  We address the question of identifying degrees of freedom for quantum
systems. Typically, quasi-particle descriptions of correlated matter are based
upon the canonical algebras of bosons or fermions. Here we highlight that a
special class of non-canonical algebras also offer useful quantum degrees of
freedom, allowing for the development of quasi-particle descriptions which go
beyond the weakly correlated paradigm. We give a broad overview of such
algebras for spin, electron and local moment systems, and outline important
test problems upon which to develop the framework.
