Representations of $\mathfrak{sl}(2,\mathbb{C})$ in category $\mathcal O$ and master symmetries

We show that the indecomposable $\mathfrak{sl}(2,\mathbb{C})$-modules in the Bernstein–Gelfand–Gelfand category $\mathcal O$ naturally arise for homogeneous integrable nonlinear evolution systems. We then develop a new approach called the $\mathcal O$ scheme to construct master symmetries for such integrable systems. This method naturally allows computing the hierarchy of time-dependent symmetries. We finally illustrate the method using both classical and new examples. We compare our approach to the known existing methods used to construct master symmetries. For new integrable equations such as a Benjamin–Ono-type equation, a new integrable Davey–Stewartson-type equation, and two different versions of $(2+1)$-dimensional generalized Volterra chains, we generate their conserved densities using their master symmetries.