nLab
Heisenberg double

Given a field kk and two kk-bialgebras AA and BB with Hopf pairing <,>:ABk\lt, \gt : A\otimes B\to k, one defines a left Hopf action \blacktriangleright of BB on AA by formulas

ba=<b,a (2)>a (1)=(<,>id)(bτΔ A(a)) 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”) ABA\sharp B associated to the Hopf action \blacktriangleright.

For example if A=S(V)A = S(V) is the symmetric (Hopf) algebra on a finite-dimensional vector space VV, and BB its algebraic dual (S(V)) *S^(V *)(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 BB is just the finite dual of S(V)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 *AA^*\sharp A has a structure of a Hopf algebroid over A *A^*; moreover A *A^* can be replaced by any module algebra over the Drinfel'd double D(A)D(A):

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)

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.
  • G. 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

  • M. Kapranov, Heisenberg doubles and derived categories, J. Alg. 202, 712–744 (1998), 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 October 19, 2019 at 09:55:51. See the history of this page for a list of all contributions to it.