The matrix-extended W 1 + ∞ $$ {\mathcal{W}}_{1+\infty } $$ algebra

Abstract We construct a quadratic basis of generators of matrix-extended W 1 + ∞ $$ {\mathcal{W}}_{1+\infty } $$ using a generalization of the Miura transformation. This makes it possible to conjecture a closed-form formula for the operator product expansions defining the algebra. We study truncations of the algebra. An explicit calculation at low levels shows that these are parametrized in a way consistent with the gluing description of the algebra. It is perhaps surprising that in spite of the fact that the algebras are rather complicated and non-linear, the structure of their truncations follows very simple gluing rules.