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Browsing by Author "Mukhin, E."
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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 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 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.