Twisted Heisenberg doubles. (Q2346845)

Let \(\Lambda\) be the monoid of \(r\)-th powers of the monoid of non-negative integers and \(\chi=(\chi',\chi'')\) a pair of biadditive maps from \(\Lambda\times\Lambda\to\mathbb Z\). Let \(q\) be an invertible element of the base commutative ring \(k\). \(\Lambda\)-graded Hopf algebra \(H\) is a twisted \((q,\chi)\)-Hopf algebra if comultiplication \(\Delta\) induces an algebra homomorphism \(H\to(H\otimes H)_\chi\) where \((H\otimes H)_\chi\) is a specific deformation of the algebra \(H\otimes H\) using powers of \(q\) depending on \(\chi\). A pair \((H^+,H^-)\) of twisted Hopf algebras is \((c,\gamma)\)-dual if there exists a compatible pairing \(H^-\times H^+\to k\). For this pair the authors introduce a twisted Heisenberg double as \(k\)-modules \(H^+\otimes H^-\) with deformed multiplication involving powers of the parameter \(q\). It is shown that there exists a Fock space representation of this Heisenberg double on \(H^+\). Some properties of this representation are established. It is shown that quantum Weyl algebra, quantum Heisenberg algebra, lattice Heisenberg algebra are particular examples of the general construction.