  A ray of photons, emitted from a laser source, is in a coherent state, where
macroscopic number of photons are degenerate in the same quantum state. The
coherent state has degrees of freedom for spin and orbital angular momentum,
which allow an arbitrary superposition state among orthogonal states with
varying their amplitudes and phases, described by a representation theory of
Lie algebra and Lie group. Here, we experimentally demonstrate that we can
construct generators of rotations for the quantum states of coherent photons,
simply by combining widely available optical components, such as half- and
quarter-wave plates and vortex lenses. We have found that a superposition state
between vortexed and no-vortex states is characterised by the motion of the
topological charge upon the rotation in the SU(3) states. We also realised
singlet and triplet states by combining rays of photons with orthogonal
polarisation states and vortexed states. This corresponds to realise an
effective SU(4) state and we have confirmed the projection to an
SU(2)$\times$SU(2) state upon passing through a polariser.
