# Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras

## TL;DR

This paper demonstrates that classical affine W-algebras associated with classical Lie algebras inherently possess integrable Hamiltonian hierarchies of Lax type equations, expanding the understanding of integrable systems in Lie algebra theory.

## Contribution

It introduces generalized Adler type operators and quasideterminants, establishing a framework for constructing integrable Hamiltonian systems from classical affine W-algebras.

## Key findings

- Proves all classical affine W-algebras have integrable Hamiltonian hierarchies.
- Develops theories of generalized Adler type operators and quasideterminants.
- Recovers all KdV type hierarchies by Drinfeld and Sokolov as special cases.

## Abstract

We prove that any classical affine W-algebra W(g,f), where g is a classical Lie algebra and f is an arbitrary nilpotent element of g, carries an integrable Hamiltonian hierarchy of Lax type equations. This is based on the theories of generalized Adler type operators and of generalized quasideterminants, which we develop in the paper. Moreover, we show that under certain conditions, the product of two generalized Adler type operators is a Lax type operator. We use this fact to construct a large number of integrable Hamiltonian systems, recovering, as a special case, all KdV type hierarchies constructed by Drinfeld and Sokolov.