Twisted quantum affinizations and their vertex representations

Chen, Fulin; Jing, Naihuan; Kong, Fei; Tan, Shaobin

doi:10.1063/1.5023790

Cited by 4 publications

(11 citation statements)

References 22 publications

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“…The second Figure given above suggests us that the classical limit of U ( g µ ) should be certain Drinfeld type presentations of g[µ] constructed in Theorem 1.4. Indeed, in [CJKT1], by the quantum vertex operators technique, we construct a new quantum algebra U ( g µ ) when g is of simply-laced affine type and µ is nontransitive. Due to Theorem 1.4, we prove in § 6 that the classical limit of this quantum algebra is isomorphic to U( g[µ]).…”

Section: Gabber-kac Presentationmentioning

confidence: 99%

“…In [CJKT1] we introduced a class of twisted quantum affinization algebras by constructing their vertex representations. As an application of Theorem 1.4, in this section we determine the classical limit of these quantum algebras.…”

Section: Classical Limit Of Twisted Quantum Affinization Algebrasmentioning

confidence: 99%

“…Throughout this section, we assume that A is simply-laced and that µ is not a transitive automorphism of A (1) ℘ , ℘ ≥ 2. We first recall the twisted quantum affinization algebra introduced in [CJKT1]. Let be an indeterminate over C, and C[[ ]] the power series ring of .…”

Section: Classical Limit Of Twisted Quantum Affinization Algebrasmentioning

confidence: 99%

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Drinfeld-type presentations of loop algebras

Chen

Jing

Kong

et al. 2020

Trans. Amer. Math. Soc.

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Let g be the derived subalgebra of a Kac-Moody Lie algebra of finite type or affine type, µ a diagram automorphism of g and L(g, µ) the loop algebra of g associated to µ. In this paper, by using the vertex algebra technique, we provide a general construction of current type presentations for the universal central extension g[µ] of L(g, µ). The construction contains the classical limit of Drinfeld's new realization for (twisted and untwisted) quantum affine algebras ([Dr]) and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras ([MRY]) as special examples. As an application, when g is of simply-laced type, we prove that the classical limit of the µ-twisted quantum affinization of the quantum Kac-Moody algebra associated to g introduced in [CJKT1] is the universal enveloping algebra of g [µ].

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Section: Gabber-kac Presentationmentioning

confidence: 99%

Section: Classical Limit Of Twisted Quantum Affinization Algebrasmentioning

confidence: 99%

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Drinfeld-type presentations of loop algebras

Chen

Jing

Kong

et al. 2020

Trans. Amer. Math. Soc.

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“…It has been a question whether the twisted quantum affinization process introduced by Drinfeld can be generalized to diagram automorphisms of any symmetrizable Kac-Moody Lie algebra in a similar way to that of the untwisted case? In [9], by using vertex operator calculations, we generalized the twisted quantum affinization to a class of diagram automorphisms on simply-laced Kac-Moody Lie algebras. One notices that in general there exist nontrivial diagram automorphisms on non-simply-laced Kac-Moody Lie algebras that are not from finite types.…”

Section: Introductionmentioning

confidence: 99%

“…When µ = Id, it coincides with the quantum affinization algebra U (ĝ). When g is of simply-laced type, U (ĝ µ ) has been realized in [9] by vertex operators. Especially, in the case when g is of untwisted affine type X…”

Section: Introductionmentioning

confidence: 99%

Twisted quantum affinizations and quantization of extended affine Lie algebras

Chen¹,

Jing²,

Kong³

et al. 2020

Preprint

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In this paper, for an arbitrary Kac-Moody Lie algebra g and a diagram automorphism µ of g satisfying two linking conditions, we introduce and study a µtwisted quantum affinization algebra U (ĝµ) of g. When g is of finite type, U (ĝµ) is Drinfeld's current algebra realization of the twisted quantum affine algebra. And, when µ = Id, U (ĝµ) is the quantum affinization algebra introduced by Ginzburg-Kapranov-Vasserot. As the main results of this paper, we first prove a triangular decomposition of U (ĝµ). Second, we give a simple characterization of the affine quantum Serre relations on restricted U (ĝµ)-modules in terms of "normal order products". Third, we prove that the category of restricted U (ĝµ)-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of U (ĝµ). Fourth, we study the classical limit of U (ĝµ) and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the -deformation of nullity 2 extended affine Lie algebras.

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Automorphisms of Quantum Toroidal Algebras from an Action of the Extended Double Affine Braid Group

Laurie

2024

Algebr Represent Theor

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We first construct an action of the extended double affine braid group $$\mathcal {\ddot{B}}$$ B ¨ on the quantum toroidal algebra $$U_{q}(\mathfrak {g}_{\textrm{tor}})$$ U q ( g tor ) in untwisted and twisted types. As a crucial step in the proof, we obtain a finite Drinfeld new style presentation for a broad class of quantum affinizations. In the simply laced cases, using our action and certain involutions of $$\mathcal {\ddot{B}}$$ B ¨ we produce automorphisms and anti-involutions of $$U_{q}(\mathfrak {g}_{\textrm{tor}})$$ U q ( g tor ) which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements C and $$k_{0}^{a_{0}}\dots k_{n}^{a_{n}}$$ k 0 a 0 ⋯ k n a n up to inverse. This can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups used by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type A due to Miki which have been instrumental in the study of the structure and representation theory of $$U_{q}(\mathfrak {sl}_{n+1,\textrm{tor}})$$ U q ( sl n + 1 , tor ) .

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Twisted quantum affinizations and their vertex representations

Abstract: In this paper we generalize Drinfeld's twisted quantum affine algebras to construct twisted quantum algebras for all simply-laced generalized Cartan matrices and present their vertex representation realizations.2010 Mathematics Subject Classification. 17B37, 17B10.

Cited by 4 publications

References 22 publications

Drinfeld-type presentations of loop algebras

Drinfeld-type presentations of loop algebras

Twisted quantum affinizations and quantization of extended affine Lie algebras

Automorphisms of Quantum Toroidal Algebras from an Action of the Extended Double Affine Braid Group

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