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 $\blacktriangleright$. If $H$ is finite-dimensional then the Heisenberg double has a structure of a scalar extension bialgebroid.
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, and has some expositional errors)
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.
Miscellaneous articles on Heisenberg doubles
F. Panaite, Doubles of (quasi) Hopf algebras and some examples of quantum groupoids and vertex groups related to them, math.QA/0101039
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
There are some generalizations or Heisenberg doubles in different setups
Mikhail Kapranov, Heisenberg doubles and derived categories, J. Alg. 202, 712–744 (1998), arXiv:q-alg/9701009.
Daniele Rosso, Alistair Savage, Twisted Heisenberg doubles, arxiv/1405.7889
Robert Laugwitz, Braided Drinfeld and Heisenberg doubles, J. Pure Appl. Alg. 219:10 (2015) 4541-4596 doi
Last revised on August 12, 2023 at 12:53:51. See the history of this page for a list of all contributions to it.