On level-zero representation of quantized affine algebras

By Masaki Kashiwara. 2000

264 citations recorded by the source corpus.

We study the properties of level zero modules over quantized affine algebras. The proof of the conjecture on the cyclicity of tensor products by Akasaka and the present author is given. Several properties of modules generated by extremal vectors are proved. The weights of a module generated by an extremal vector are contained in the convex hull of the Weyl group orbit of the extremal weight. The universal extremal weight module with level zero fundamental weight as an extremal weight is irreducible, and isomorphic to the affinization of an irreducible finite-dimensional module.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/381f3560179173eb04c34b2ceee4324b2b54cd46

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In Algebra and Number Theory

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In Mathematical Physics

Citations in the corpus

Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions (2005, background): “The results of Kashiwara [24,25] imply that the Weyl modules are specializations (the q = 1 limit) of certain finite-dimensional quotients of the extremal weight modules for the quantum affine”
A categorical approach to Weyl modules (2009, methodology): “A with Uλ = 0. As in the proof of Theorem 3.7 we have V = U(g ⊗ A)Vλ and hence a short exact sequence 0 → K → Wλ AVλ → V → 0.”
Existence of Kirillov–Reshetikhin crystals for nonexceptional types (2007, background): “Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Perfect crystals of quantum affine Lie algebras, Duke Math.”
Quantum toroidal algebras and their representations (2008, background): “quantum ane algebras, certain tensor products of l-highest weight modules are lhighest weight (see [Cha2, Theorem 4] and [ Kash2 , Theorem 9.1]), but not all.”
Simple tensor products (2009, methodology): “That is why in our context the condition is translated as apa −1 r ∈ � Z q N . The statement in [ 24 ] involves representations W( ωi) ∼ Vi(a) for some a ∈ C ∗ computed in [4, Lemma 4.6] and [28, Remark 3.3]”