  An irreducible weight module of an affine Kac-Moody algebra $\mathfrak{g}$ is
called dense if its support is equal to a coset in $\mathfrak{h}^{*}/Q$.
Following a conjecture of V. Futorny about affine Kac-Moody algebras
$\mathfrak{g}$, an irreducible weight $\mathfrak{g}$-module is dense if and
only if it is cuspidal (i.e. not a quotient of an induced module). The
conjecture is confirmed for $\mathfrak{g}=A_{2}^{\left(1\right)}$,
$A_{3}^{\left(1\right)}$ and$A_{4}^{\left(1\right)}$ and a classification of
the supports of the irreducible weight $\mathfrak{g}$-modules obtained. For all
$A_{n}^{\left(1\right)}$ the problem is reduced to finding primitive elements
for only finitely many cases, all lying below a certain bound. For the
left-over finitely many cases an algorithm is proposed, which leads to the
solution of Futorny's conjecture for the cases $A_{2}^{\left(1\right)}$ and
$A_{3}^{\left(1\right)}$. Yet, the solution of the case
$A_{4}^{\left(1\right)}$ required additional combinatorics.
  For the proofs, a new category of hypoabelian Lie subalgebras,
pre-prosolvable subalgebras, and a subclass thereof, quasicone subalgebras, is
introduced and its tropical matrix algebra structure outlined.
