Conceptual origin of the comultiplication map in the S-module structure on E_x

Determine a conceptual explanation for why the Hopf algebra comultiplication map Δ: S → S ⊗_K S, defined by Δ(t_z) = t_z ⊗ 1 + 1 ⊗ t_z for all z ∈ P*, naturally arises in the construction of the Z^n-graded injective envelope {}^*_S(S/𝔭_x) via the S-module structure on E_x = E^x ⊗_K E^{-x}. Clarify the structural reason in this setting—e.g., through a categorical, Hopf-algebraic, or geometric framework—explaining why Δ governs the action used to define the S-module structure on E_x.

Background

In Section 2, the authors explicitly construct the Z^n-graded injective envelope {}^*_S(S/𝔭_x) for x ∈ P by setting E_x := E^x ⊗_K E^{-x} and then endowing E_x with an S-module structure via the ring homomorphism Δ: S → S ⊗_K S defined by Δ(t_z) = t_z ⊗ 1 + 1 ⊗ t_z. This Δ is recognized as the comultiplication map that makes S a Hopf algebra (the coordinate ring of the additive group K^{#P*}).

Despite this concrete use of Δ to define the S-action on E_x, the authors explicitly note that they do not understand why the comultiplication appears here, indicating a conceptual gap in the current framework. This motivates an open problem to provide a structural explanation for the role of Δ in this injective-envelope construction.