  General dynamic properties like controllability and simulability of spin
systems, fermionic and bosonic systems are investigated in terms of symmetry.
Symmetries may be due to the interaction topology or due to the structure and
representation of the system and control Hamiltonians. In either case, they
obviously entail constants of motion. Conversely, the absence of symmetry
implies irreducibility and provides a convenient necessary condition for full
controllability much easier to assess than the well-established Lie-algebra
rank condition. We give a complete lattice of irreducible simple subalgebras of
su(2^n) for up to n=15 qubits. It complements the symmetry condition by
allowing for easy tests solving homogeneous linear equations to filter
irreducible unitary representations of other candidate algebras of classical
type as well as of exceptional types. --- The lattice of irreducible simple
subalgebras given also determines mutual simulability of dynamic systems of
spin or fermionic or bosonic nature. We illustrate how controlled quadratic
fermionic (and bosonic) systems can be simulated by spin systems and in certain
cases also vice versa.
