Duality for the quantum \(E(2)\) group
Let \(\mu\) be any number of \(]0,1[\). Consider the *-algebra \(A_0\) over \(\mathbb{C}\) with an identity, generated by a unitary element \(v\) and a normal element \(n\) satisfying \(vn= \mu nv\). This *-algebra can be made into a Hopf *-algebra when the comultiplication \(\Phi\) is defined by \(\Phi (v)= v\otimes v\) and \(\Phi (n)= v\otimes n+ n\otimes v^*\). The algebra \(A_0\) can be considered as a quantized version -- on the Hopf *-algebra level -- of the group \(E(2)\) of motions of the plane. By taking a suitable representation of the *-algebra \(A_0\) on a Hilbert space, one can associate a \(C^*\)-algebra \(A\) to it in a more or less canonical way and lift the comultiplication from the *-algebra \(A_0\) to the \(C^*\)-algebra \(A\). Thus, one obtains the quantum group \(E_\mu (2)\) on the \(C^*\)-level. On the other hand, consider the *-algebra \(B_0\) over \(\mathbb{C}\) with identity, generated by a selfadjoint invertible element \(a\) and a normal element \(b\) satisfying \(ab= \mu ba\). Again \(B_0\) can be made into a Hopf *-algebra when \(\Phi\) is defined by \(\Phi (a)= a\otimes a\) and \(\Phi (b)= a\otimes b+b \otimes a^{-1}\). Also here it is possible to associate a \(C^*\)-algebra \(B\) and to lift the comultiplication from \(B_0\) to \(B\). One can show that \((A_0, B_0)\) is a dual pair of Hopf *-algebras and that the pairing is given by \[ \begin{alignedat}{2} \langle v,a \rangle &= \mu^{1\over 2}, &\qquad \langle n,a \rangle &= 0,\\ \langle v,b \rangle &= 0, &\qquad \langle n,b \rangle &=1. \end{alignedat} \] The pairing is non-degenerate [see e.g. the first author, Lect. Notes K. U. Leuven (1991) (first version), Bull. Lond. Math. Soc. 25, 209-230 (1993; Zbl 0796.16034) (second version)]. It was expected that also this pairing could be lifted to the corresponding \(C^*\)-algebras in a canonical way. Indeed, it has been shown in [\textit{S. L. Woronowicz}, Lett. Math. Phys. 23, 251-263 (1991; Zbl 0752.17017)] that the second pair \((B, \Phi)\) can be identified with the Pontryagin dual \(\widehat {E}_\mu (2)\) of \(E_\mu (2)\). In this paper, it is shown that also the dual of \(\widehat {E}_\mu (2)\) is again \(E_\mu (2)\). This result establishes a nice example of `biduality' in the case of a nontrivial noncompact, nondiscrete quantum group.
scientific article; zbMATH DE number 950231