Title: On the tensor semigroup of affine kac-moody lie algebras,
Abstract: In this paper, we are interested in the decomposition of the tensor product
of two representations of a symmetrizable Kac-Moody Lie algebra $\mathfrak g$.
Let $P\_+$ be the set of dominant integral weights. For $\lambda\in P\_+$ ,
$L(\lambda)$ denotes the irreducible, integrable, highest weight representation
of g with highest weight $\lambda$. Let $P\_{+,\mathbb Q}$ be the rational
convex cone generated by $P\_+$. Consider the tensor cone $\Gamma(\mathfrak g)
:= \{(\lambda\_1 ,\lambda\_2, \mu) $\in$ P\_{+,\mathbb Q}^3\,| \exists N
\textgreater{} 1 L(N\mu) \subset L(N \lambda\_1)\otimes L(N \lambda\_2)\}$. If
$\mathfrak g$ is finite dimensional, $\Gamma(\mathfrak g)$ is a polyhedral
convex cone described in 2006 by Belkale-Kumar by an explicit finite list of
inequalities. In general, $\Gamma(\mathfrak g)$ is nor polyhedral, nor closed.
In this article we describe the closure of $\Gamma(\mathfrak g)$ by an explicit
countable family of linear inequalities, when $\mathfrak g$ is untwisted
affine. This solves a Brown-Kumar's conjecture in this case. We also obtain
explicit saturation factors for the semigroup of triples $(\lambda\_1,
\lambda\_2 , \mu) $\in$ P\_+^3$ such that $L(\mu) $\subset$ L(\lambda\_1)
\otimes L(\lambda\_2)$. Note that even the existence of such saturation factors
is not obvious since the semigroup is not finitely generated. For example, in
type $A , we prove that any integer $d\geq 2$ is a saturation factor,
generalizing the case ${\tilde A}\_1$ shown by Brown-Kumar.