nLab Heisenberg double

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

$$b\blacktriangleright a = \sum \lt b, a_{(2)}\gt a_{(1)}= (\lt,\gt \otimes \id)(b\otimes \tau\Delta_A(a))$$

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.

• Jiang-Hua Lu, On the Drinfeld double and the Heisenberg double of a Hopf algebra, Duke Math. J. 74 (1994) 763–776.

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 $D(A)$:

• Jiang-Hua Lu, Hopf algebroids and quantum groupoids, Int. J. Math. 7, 1 (1996) pp. 47-70, q-alg/9505024, MR95e:16037, doi

An example of an infinite dimensional analogue coming from Lie algebras is treated in

• S. Meljanac, Z. Škoda, Lie algebra type noncommutative phase spaces are Hopf algebroids, arxiv:1409.8188

which partly refers to previous paper (which however neglects the issues related to completions, and has some expositional errors)

• Zoran Škoda, Heisenberg double versus deformed derivatives, Int. J. of Modern Physics A 26, Nos. 27 & 28 (2011) 4845–4854, arXiv:0909.3769, doi

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.

• R.M. Kashaev, Heisenberg double and the pentagon relation, St. Petersburg Math. J. 8 (1997) 585–592 q-alg/9503005.
• Gigel Militaru, Heisenberg double, pentagon equation, structure and classification of finite-dimensional Hopf algebras, J. London Math. Soc. (2) 69 (2004) 44–64 (doi).

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

category: algebra