On the Support of Weight Modules for Affine Kac-Moody-Algebras

An irreducible weight module of an affine Kac-Moody algebra $\mathfrak{g}$ is called dense if its support is equal to a coset in ${\mathfrak{h}}^{\ast }\mathrm{/}Q$. 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.