Finite Type Modules and Bethe Ansatz Equations

Feigin, Boris; Jimbo, Michio; Miwa, Tetsuji; Mukhin, E.

doi:10.1007/s00023-017-0577-y

Cited by 27 publications

(20 citation statements)

References 24 publications

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“…Many results for E m|n (q 1 , q 2 , q 3 ) need to be established, and we plan to address it in the followup papers. In particular, similar to the even case, we expect to obtain the Miki automorphism, see [M1], the shuffle algebra realization, see [N], the PBW type theorem, see [T2], the category O, see [FJMM1], the fusion subalgebras, see [FJMM2], the integrable systems and Bethe ansatz, see [FJMM3] with proper modifications.…”

Section: Introductionmentioning

confidence: 99%

Quantum Toroidal Algebra Associated with $\mathfrak {gl}_{m|n}$

Bezerra

Mukhin

2020

Algebr Represent Theor

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We introduce and study the quantum toroidal algebra E m|n (q 1 , q 2 , q 3 ) associated with the superalgebra gl m|n with m = n, where the parameters satisfy q 1 q 2 q 3 = 1.We give an evaluation map. The evaluation map is a surjective homomorphism of algebras E m|n (q 1 , q 2 , q 3 ) → U q gl m|n to the quantum affine algebra associated with the superalgebra gl m|n at level c completed with respect to the homogeneous grading, where q 2 = q 2 and q m−n 3 = c 2 . We also give a bosonic realization of level one E m|n (q 1 , q 2 , q 3 )-modules. Quantum toroidal gl m|nAssume m, n ≥ 1 and m = n. In this section we introduce the quantum toroidal algebra associated with gl m|n , denoted by E m|n , and collect a few properties.2.1. Definition of E m|n . Fix d, q ∈ C × and define

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Section: Introductionmentioning

confidence: 99%

Quantum Toroidal Algebra Associated with $\mathfrak {gl}_{m|n}$

Bezerra

Mukhin

2020

Algebr Represent Theor

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“…This leads to interpret the automorphism S as a sort of square root of the R-matrix. This result should have tremendous consequences for integrable systems built upon DIM algebra [48][49][50]. We hope to come back to this important issue in a future publication.…”

Section: Twist By An Automorphismmentioning

confidence: 93%

Fiber-base duality from the algebraic perspective

Bourgine

2019

J. High Energ. Phys.

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Quiver 5D N = 1 gauge theories describe the low-energy dynamics on webs of (p, q)-branes in type IIB string theory. S-duality exchanges NS5 and D5 branes, mapping (p, q)-branes to branes of charge (−q, p), and, in this way, induces several dualities between 5D gauge theories. On the other hand, these theories can also be obtained from the compactification of topological strings on a Calabi-Yau manifold, for which the S-duality is realized as a fiber-base duality. Recently, a third point of view has emerged in which 5D gauge theories are engineered using algebraic objects from the Ding-Iohara-Miki (DIM) algebra. Specifically, the instanton partition function is obtained as the vacuum expectation value of an operator T constructed by gluing the algebra's intertwiners (the equivalent of topological vertices) following the rules of the toric diagram/brane web. Intertwiners and Toperators are deeply connected to the co-algebraic structure of the DIM algebra. We show here that S-duality can be realized as the twist of this structure by Miki's automorphism.

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“…We call this qq-character the truncation of χY −1 i,σ and denote it by Trn(χY −1 i,σ ). The truncation procedure is an analog of the construction of finite type modules which are obtained by multiplying known modules by polynomial modules and taking the irreducible submodule, see [FJMM1], [FJMM2]. The finite type modules have properties similar to finite-dimensional ones, but they are in general infinite-dimensional.…”

Section: The Qq-charactersmentioning

confidence: 99%

“…We follow notation of[FJMV] which is different from the usual q-character notation. Variable Y i should be compared to variable X −1 i in[FJMM1],[FJMM2] and usual Y i variables are ratios of two X i variables.…”

mentioning

confidence: 99%

Combinatorics of vertex operators and deformed $W$-algebra of type D$(2,1;α)$

Feigin¹,

Jimbo²,

Mukhin³

2021

Preprint

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We consider sets of screening operators with fermionic screening currents. We study sums of vertex operators which formally commute with the screening operators assuming that each vertex operator has rational contractions with all screening currents with only simple poles. We develop and use the method of qq-characters which are combinatorial objects described in terms of deformed Cartan matrix. We show that each qq-character gives rise to a sum of vertex operators commuting with screening operators and describe ways to understand the sum in the case it is infinite.We discuss combinatorics of the qq-characters and their relation to the q-characters of representations of quantum groups.We provide a number of explicit examples of the qq-characters with the emphasis on the case of D(2, 1; α). We describe a relationship of the examples to various integrals of motion.

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Finite Type Modules and Bethe Ansatz Equations

Cited by 27 publications

References 24 publications

Quantum Toroidal Algebra Associated with $\mathfrak {gl}_{m|n}$

Quantum Toroidal Algebra Associated with $\mathfrak {gl}_{m|n}$

Fiber-base duality from the algebraic perspective

Combinatorics of vertex operators and deformed $W$-algebra of type D$(2,1;α)$

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