Tensor product stabilization in Kac–Moody algebras

We consider a large class of series of symmetrizable Kac–Moody algebras (generically denoted Xn). This includes the classical series An as well as others like En whose members are of Indefinite type. The focus is to analyze the behavior of representations in the limit n→∞. Motivated by the classical theory of An=sln+1C, we consider tensor product decompositions of irreducible highest weight representations of Xn and study how these vary with n. The notion of “double-headed” dominant weights is introduced. For such weights, we show that tensor product decompositions in Xn do stabilize, generalizing the classical results for An. The main tool used is Littelmann's celebrated path model. One can also use the stable multiplicities as structure constants to define a multiplication operation on a suitable space. We define this so-called stable representation ring and show that the multiplication operation is associative.