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\begin{document}

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\section*{Heisenberg double}

Given a field $k$ and two $k$-[[bialgebra]]s $A$ and $B$ with [[Hopf pairing]] $\lt, \gt : A\otimes B\to k$, one defines a left [[Hopf action]] $\blacktriangleright$ of $B$ on $A$ by formulas

\begin{displaymath}
b\blacktriangleright a = \sum \lt b, a_{(2)}\gt a_{(1)}= (\lt,\gt \otimes \id)(b\otimes \tau\Delta_A(a))
\end{displaymath}
one forms the \textbf{Heisenberg double} corresponding to these data as the crossed product algebra (``smash product'') $A\sharp B$ associated to the Hopf action $\blacktriangleright$.

For example if $A = S(V)$ is the symmetric (Hopf) algebra on a finite-dimensional vector space $V$, and $B$ its algebraic dual $(S(V))^*\cong \hat{S}(V^*)$, considered as its dual topological Hopf algebra, the result is the [[Weyl algebra]] of [[regular differential operators]], completed with respect to the filtration corresponding to the degree of differential operator. If $B$ is just the finite dual of $S(V)$ which is a usual Hopf algebra, then there is no completion, of course.

\begin{itemize}%
\item J.-H. Lu, \emph{On the Drinfeld double and the Heisenberg double of a Hopf algebra}, Duke Math. J. \textbf{74} (1994) 763--776.

\end{itemize}
In the following paper there is an example showing that the Heisenberg double $A^*\sharp A$ has a structure of a Hopf algebroid over $A^*$; moreover $A^*$ can be replaced by any module algebra over the [[Drinfel'd double]] $D(A)$:

\begin{itemize}%
\item Jiang-Hua Lu, \emph{Hopf algebroids and quantum groupoids}, Int. J. Math. \textbf{7}, 1 (1996) pp. 47-70, \href{http://arxiv.org/abs/q-alg/9505024}{q-alg/9505024}, \href{http://www.ams.org/mathscinet-getitem?mr=95e:16037}{MR95e:16037}, \href{http://dx.doi.org/10.1142/S0129167X96000050}{doi}

\end{itemize}
An example of an infinite dimensional analogue coming from Lie algebras is treated in

\begin{itemize}%
\item S. Meljanac, Z. \v{S}koda, \emph{Lie algebra type noncommutative phase spaces are Hopf algebroids}, \href{http://arxiv.org/abs/1409.8188}{arxiv/1409.8188}

\end{itemize}
which partly refers to previous paper (which however neglects the issues related to completions)

\begin{itemize}%
\item [[Zoran Škoda]], \emph{Heisenberg double versus deformed derivatives}, Int. J. of Modern Physics A \textbf{26}, Nos. 27 \& 28 (2011) 4845--4854, \href{http://arxiv.org/abs/0909.3769}{arXiv:0909.3769}, \href{http://dx.doi.org/10.1142/S0217751X11054772}{doi}

\end{itemize}
The canonical element in the Heisenberg double satisfies a [[pentagon relation]], which is a version of pentagon relation for [[multiplicative unitaries]] of Baaj-Skandalis. Kashaev has explained the pentagon relations for quantum [[dilogarithm]] as coming from the pentagon for the canonical element in the double.

\begin{itemize}%
\item R.M. Kashaev, \emph{Heisenberg double and the pentagon relation}, St. Petersburg Math. J. 8 (1997) 585-- 592 \href{http://arxiv.org/abs/q-alg/9503005}{q-alg/9503005}.
\item G. Militaru, \emph{Heisenberg double, pentagon equation, structure and classification of finite-dimensional Hopf algebras}, J. London Math. Soc. (2) 69 (2004) 44--64 (\href{http://dx.doi.org/10.1112/S0024610703004897}{doi}).

\end{itemize}
Miscellaneous articles on Heisenberg doubles

\begin{itemize}%
\item F. Panaite, \emph{Doubles of (quasi) Hopf algebras and some examples of quantum groupoids and vertex groups related to them}, \href{http://arxiv.org/abs/math/0101039}{math.QA/0101039}


\item A.M. Semikhatov, \emph{A Heisenberg double addition to the logarithmic Kazhdan--Lusztig duality}, \href{http://arxiv.org/abs/0905.2215}{arXiv:0905.2215}.


\item A.M. Semikhatov, \emph{Yetter--Drinfeld structures on Heisenberg doubles and chains}, \href{http://arxiv.org/abs/0908.3105}{arXiv:0908.3105}



\end{itemize}
There are some generalizations or Heisenberg doubles in different setups

\begin{itemize}%
\item M. Kapranov, \emph{Heisenberg doubles and derived categories}, J. Alg. 202, 712--744 (1998), \href{http://arxiv.org/abs/q-alg/9701009}{q-alg/9701009}.


\item Daniele Rosso, Alistair Savage, \emph{Twisted Heisenberg doubles}, \href{http://arxiv.org/abs/1405.7889}{arxiv/1405.7889}


\item Robert Laugwitz, \emph{Braided Drinfeld and Heisenberg doubles}, J. Pure Appl. Alg. \textbf{219}:10 (2015) 4541-4596 \href{https://doi.org/10.1016/j.jpaa.2015.02.031}{doi}



\end{itemize}


\end{document}