Integrability of Nonabelian Differential–Difference Equations: The Symmetry Approach

We extend the approach proposed in Mikhailov et al. (Commun Math Phys 393:1063–1104, 2022) to tackle the integrability problem for evolutionary differential–difference equations (DΔEs) on free associative algebras, also referred to as nonabelian DΔEs. This approach enables us to derive necessary integrability conditions, determine the integrability of a given equation, and make progress in the classification of integrable nonabelian DΔEs. This work involves establishing symbolic representations for the nonabelian difference algebra, difference operators, and formal series, as well as introducing a quasi-local extension for the algebra of formal series within the context of symbolic representations. Applying this formalism, we solve the classification problem of integrable skew-symmetric quasi-linear nonabelian equations of orders (-1, 1), (-2, 2), and (-3, 3), consequently revealing some new equations in the process.