^i\otimes e_i .
$
Here $\{e_i\}$ is a homogeneous basis of $U$ with dual basis $\{e^i\}$ of $U^*$.
Definitions
The Drinfeld double $D(U)$ is the vector space
$$
D(U)=U^{*\mathrm{cop}}\bowtie U=(U^{*\mathrm{cop}}\otimes U,M_{D(U)},1_{D(U))})
$$
equipped with multiplication
$$
\begin{aligned}
M_{D(U)}(f\bowtie u)\otimes(g\bowtie v)
&=
(-1)^{
|g_{(3)}||u|
+|g_{(1)}||g_{(2)}|
+|u_{(3)}||g_{(1)}|
+|u_{(2)}||g_{(2)}|
}
\\
&\qquad
\langle g_{(3)},u_{(1)}\rangle
\langle g_{(1)},S^{-1}(u_{(3)})\rangle\;
(f g_{(2)})\bowtie (u_{(2)}v),
\end{aligned}
$$
and unit $1_{D(U)}=\varepsilon\bowtie 1$
Here, $\Delta^{(2)}(u)=u_{(1)}\otimes u_{(2)}\otimes u_{(3)}$ and $M^{*(2)}(g)=g_{(3)}\otimes g_{(2)}\otimes g_{(1)}$ is a comultiplication.
Note that $\Delta(g)=g_{(1)}\otimes g_{(2)}$ in $U^*$
The Heisenberg double $\mathcal{H}(U)$ is the vector space
$$
\mathcal{H}(U)=(U^*\# U,M_{H(U)},1_{H(U)})
$$
equipped with multiplication
$$
\begin{aligned}
M_{H(U)}(f\# u)\otimes(g\# v)
&=
(-1)^{|u_{(2)}||g_{(2)}|+|u_{(1)}||g_{(2)}|+|u_{(2)}||g_{(1)}|}\langle g_{(2)},u_{(1)}\rangle\;
(f g_{(1)})\# (u_{(2)}v),
\end{aligned}
$$
and unit $1_{H(U)}=\varepsilon\#1$
Drinfeld Double
We compute the product in the Drinfeld double starting from
$f\otimes u\otimes g\otimes v$.
$$
\begin{aligned}
f\otimes u\otimes g\otimes v
&\;\overset{\Delta^{(2)}\otimes\Delta^{(2)}_{\mathrm{cop}}}{\longrightarrow}\;
f\otimes
u_{(1)}\otimes u_{(2)}\otimes u_{(3)}
\otimes
g_{(3)}\otimes g_{(2)}\otimes g_{(1)}
\otimes v
\\
&\;\overset{S^{-1}}{\longrightarrow}\;
f\otimes
u_{(1)}\otimes u_{(2)}\otimes S^{-1}(u_{(3)})
\otimes
g_{(3)}\otimes g_{(2)}\otimes g_{(1)}
\otimes v
\\
&\;\overset{\tau}{\longrightarrow}\;
(-1)^{|g_{(3)}|(|u_{(2)}|+|u_{(3)}|)}
f\otimes
u_{(1)}\otimes g_{(3)}\otimes
u_{(2)}\otimes S^{-1}(u_{(3)})
\otimes g_{(2)}\otimes g_{(1)}\otimes v
\\
&\;\overset{\overrightarrow{\mathrm{ev}}}{\longrightarrow}\;
(-1)^{|g_{(3)}|(|u_{(2)}|+|u_{(3)}|)+|u_{(1)}||g_{(3)}|}
\langle g_{(3)},u_{(1)}\rangle\;
f\otimes
u_{(2)}\otimes S^{-1}(u_{(3)})
\otimes g_{(2)}\otimes g_{(1)}\otimes v
\\
&\;\overset{\tau}{\longrightarrow}\;
(-1)^{\cdots+|g_{(1)}||g_{(2)}|}
\langle g_{(3)},u_{(1)}\rangle\;
f\otimes
u_{(2)}\otimes S^{-1}(u_{(3)})
\otimes g_{(1)}\otimes g_{(2)}\otimes v
\\
&\;\overset{\tau}{\longrightarrow}\;
(-1)^{\cdots+|u_{(3)}||g_{(1)}|}
\langle g_{(3)},u_{(1)}\rangle\;
f\otimes
u_{(2)}\otimes g_{(1)}\otimes S^{-1}(u_{(3)})
\otimes g_{(2)}\otimes v
\\
&\;\overset{\overrightarrow{\mathrm{ev}}}{\longrightarrow}\;
(-1)^{\cdots+|u_{(3)}||g_{(1)}|}
\langle g_{(3)},u_{(1)}\rangle
\langle g_{(1)},S^{-1}(u_{(3)})\rangle\;
f\otimes u_{(2)}\otimes g_{(2)}\otimes v
\\
&\;\overset{\tau}{\longrightarrow}\;
(-1)^{\cdots+|u_{(2)}||g_{(2)}|}
\langle g_{(3)},u_{(1)}\rangle
\langle g_{(1)},S^{-1}(u_{(3)})\rangle\;
f\otimes g_{(2)}\otimes u_{(2)}\otimes v
\\
&\;\overset{M\otimes M}{\longrightarrow}\;
(-1)^{
|g_{(3)}||u_{(2)}|+|g_{(3)}||u_{(3)}|
+|u_{(1)}||g_{(3)}|
+|g_{(1)}||g_{(2)}|
+|u_{(3)}||g_{(1)}|
+|u_{(2)}||g_{(2)}|
}
\\
&\hspace{5em}\langle g_{(3)},u_{(1)}\rangle
\langle g_{(1)},S^{-1}(u_{(3)})\rangle\;
(f g_{(2)})\otimes (u_{(2)}v)
\\
&=
(-1)^{
|g_{(3)}||u|
+|g_{(1)}||g_{(2)}|
+|u_{(3)}||g_{(1)}|
+|u_{(2)}||g_{(2)}|
}
\\
&\hspace{5em}\langle g_{(3)},u_{(1)}\rangle
\langle g_{(1)},S^{-1}(u_{(3)})\rangle\;
(f g_{(2)})\otimes (u_{(2)}v).
\end{aligned}
$$
This is the Drinfeld double multiplication written purely as a composition of
$\Delta$, $\Delta_{\mathrm{cop}}$, $\tau$, $\overrightarrow{\mathrm{ev}}$, $S^{-1}$, and $M$.
Heisenberg Double
We next record the corresponding computation for the Heisenberg double.
$$
\begin{aligned}
f\otimes u\otimes g\otimes v
&\;\overset{\Delta\otimes\Delta^{\mathrm{op}}}{\longrightarrow}\;
f\otimes
u_{(1)}\otimes u_{(2)}
\otimes
g_{(2)}\otimes g_{(1)}
\otimes v
\\
&\;\overset{\tau}{\longrightarrow}\;
(-1)^{|u_{(2)}||g_{(2)}|}
f\otimes
u_{(1)}\otimes g_{(2)}\otimes u_{(2)}
\otimes g_{(1)}\otimes v
\\
&\;\overset{\overrightarrow{\mathrm{ev}}}{\longrightarrow}\;
(-1)^{|u_{(2)}||g_{(2)}|+|u_{(1)}||g_{(2)}|}
\langle g_{(2)},u_{(1)}\rangle\;
f\otimes u_{(2)}\otimes g_{(1)}\otimes v
\\
&\;\overset{\tau}{\longrightarrow}\;
(-1)^{|u_{(2)}||g_{(2)}|+|u_{(1)}||g_{(2)}|+|u_{(2)}||g_{(1)}|}
\langle g_{(2)},u_{(1)}\rangle\;
f\otimes g_{(1)}\otimes u_{(2)}\otimes v
\\
&\;\overset{M\otimes M}{\longrightarrow}\;
(-1)^{|u_{(2)}||g_{(2)}|+|u_{(1)}||g_{(2)}|+|u_{(2)}||g_{(1)}|}
\langle g_{(2)},u_{(1)}\rangle\;
(f g_{(1)})\otimes (u_{(2)}v).
\end{aligned}
$$
String diagram for multiplication
