# Bosonic symmetries of the extended fermionic $(2N,2M)$-Toda hierarchy

## TL;DR

This paper constructs and analyzes the additional symmetries of the fermionic (2N,2M)-Toda hierarchy, revealing a rich algebraic structure and a new extended hierarchy with superconformal properties.

## Contribution

It generalizes the supersymmetric Toda hierarchy to include additional symmetries forming a $w_{ty} imes w_{ty}$ algebra and introduces a new extended hierarchy with Bosonic Block superconformal structure.

## Key findings

- Additional symmetries form a $w_{ty} imes w_{ty}$ algebra.
- Defined a new extended fermionic (2N,2M)-Toda hierarchy.
- Hierarchy admits a Bosonic Block superconformal structure.

## Abstract

In this paper, we construct the additional symmetries of the fermionic $(2N,2M)$-Toda hierarchy basing on the generalization of the $N{=}(1|1)$ supersymmetric two dimensional Toda lattice hierarchy. These additional flows constitute a $w_{\infty}\times w_{\infty}$ Lie algebra. As a Bosonic reduction of the $N{=}(1|1)$ supersymmetric two dimensional Toda lattice hierarchy and the fermionic $(2N,2M)$-Toda hierarchy, we define a new extended fermionic $(2N,2M)$-Toda hierarchy which admits a Bosonic Block type superconformal structure.