Deformations of \mathcal W algebras via quantum toroidal algebras

2020/03/09 by Boris Feigin, M. Jimbo, Feigin, B. +5 · 2 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2003.04234

Abstract

The deformed \mathcal W algebras of type \textsfA have a uniform description in terms of the quantum toroidal \mathfrakgl₁ algebra \mathcal E. We introduce a comodule algebra \mathcal K over \mathcal E which gives a uniform construction of basic deformed \mathcal W currents and screening operators in types \textsfB,\textsfC,\textsfD including twisted and supersymmetric cases. We show that a completion of algebra \mathcal 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 \textsfD⁽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.

Cited by

• Combinatorics of vertex operators and deformed W-algebra of type D(2,1;α)
• Direct Sum Structure of the Super Virasoro Algebra and a Fermion Algebra Arising from the Quantum Toroidal \mathfrakgl₂

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