nLab
Heisenberg double

Given two bialgebras $A$ and $B$ in Hopf pairing $<,>$ (i.e. making comultiplication on one transposed to multiplication to another and viceversa), one define a left Hopf action $▹$ of $B$ on $A$ by formulas

$$b▹a=\sum <b,a_{(2)}>a_{(1)}=(<,>\otimes id)(b\otimes \tau {\Delta }_A(a))$$b\triangleright 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”) $A♯B$ associated to the Hopf action $▹$.

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.

• J.-H. 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 $A^{*}♯A$ 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

• 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

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