Given a field $k$ and two $k$-bialgebras $A$ 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
one forms the 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.
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 $D(A)$:
An example of an infinite dimensional analogue coming from Lie algebras is treated in
which partly refers to previous paper (which however neglects the issues related to completions)
Miscellaneous articles on Heisenberg doubles
R.M. Kashaev, Heisenberg double and the pentagon relation, St. Petersburg Math. J. 8 (1997) 585– 592 q-alg/9503005.
M. Kapranov, Heisenberg doubles and derived categories, J. Alg. 202, 712–744 (1998), q-alg/9701009.
F. Panaite, Doubles of (quasi) Hopf algebras and some examples of quantum groupoids and vertex groups related to them, math.QA/0101039
G. Militaru, Heisenberg double, pentagon equation, structure and classification of finite-dimensional Hopf algebras, J. London Math. Soc. (2) 69 (2004) 44–64 (doi).
A.M. Semikhatov, A Heisenberg double addition to the logarithmic Kazhdan–Lusztig duality, arXiv:0905.2215.
A.M. Semikhatov, Yetter–Drinfeld structures on Heisenberg doubles and chains, arXiv:0908.3105
Daniele Rosso, Alistair Savage, Twisted Heisenberg doubles, arxiv/1405.7889