Unique factorization of tensor products for Kac–Moody algebras

On tensor products of irreducible integrable representations

“…One of his motivations was to show that given that End(V ) = End(W ) for finite-dimensional irreducible modules V, W , using the natural isomorphism End(V ) ∼ = V ⊗ V * , one has that either V ∼ = W or V ∼ = W * . A more direct and simpler proof of Rajan's theorem is obtained by the second author and Viswanath in [12] and they also obtained a natural generalization of Rajan's theorem to Kac-Moody algebras setting. A natural category of representations to consider for Kac-Moody algebras is the category O int , whose objects are integrable g-modules in category O, since both proofs of [11,12] heavily use the Kac-Weyl character formula.…”

“…A more direct and simpler proof of Rajan's theorem is obtained by the second author and Viswanath in [12] and they also obtained a natural generalization of Rajan's theorem to Kac-Moody algebras setting. A natural category of representations to consider for Kac-Moody algebras is the category O int , whose objects are integrable g-modules in category O, since both proofs of [11,12] heavily use the Kac-Weyl character formula.…”

Spectral Characters of a Class of Integrable Representations of Toroidal Lie Algebras

“…Using the definition of the action of T (g) on L(X π ) it follows that if χ is an isomorphism, then the g af f module generated by v Mi is isomorphic to the g af f -module generated by v s b (Mi) . Hence by [VV,Theorem 3,Lemma 1] for every M ∈ supp(π), X(π(M )) is isomorphic to X(π ′ (s b (M ))) as a g f in ⊗ C[t 1 , t −1 1 ] ⊕ CK 1 -module. As a consequence of conditions (i) and (ii), G π = G π ′ whenever X g π is isomorphic to X g ′ π ′ implying that g, g ′ ∈ Z k−1 /G π .…”

Chromatic polynomials of graphs from Kac–Moody algebras

“…Let I := {w ∈ W \{e} : I(w) is an independent set}. The following lemma is a special case of lemma 2 of [11].…”

“…Theorem 1.1 and its corollaries are proved in section 2. The key ingredient in the proof is Lemma 2.3 below, which is a special case of a result proved in [11] in the context of unique factorization of tensor products for Kac-Moody algebras. In fact, it was the occurrence of the deletion-contraction recurrence in [11] that suggested a possible connection of those ideas to chromatic polynomials.…”