Quantum co-adjoint orbits of MD\(_4\)-groups. (Q1862713)

Using the \(\star\)-product on co-adjoint orbits of the MD\(_4\)-groups the author obtains quantum co-adjoint orbits via Fedosov deformation quantization. MD-groups were introduced by Do Ngoc Diep. They are solvable Lie groups whose co-adjoint orbits are either \(0\)-dimensional or maximal. If their corresponding Lie algebra is \(4\)-dimensional then they are called MD\(_4\) groups; these were classified by Le Anh Vu. The author of the present paper relies on this classification to construct \(\star\)-products on the co-adjoint orbits. He obtains the full list of corresponding irreducible unitary representation of the exponential MD\(_4\)-groups. Of interest is the case of the groups \(G_{4,2,3(\frac{\pi}{2})}\), \(G_{4,2,4}\), and \(G_{4,4,1}\), which are neither exponential nor nilpotent. The author obtains explicit formulas for these, too.