A study of saturated tensor cone for symmetrizable Kac-Moody algebras

Abstract

Let $\fg$ be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra $\fh$ and the Weyl group $W$. Let $P_{+}$ be the set of dominant integral weights. For $\hat{I}»\in P_{+}$, let $L(\hat{I}»)$ be the irreducible, integrable, highest weight representation of $\fg$ with highest weight $\hat{I}»$. For a positive integer $s$, define the {\em saturated tensor semigroup} as \begin{align*} Γ_s:= \{(λ_1, \dots, λ_s,μ)\in P_+^{s+1}: \exists\, N>1 \,\,\text{with}\,\, L(Nμ)\subset L(Nλ_1)\otimes \dots \otimes L(Nλ_s)\}. \end{align*} The aim of this paper is to begin a systematic study of $\hat{I}{“}_s$ in the infinite dimensional symmetrizable Kac-Moody case. In this paper, we produce a set of necessary inequalities satisfied by $\hat{I}{“}_s$. We further prove that any integer $d>0$ is a saturation factor for $A_1^{(1)}$ and 4 is a saturation factor for $A_2^{(2)}$.

30 pages