  We propose that Miura operators are R-matrices of certain
infinite-dimensional quantum algebras. We test our proposal by realizing Miura
operators of $q$-deformed $W$- and $Y$-algebras in terms of R-matrices of the
quantum toroidal algebra of $\mathfrak{gl}(1)$. Physically, the representations
of this toroidal algebra arise from the algebra of local operators on M2-branes
and M5-branes, in M-theory subject to an $\Omega$-background. We associate an
R-matrix to each M2-M5 brane crossing, by studying its description as a
gauge-invariant intersection of a topological line defect and a holomorphic
surface defect in 5-dimensional non-commutative Chern-Simons theory. The Miura
transformation is engineered using multiple M2-M5 intersections, relying
crucially on the properties of the underlying R-matrices. We thereby identify
each R-matrix with a Miura operator. In a dual Type IIB frame, the components
of the Miura transformation are shown to coincide with the half-index of a 3d
supersymmetric gauge theory on a Hanany-Witten system of D3-NS5 branes. As a
further application, we demonstrate that $qq$-characters can be algebraically
constructed from the Miura transformation.
