Browsing by Author "Jimbo, M."
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Item Branching rules for quantum toroidal gl (n)(2013) Feigin, B.; Jimbo, M.; Miwa, T.; Mukhin, Eugene; Department of Mathematical Sciences, School of ScienceWe construct an analog of the subalgebra Ugl(n) ⊗ Ugl(m) ⊂ Ugl(m + n) in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra.Item Combinatorics of vertex operators and deformed W-algebra of type D(2,1;α)(Elsevier, 2022-07-16) Feigin, B.; Jimbo, M.; Mukhin, E.; Mathematical Sciences, School of ScienceWe 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. We describe a relationship of the examples to various integrals of motion.Item Commutative subalgebra of a shuffle algebra associated with quantum toroidal glm|n(Elsevier, 2024-06) Feigin, B.; Jimbo, M.; Mukhin, E.; Mathematical Sciences, School of ScienceWe define and study the shuffle algebra Shm|n of the quantum toroidal algebra Em|n associated to Lie superalgebra glm|n. We show that Shm|n contains a family of commutative subalgebras Bm|n(s) depending on parameters s=(s1,…,sm+n), ∏isi=1, given by appropriate regularity conditions. We show that Bm|n(s) is a free polynomial algebra and give explicit generators which conjecturally correspond to the traces of the s-weighted R-matrix computed on the degree zero part of Em|n modules of levels ±1.Item Deformations of W algebras via quantum toroidal algebras(Springer, 2021-06) Feigin, B.; Jimbo, M.; Mukhin, E.; Vilkovisky, I.; Mathematical Sciences, School of ScienceWe study the uniform description of deformed W algebras of type A including the supersymmetric case in terms of the quantum toroidal gl1 algebra E. In particular, we recover the deformed affine Cartan matrices and the deformed integrals of motion. We introduce a comodule algebra K over E which gives a uniform construction of basic deformed W currents and screening operators in types B,C,D including twisted and supersymmetric cases. We show that a completion of algebra K contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine Dynkin diagrams of all non-exceptional types except D(2)ℓ+1. We also obtain in a uniform way deformed finite and affine Cartan matrices in all classical types together with a number of new examples, and discuss the corresponding screening operators.Item Finite Type Modules and Bethe Ansatz for Quantum Toroidal gl1(Springer, 2017-11) Feigin, B.; Jimbo, M.; Miwa, T.; Mukhin, Evgeny; Mathematical Sciences, School of ScienceWe study highest weight representations of the Borel subalgebra of the quantum toroidal gl1 algebra with finite-dimensional weight spaces. In particular, we develop the q-character theory for such modules. We introduce and study the subcategory of ‘finite type’ modules. By definition, a module over the Borel subalgebra is finite type if the Cartan like current ψ+(z) has a finite number of eigenvalues, even though the module itself can be infinite dimensional. We use our results to diagonalize the transfer matrix T V,W (u; p) analogous to those of the six vertex model. In our setting T V,W (u; p) acts in a tensor product W of Fock spaces and V is a highest weight module over the Borel subalgebra of quantum toroidal gl1 with finite-dimensional weight spaces. Namely we show that for a special choice of finite type modules V the corresponding transfer matrices, Q(u; p) and T(u;p) , are polynomials in u and satisfy a two-term TQ relation. We use this relation to prove the Bethe Ansatz equation for the zeroes of the eigenvalues of Q(u; p). Then we show that the eigenvalues of T V,W (u; p) are given by an appropriate substitution of eigenvalues of Q(u; p) into the q-character of V.Item Integrals of motion from quantum toroidal algebras(IOP, 2017) Feigin, B.; Jimbo, M.; Mukhin, Eugene; Mathematical Sciences, School of ScienceWe identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors. That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model. We also discuss the $({\mathfrak {gl}}_m, {\mathfrak {gl}}_n)$ duality of XXZ models in quantum toroidal setting and the implications for the quantum KdV model. In particular, we conjecture that the spectrum of non-local integrals of motion of Bazhanov, Lukyanov, and Zamolodchikov is described by Gaudin Bethe ansatz equations associated to affine ${\mathfrak{sl}}_2$ .Item Towards trigonometric deformation of 𝔰𝔩ˆ2 coset VOA(AIP, 2019) Feigin, B.; Jimbo, M.; Mukhin, E.; Mathematical Sciences, School of ScienceWe discuss the quantization of the 𝔰𝔩ˆ2 coset vertex operator algebra 𝒲D(2,1;α) using the bosonization technique. We show that after quantization, there exist three families of commuting integrals of motion coming from three copies of the quantum toroidal algebra associated with 𝔤𝔩2.