2 clarification

Let $G$ be a finite group with identity element $e$, and $C[G]$ the ring of complex-valued functions on $G$, with pointwise addition and multiplication. Then $C[G]$ is naturally a Hopf algebra, with comultiplication, counit, and antipode given by $$[\Delta(f)](g_1, g_2) = f(g_1 \cdot g_2), [\epsilon(f)] = f(e), S(f) = f(g^{-1}),$$ for all $f \in C[G]$, $g, g_1, g_2 \in G$. Here we identify $C[G] \otimes_C C[G]$ with $C[G \times G]$ in the usual way.

If $\chi \in Z^2(G, C^\times)$ is a normalized (meaning $\chi(g,1) = \chi(1,g) = 1$) two-cocycle, then one can "twist" the coalgebra structure on $C[G]$, defining new comultiplication and antipode by $$\Delta_\chi(f) = \chi \cdot \Delta(f) \cdot \chi^{-1}, S_\chi f = U (S f) U^{-1},$$ where $U = \sum_i \chi_i^{(1)} (S \chi_i^{(2)})$.

This, according to Theorem 2.3.4 of Shahn Majid's "Foundations of quantum group theory," produces a new Hopf algebra structure. It's typically called $C_\chi[G]$ (though my notation differs slightly from Majid's). Here, I think that we are viewing $\chi$ -- a priori a function from $G \times G$ to $C^\times$ -- as an element of the ring $\sum_i \chi_i^{(1)} \otimes \chi_i^{(2)} \in C[G] \otimes_C C[G]$, identifying complex-valued functions on $G \times G$ with elements of $C[G] \otimes C[G]$ as usual, and for some reason $U$ is invertible in the ring $C[G] \otimes C[G]$.

Now for the question...

The Hopf algebra $C_\chi[G]$ obtained through this process is still a commutative Hopf algebra over $C$; only the antipode and comultiplication were changed. So $Spec(C_\chi[G])$ is an affine group scheme over $C$, whose $C$-points are in bijection with the elements of $G$.

So... what is this group scheme?! Or have I messed something up in this construction? It seems very odd to me, since the cocycle should -- group theoretically -- produce a central extension of $G$ by $C^\times$, and I don't know how such a thing would be encoded in a group scheme whose $C$-points are in bijection with $G$. Any references for this group scheme?

1

# Twist of a group Hopf-algebra

Let $G$ be a finite group with identity element $e$, and $C[G]$ the ring of complex-valued functions on $G$, with pointwise addition and multiplication. Then $C[G]$ is naturally a Hopf algebra, with comultiplication, counit, and antipode given by $$[\Delta(f)](g_1, g_2) = f(g_1 \cdot g_2), [\epsilon(f)] = f(e), S(f) = f(g^{-1}),$$ for all $f \in C[G]$, $g, g_1, g_2 \in G$. Here we identify $C[G] \otimes_C C[G]$ with $C[G \times G]$ in the usual way.

If $\chi \in Z^2(G, C^\times)$ is a normalized (meaning $\chi(g,1) = \chi(1,g) = 1$) two-cocycle, then one can "twist" the coalgebra structure on $C[G]$, defining new comultiplication and antipode by $$\Delta_\chi(f) = \chi \cdot \Delta(f) \cdot \chi^{-1}, S_\chi f = U (S f) U^{-1},$$ where $U = \sum_i \chi_i^{(1)} (S \chi_i^{(2)})$.

This, according to Theorem 2.3.4 of Shahn Majid's "Foundations of quantum group theory," produces a new Hopf algebra structure. It's typically called $C_\chi[G]$ (though my notation differs slightly from Majid's). Here, I think that we are viewing $\chi$ -- a priori a function from $G \times G$ to $C^\times$ -- as an element of the ring $\sum_i \chi_i^{(1)} \otimes \chi_i^{(2)} \in C[G] \otimes_C C[G]$, and for some reason $U$ is invertible in the ring $C[G] \otimes C[G]$.

Now for the question...

The Hopf algebra $C_\chi[G]$ obtained through this process is still a commutative Hopf algebra over $C$; only the antipode and comultiplication were changed. So $Spec(C_\chi[G])$ is an affine group scheme over $C$, whose $C$-points are in bijection with the elements of $G$.

So... what is this group scheme?! Or have I messed something up in this construction? It seems very odd to me, since the cocycle should -- group theoretically -- produce a central extension of $G$ by $C^\times$, and I don't know how such a thing would be encoded in a group scheme whose $C$-points are in bijection with $G$. Any references for this group scheme?