Lie Algebra Expansion and Integrability in Superstring Sigma-Models

Lie algebra expansion is a technique to generate new Lie algebras from a given one. In this paper, we apply the method of Lie algebra expansion to superstring $\sigma$-models with a $\mathbb{Z}_4$ coset target space. By applying the Lie algebra expansion to the isometry algebra, we obtain different $\sigma$-models, where the number of dynamical fields can change. We reproduce and extend in a systematic way actions of some known string regimes (flat space, BMN and non-relativistic in AdS$_5 \times$S$^5$). We define a criterion for the algebra truncation such that the equations of motion of the expanded action of the new $\sigma$-model are equivalent to the vanishing curvature condition of the Lax connection obtained by expanding the Lax connection of the initial model.