On Tensor Products of a Minimal Affinization With an Extreme Kirillov-Reshetikhin Module for Type A

Moura, Adriano; Pereira, Fernanda de Andrade

doi:10.1007/s00031-017-9462-5

Cited by 5 publications

(8 citation statements)

References 27 publications

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“…From type A we need tensor products of a general minimal affinization with a Kirillov-Reshetikhin module supported on an extremal node. This was exactly the topic of our first paper in this series, [26], where we described a necessary and sufficient condition for the irreducibility of such tensor products. In fact, this criterion for irreducibility is half of the main result of [26].…”

Section: Introductionmentioning

confidence: 87%

“…This was exactly the topic of our first paper in this series, [26], where we described a necessary and sufficient condition for the irreducibility of such tensor products. In fact, this criterion for irreducibility is half of the main result of [26]. The other half will be crucial in the proof of the final classification as it also provides a tool to compare certain affinizations or to show that they are not comparable.…”

Section: Introductionmentioning

confidence: 87%

“…is reducible if and only if there exist s ∈ Z, j ′ ∈ supp(λ), and η ′ ∈ Z >0 such that b = aq s and either one of the following options hold: We will also need a criterion that guarantees the irreducibility of tensor products of KR modules associated to nodes in ∂I when g is of type D. To deduce it, we begin by recalling some facts about duality (a slightly more complete review was given in [26,Section 4.1]). For any two finite-dimensional U q (g)-modules V and W , we have…”

Section: It Follows From the Cases Already Considered Thatmentioning

confidence: 99%

“…The proof of Theorem 2.4.4, given in Section 4.5, relies on tensor product results from [3,26], which will be reviewd in Section 3.4, and on the computation of certain outer multiplicities for preminimal affinizations satisfying (2.4.4) mo(ω I λ ) = 2 and λ(h i * ) = 0, where λ = wt(ω), which we now explain. Thus, let ω ∈ P + be preminimal satisfying (2.4.4), let k ∈ ∂I be such that (2.4.5) ω I λ is not k λ -minimal, and set V = V q (ω).…”

Section: 4mentioning

confidence: 99%

“…This is the second paper of a series based on a project aiming at describing the classification of the Drinfeld polynomials of the irregular minimal affinizations of type D. The theory of minimal affinizations, initiated in [1,10], is object of intensive study due to its rich structure and connections to other areas such as mathematical physics and combinatorics [5,13,17,18,19,22,28,29]. We refer to the first two paragraphs of the first paper of the series [26] for an account of the status of this classification problem when this project started. The main result of the present paper (Theorem 2.4.4) is one of the crucial steps towards the final classification: it will provide one of the tools we shall use to compare certain affinizations or to show that they are not comparable.…”

Section: Introductionmentioning

confidence: 99%

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Non-minimality of certain irregular coherent preminimal affinizations

Moura

Pereira

2018

Pacific J. Math.

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Let g be a finite-dimensional simple Lie algebra of type D or E and λ be a dominant integral weight whose support bounds the subdiagram of type D4. We study certain quantum affinizations of the simple g-module of highest weight λ which we term preminimal affinizations of order two (this is the maximal order for such λ). This class can be split in two: the coherent and the incoherent affinizations. If λ is regular, Chari and Pressley proved that the associated minimal affinizations belong to one of the three equivalent classes of coherent preminimal affinizations. In this paper we show that, if λ is irregular, the coherent preminimal affinizations are not minimal under certain hypotheses. Since these hypotheses are always satisfied if g is of type D4, this completes the classification of minimal affinizations for type D4 by giving a negative answer to a conjecture of Chari-Pressley stating that the coherent and the incoherent affinizations were equivalent in type D4 (this corrects the opposite claim made by the first author in a previous publication).

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 87%

Section: It Follows From the Cases Already Considered Thatmentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Non-minimality of certain irregular coherent preminimal affinizations

Moura

Pereira

2018

Pacific J. Math.

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Three-vertex prime graphs and reality of trees

Moura

Silva

2023

Communications in Algebra

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Elliptic quantum groups and Baxter relations

Zhang

2018

Alg. Number Th.

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We introduce a category O of modules over the elliptic quantum group of sl N with well-behaved q-character theory. We construct asymptotic modules as analytic continuation of a family of finite-dimensional modules, the Kirillov-Reshetikhin modules. In the Grothendieck ring of this category we prove two types of identities: generalized Baxter relations in the spirit of Frenkel-Hernandez between finite-dimensional modules and asymptotic modules; three-term Baxter TQ relations of infinite-dimensional modules.1 2 HUAFENG ZHANG Such R-matrices R(z; λ) appeared previously in face-type integrable models [20,38]; for instance, the R-matrix of the Andrews-Baxter-Forrester model comes from two-dimensional irreducible modules of E τ, (sl 2 ), as does the 6-vertex model from the affine quantum group U (Lsl 2 ). The definition of E τ, (sl N ) in [17], by RLL exchange relations, is in the spirit of Faddeev-Reshetikhin-Takhatajan, originated from Quantum Inverse Scattering Method. We mention that elliptic R-matrices describe the monodromy of the quantized Knizhnik-Zamolodchikov equation associated with representations of affine quantum groups, e.g. [28,29,43,50].Recently Aganagic-Okounkov [1] proposed the elliptic stable envelope in equivariant elliptic cohomology, as a geometric framework to obtain elliptic R-matrices. This was made explicit [18] for cotangent bundles of Grassmannians, resulting in tensor products of two-dimensional irreducible representations of E τ, (sl 2 ). The higher rank case of sl N was studied later by H. Konno [46].Meanwhile, Nekrasov-Pestun-Shatashvili [49] from the 6d quiver gauge theory predicted the elliptic quantum group associated to an arbitrary Kac-Moody algebra, the precise definition of which (as an associative algebra) was proposed by . See also [52] in the context of quiver geometry.We are interested in the representation theory of E τ, (g) with ∈ C generic. The formal twist constructions [11,40] from U (Lg) might reduce the problem to the representation theory of affine quantum groups, which is a subject developed intensively in the last three decades from algebraic, geometric and combinatorial aspects. However loc.cit. involve formal power series of and infinite products in the comultiplication of E τ, (g). Some of these divergence issues was addressed [12] by Etingof-Moura, who defined a fully faithful tenor functor between representation categories of BGG type for U (Lsl N ) and E τ, (sl N ). Towards this functor not much is known: its image, the induced homomorphism of Grothendieck rings, etc.In this paper we study representations of E τ, (sl N ) via the RLL presentation [17] so as to bypass affine quantum groups, yet along the way we borrow ideas from the affine case. Compared to other works [7,12,19,32,44,45,51,52], our approach emphasizes more on the Grothendieck ring structure of representation category. It is a higher rank extension of a recent joint work with G. Felder [21].The presence of the dynamical parameter λ is one of the technical difficulties of elliptic quantum groups. ...

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On Tensor Products of a Minimal Affinization With an Extreme Kirillov-Reshetikhin Module for Type A

Abstract: For a quantum affine algebra of type A, we describe the composition series of the tensor product of a general minimal affinization with a Kirillov-Resehtikhin module associated to an extreme node of the Dynkin diagram of the underlying simple Lie algebra.

Cited by 5 publications

References 27 publications

Non-minimality of certain irregular coherent preminimal affinizations

Non-minimality of certain irregular coherent preminimal affinizations

Three-vertex prime graphs and reality of trees

Elliptic quantum groups and Baxter relations

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