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.
Categories: math.RT