The quantity you introduced on MO—denoted n(A)n(A)n(A), the maximal dimension of an Abelian Lie group acting effectively via homogeneous morphisms on a coalgebra AAA—is already known in a classical form. In his answer on MO, Will Sawin shows that such actions correspond exactly to invertible elements in the cocenter of AAA. Equivalently, for finite-dimensional coalgebras, this relates to the center of the dual algebra, giving a familiar group-theoretic interpretation mathoverflow.net.

Emmanuel Ansong Adjei Thank you very much for your attention to my question.

I think that the Will Sawin answer in MO is good but is NOT a complete answer to my question. It is a translation of my question into cocenter language. For example can one say that: what is this quantity $n(A)$ for $A=C(G)$ where $G$ is a low dimensional Lie group say $S^1, S^3, SO(n) or Sp(n)$ for low $n$? I appreciate your attention to my question but does your answer have some thing more than the content of MO link I already mentioned in my post?