  The structure of half-BPS representations of psu$(2,2|4)$ leads to the
definition of a super-polynomial ring $\mathcal{R}(8|8)$ which admits a
realisation of psu$(2,2|4)$ in terms of differential operators on the
super-ring. The character of the half-BPS fundamental field representation
encodes the resolution of the representation in terms of an exact sequence of
modules of $\mathcal{R}(8|8)$. The half-BPS representation is realized by
quotienting the super-ring by a quadratic ideal, equivalently by setting to
zero certain quadratic polynomials in the generators of the super-ring. This
description of the half-BPS fundamental field irreducible representation of
psu$(2,2|4)$ in terms of a super-polynomial ring is an example of a more
general construction of lowest-weight representations of Lie (super-) algebras
using polynomial rings generated by a commuting subspace of the standard
raising operators, corresponding to positive roots of the Lie (super-) algebra.
We illustrate the construction using simple examples of representations of
su(3) and su(4). These results lead to the definition of a notion of quantum
mechanical emergence for oscillator realisations of symmetries, which is based
on ideals in the ring of polynomials in the creation operators.
