Some quantities associated to finite dimensional Hopf algebras

Question

let $(H,\Delta,m,s)$ be a Hopf algebra. To this Hopf algebra one can associate two obvious linear maps $T_H, S_H: H \to H $ with $T_H=m\circ \Delta,\quad S_H=s$.

Are there two finite dimensional Hopf algebras $H,H'$ which are not isomorphic Hopf algebras but they are equivalents in the following sense:

Each coefficient of characteristic polynomial $Det (T_H-\lambda I)$ equals to the corresponding coefficient of $Det (T_{H'}-\lambda I)$. Similarly, Each coefficients of $Det (S_H-\lambda I)$ equals to the corresponding coefficients in $Det (S_{H'}-\lambda I)$.That is:$Det(T_H)=Det(T_{H'}),\ldots,trace(T_H)=traceT_{H'}$ and likewise for $S_H$ and $S_{H'}$.

In the other words, to what extent the set of these coefficients can determine the nature of Hopf algebras. Are there some infinite dimensional analogy for these quantities?

Are there two finite groups $G,G'$ of the same orders which are not isomorphic groups but their corresponding Hopf algebras $CG, CG'$ have the same quantities described above?

Answer 1

For a finite group algebra, $T_H(g)=g^2$ and $S_H(g)=g^{-1}$ for any group element $g$.

So for any two finite groups of exponent three and the same order, the quantities will be the same. For example, there's a non-abelian group of order 27 and exponent 3, and also an abelian one.