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Generalizing the polynomial web category, we introduce a diagrammatic -linear monoidal category, the affine web category, for any commutative ring. Integral bases consisting of elementary diagrams are obtained for the affine web category and its cyclotomic quotient categories. Connections between cyclotomic web categories and finite W-algebras are established, leading to a diagrammatic presentation of idempotent subalgebras of W-Schur algebras introduced by Brundan-Kleshchev. The affine web category will be used as a basic building block of another -linear monoidal category, the affine Schur category, formulated in a sequel.
Song et al. (Tue,) studied this question.