Integrable deformations of the mKdV and SG hierarchies and quasigraded Lie algebras

We construct a new family of quasigraded Lie algebras that admit the Kostant–Adler scheme. They coincide with special quasigraded deformations of twisted subalgebras of the loop algebras. Using them we obtain new hierarchies of integrable equations in partial derivatives. They coincide with the deformations of integrable hierarchies associated with the loop algebras. We consider the case g=gl(2)g=gl(2) in detail and obtain integrable hierarchies that could be viewed as deformations of mKdV, sine-Gordon and derivative non-linear Shrödinger hierarchies and some other integrable hierarchies, such as the (w3) non-linear Shrödinger hierarchy and its doubled form.