  These notes describe some links between the group
$\mathrm{SL}_2(\mathbb{R})$, the Heisenberg group and hypercomplex
numbers---complex, dual and double numbers. Relations between quantum and
classical mechanics are clarified in this framework. In particular, classical
mechanics can be obtained as a theory with noncommutative observables and a
non-zero Planck constant if we replace complex numbers in quantum mechanics by
dual numbers. Our consideration is based on induced representations which are
build from complex-/dual-/double-valued characters. Dynamic equations, rules of
additions of probabilities, ladder operators and uncertainty relations are
discussed. Finally, we prove a Calder\'on--Vaillancourt-type norm estimation
for relative convolutions.
