Braid actions on quantum toroidal superalgebras

TL;DR

This paper demonstrates the isomorphism of quantum toroidal superalgebras for different root systems and establishes a braid group action that exchanges subalgebras, revealing deep symmetries in these algebraic structures.

Contribution

It introduces the isomorphism between quantum toroidal superalgebras of different root systems and constructs a braid group action that exchanges their subalgebras.

Findings

Quantum toroidal superalgebras are isomorphic for different root systems.

Existence of Miki automorphism exchanging vertical and horizontal subalgebras.

Braid group action on the direct sum of all such algebras.

Abstract

We prove that the quantum toroidal algebras $\mathcal{E}_\mathbf{s}$ associated with different root systems $\mathbf{s}$ of $\mathfrak{gl}_{m|n}$ type are isomorphic. We also show the existence of Miki automorphism of $\mathcal{E}_\mathbf{s}$, which exchanges the vertical and horizontal subalgebras. To obtain these results, we establish an action of the toroidal braid group on the direct sum $\oplus_\mathbf{s} \mathcal{E}_\mathbf{s}$ of all such algebras.