The comultiplication on $\mathbb{C} Q $ for a matrix basis?

Question

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra.
The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$.

For $G=Q$ the quaternion group, the algebra structure of $\mathcal{A}$ is $\mathbb{C} \oplus \mathbb{C} \oplus \mathbb{C} \oplus \mathbb{C} \oplus M_2(\mathbb{C})$.
Let $\{ e_1,e_2,e_3,e_4,a_{11}, a_{12}, a_{21}, a_{22} \}$ be a matrix basis of $\mathcal{A}$.

Question: What are the formulas for the comultiplication computed on a matrix basis?

I would also be interested by generic formulas for every finite group $G$.

Answer 1

By choosing a specific matrix basis of $\mathcal{A}$ (see here):

$1 \to$ $(1) \oplus (1) \oplus (1) \oplus (1) \oplus \begin{pmatrix} 1& 0 \newline 0& 1 \end{pmatrix}$
$i\to$ $(1) \oplus (1) \oplus (-1) \oplus (-1) \oplus \begin{pmatrix} i& 0 \newline 0& -i \end{pmatrix}$

$j \to$ $(1) \oplus (-1) \oplus (1) \oplus (-1) \oplus \begin{pmatrix} 0& -1 \newline 1& 0 \end{pmatrix}$

$k \to$ $(1) \oplus (-1) \oplus (-1) \oplus (1) \oplus \begin{pmatrix} 0& -i \newline -i& 0 \end{pmatrix}$

and by the method of this answer, we obtain:

sage: F=matrix([[1,1,1,1,1,1,1,1],[1,1,1,1,-1,-1,-1,-1],[1,1,-1,-1,1,1,-1,-1],[1,1,-1,-1,-1,-1,1,1],[1,-1,I,-1,0,0,0,0],[0,0,0,0,-1,1,-I,I],[0,0,0,0,1,-1,-I,I],[1,-1,-I,I,0,0,0,0]])
sage: FI=F^(-1)
sage: BF=[matrix([[sum([F[a][i]*F[b][i]*FI[i][c] for i in range(8)]) for a in range(8)] for b in range(8)]) for c in range(8)]
sage: for m in BF:
....:     print(m)
....:     print(',')
....:
[            1             0             0             0             0             0             0             0]
[            0             1             0             0             0             0             0             0]
[            0             0             1             0 -1/10*I + 1/5             0             0 -1/10*I + 1/5]
[            0             0             0             1 -1/10*I + 1/5             0             0 -1/10*I + 1/5]
[            0             0 -1/10*I + 1/5 -1/10*I + 1/5 1/10*I + 3/10             0             0  -1/5*I + 2/5]
[            0             0             0             0             0             0          -1/2             0]
[            0             0             0             0             0          -1/2             0             0]
[            0             0 -1/10*I + 1/5 -1/10*I + 1/5  -1/5*I + 2/5             0             0             0]
,
[            0             1             0             0             0             0             0             0]
[            1             0             0             0             0             0             0             0]
[            0             0             0             1 -1/10*I + 1/5             0             0 -1/10*I + 1/5]
[            0             0             1             0 -1/10*I + 1/5             0             0 -1/10*I + 1/5]
[            0             0 -1/10*I + 1/5 -1/10*I + 1/5 1/10*I + 3/10             0             0  -1/5*I + 2/5]
[            0             0             0             0             0             0           1/2             0]
[            0             0             0             0             0           1/2             0             0]
[            0             0 -1/10*I + 1/5 -1/10*I + 1/5  -1/5*I + 2/5             0             0             0]
,
[            0             0             1             0             0             0             0             0]
[            0             0             0             1             0             0             0             0]
[            1             0             0             0  1/10*I - 1/5             0             0  1/10*I - 1/5]
[            0             1             0             0  1/10*I - 1/5             0             0  1/10*I - 1/5]
[            0             0  1/10*I - 1/5  1/10*I - 1/5 -1/10*I + 1/5             0             0  1/5*I + 1/10]
[            0             0             0             0             0           1/2             0             0]
[            0             0             0             0             0             0           1/2             0]
[            0             0  1/10*I - 1/5  1/10*I - 1/5  1/5*I + 1/10             0             0           1/2]
,
[            0             0             0             1             0             0             0             0]
[            0             0             1             0             0             0             0             0]
[            0             1             0             0  1/10*I - 1/5             0             0  1/10*I - 1/5]
[            1             0             0             0  1/10*I - 1/5             0             0  1/10*I - 1/5]
[            0             0  1/10*I - 1/5  1/10*I - 1/5 -1/10*I + 1/5             0             0  1/5*I + 1/10]
[            0             0             0             0             0          -1/2             0             0]
[            0             0             0             0             0             0          -1/2             0]
[            0             0  1/10*I - 1/5  1/10*I - 1/5  1/5*I + 1/10             0             0           1/2]
,
[           0            0            0            0            1            0            0            0]
[           0            0            0            0            1            0            0            0]
[           0            0            0            0  2/5*I + 1/5            0            0  2/5*I + 6/5]
[           0            0            0            0  2/5*I + 1/5            0            0  2/5*I + 6/5]
[           1            1  2/5*I + 1/5  2/5*I + 1/5  3/5*I - 1/5            0            0 -1/5*I + 2/5]
[           0            0            0            0            0            0            0            0]
[           0            0            0            0            0            0            0            0]
[           0            0  2/5*I + 6/5  2/5*I + 6/5 -1/5*I + 2/5            0            0            0]
,
[ 0  0  0  0  0  1  0  0]
[ 0  0  0  0  0 -1  0  0]
[ 0  0  0  0  0  0 -1  0]
[ 0  0  0  0  0  0  1  0]
[ 0  0  0  0  0  0  0  0]
[ 1 -1  0  0  0  0  0  0]
[ 0  0 -1  1  0  0  0  0]
[ 0  0  0  0  0  0  0  0]
,
[ 0  0  0  0  0  0  1  0]
[ 0  0  0  0  0  0 -1  0]
[ 0  0  0  0  0 -1  0  0]
[ 0  0  0  0  0  1  0  0]
[ 0  0  0  0  0  0  0  0]
[ 0  0 -1  1  0  0  0  0]
[ 1 -1  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0  0]
,
[           0            0            0            0            0            0            0            1]
[           0            0            0            0            0            0            0            1]
[           0            0            0            0 -2/5*I + 4/5            0            0 -2/5*I - 1/5]
[           0            0            0            0 -2/5*I + 4/5            0            0 -2/5*I - 1/5]
[           0            0 -2/5*I + 4/5 -2/5*I + 4/5 -3/5*I + 1/5            0            0  1/5*I - 2/5]
[           0            0            0            0            0            0            0            0]
[           0            0            0            0            0            0            0            0]
[           1            1 -2/5*I - 1/5 -2/5*I - 1/5  1/5*I - 2/5            0            0            0]