Bialgebraic Structure on a Coder Pair of Any Weight

In this paper, we establish a bialgebraic structure on a Coder pair of any weight \(\lambda \) $\lambda$ (i.e., a coassociative coalgebra with a coderivation of weight \(\lambda \) $\lambda$ ), which is consistent with the differential antisymmetric infinitesimal bialgebra in (SIGMA 19:018, 2023) when the weight \(\lambda =0\) $\lambda =0$ .