W-algebra is a class of vertex algebras attached to a complex reductive Lie algebra, a nilpotent element in the Lie algebra, and a complex number. We consider the case of the general linear Lie algebra $\mathfrak{gl}_N$ with $N=l \times n$ and a nilpotent element whose Jordan form corresponds to the partition $(l^n)$. We call it rectangular W-algebra. Its current algebra (or enveloping algebra) is defined as the associative algebra generated by the Fourier modes of generating fields.

The goal of this talk is to construct an algebra homomorphism from the affine Yangian of type A to the current algebra of the rectangular W-algebra. We use the coproduct and the evaluation map of the affine Yangian to construct it. We hope that the homomorphism will be applied to the study of the AGT correspondence for parabolic sheaves and of integrable systems associated with the W-algebra. The talk is based on a joint work with Mamoru Ueda.

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