Deformed Double Current Algebras, Matrix Extended ${\mathcal{W}}_{\mathrm{\infty }}$ Algebras, Coproducts, and Intertwiners from the M2-M5 Intersection

We study the algebraic structures which govern the deformation of supersymmetric intersections of M2 and M5 branes. The universal algebras on M2 and M5 branes are deformed double current algebra of ${\mathfrak{g}\mathfrak{l}}_K$ and ${\mathfrak{g}\mathfrak{l}}_K$-extended ${\mathcal{W}}_{\mathrm{\infty }}$-algebra respectively. We give a new presentation of the deformed double current algebra of ${\mathfrak{g}\mathfrak{l}}_K$, and we give a rigorous mathematical construction of the ${\mathfrak{g}\mathfrak{l}}_K$-extended ${\mathcal{W}}_{\mathrm{\infty }}$-algebra. A new presentation of the affine Yangian of ${\mathfrak{g}\mathfrak{l}}_K$ is also obtained. We construct various coproducts of these algebras, which are expected to encode the fusions of defects in twisted M-theory. The matrix extended Miura operators are identified as intertwiners in certain bimodules of these algebras.