Representations of Quantum Affinizations and Fusion Product

By D. Hernandez. 2003

116 citations recorded by the source corpus.

AbstractIn this paper we study general quantum affinizations $\mathcal{U}_q(\widehat{\mathfrak{g}})$ of symmetrizable quantum Kac-Moody algebras and we develop their representation theory. We prove a triangular decomposition and we give a classication of (type 1) highest weight simple integrable representations analog to Drinfel'd-Chari-Presley one. A generalization of the q-characters morphism, introduced by Frenkel-Reshetikhin for quantum affine algebras, appears to be a powerful tool for this investigation. For a large class of quantum affinizations (including quantum affine algebras and quantum toroidal algebras), the combinatorics of q-characters give a ring structure * on the Grothendieck group $\text{Rep}(\mathcal{U}_q(\widehat{\mathfrak{g}}))$ of the integrable representations that we classified. We propose a new construction of tensor products in a larger category by using the Drinfel'd new coproduct (it cannot directly be used for $\text{Rep}(\mathcal{U}_q(\widehat{\mathfrak{g}}))$ because it involves infinite sums). In particular, we prove that * is a fusion product (a product of representations is a representation).

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/179bcfc1496b60af574fdf18a7967a1f1ab5e5cd

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Citations in the corpus

Yangians, quantum loop algebras and abelian difference equations (2013, methodology): “The treatment in [31, 32] readily carries over to the Yangian and allows them to extend their deﬁnition to the case of a symmetrizable Kac-Moody algebra.”
Quantum Geometry and Quiver Gauge Theories (2013, methodology): “5Here q 1 2 = q[49, 50, 134] and q 1 2 c = c[50] and hi,nq −c|n|/4 = ai,n|[121] 6Here ξ = z[121] = u −1 [49, 50] = z −1 [134] 7It is convenient to compute [logψ i (ξ), e ± j (ξ )] from (A.”
T-systems and Y-systems in integrable systems (2010, methodology): “The T-systems have been generalized by Hernandez [8] to the quantum affinizations of a wide class of quantum Kac–Moody algebras studied in [63, 73, 74, 75, 76, 77].”
The Kirillov-Reshetikhin conjecture and solutions of T-systems (2005, unknown): “Note that for m ∈ A and a ∈ C ∗ we have ui,a(m) = ui,a(τi(m)) and : ν(ω(m)) = μ(τi(m)) + ui(m)riα ∨ i /2 = μ(τi(m)) + ui(τi(m))riα ∨ i /2, or equivalently : μ(τi(m)) = ν(ω(m)) − αi(ν(ω(m)))α ∨ i /2 (see the definition of [13], section 5.”
Quantum toroidal algebras and their representations (2008, background): “These modules are not nitedimensional, but by using the same proof as for quantum anizations [ He3 ], they are integrable.”
On Minimal Affinizations of Representations of Quantum Groups (2006, methodology): “For example let us consider the type Bn, p (n ≥ 2, p ≥ 2) corresponding to the Cartan matrix (Ci, j )1≤i, j≤n defined as the Cartan matrix of type Bn except that we replace Cn,n−1 =− 2b yCn,n−1 =− p. Then one can prove exactly as for Lemma 5.6 that (an analog of Theorem 4.7 is proved by using [ H2 ,”
Affine Yangians and deformed double current algebras in type A (2007, unknown): “…motivation for studying the representation theory of these algebras is that we hope that it will be easier to understand, using classical methods, than the representation theory of quantum toroidal algebras, which is still quite mysterious - for some important results, see [16],[28, 29],[18, 19].”