Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 051, 31 pages      arXiv:2112.14631      https://doi.org/10.3842/SIGMA.2022.051

Quantum Toroidal Comodule Algebra of Type $A_{n-1}$ and Integrals of Motion

Boris Feigin ^ab, Michio Jimbo ^c and Evgeny Mukhin ^d
a) National Research University Higher School of Economics, 20 Myasnitskaya Str., Moscow, 101000, Russia
^b) Landau Institute for Theoretical Physics, 1a Akademika Semenova Ave., Chernogolovka, 142432, Russia
^c) Department of Mathematics, Rikkyo University, Toshima-ku, Tokyo 171-8501, Japan
^d) Department of Mathematics, Indiana University Purdue University Indianapolis, 402 N. Blackford St., LD 270, Indianapolis, IN 46202, USA

Received March 02, 2022, in final form June 27, 2022; Published online July 07, 2022

Abstract
We introduce an algebra $\mathcal{K}_n$ which has a structure of a left comodule over the quantum toroidal algebra of type $A_{n-1}$. Algebra $\mathcal{K}_n$ is a higher rank generalization of $\mathcal{K}_1$, which provides a uniform description of deformed $W$ algebras associated with Lie (super)algebras of types BCD. We show that $\mathcal{K}_n$ possesses a family of commutative subalgebras.

Key words: quantum toroidal algebras; comodule; integrals of motion.

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