Is $\mathcal{C}_{G}$ an algebra?
Let $\mathcal{C}_{G}$ be the group algebra of $G=F_{2}$ over a field $k$ and let $U$ be the copy of $\mathbb{Z}_{2}$ subgroup of $G$ given by $U= \{x \in G | x^{2}=0\}$. Let $X= \{(g,g) \in G \times G | g \in G \}$ and \$Y= \{(g,gx