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

SIGMA 18 (2022), 051, 31 pages      arXiv:2112.14631

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

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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