Coaction of a product.

Question

Let $G$ be a group and let $X$, $Y$ be two algebraic varieties on which $G$ acts. Let $\delta_1: \mathbb{C}[X] \to \mathbb{C}[G] \otimes \mathbb{C}[X]$ be a coaction given by $\delta_1(f) = \sum f_{(-1)} \otimes f_{(0)}$ and $\delta_2: \mathbb{C}[Y] \to \mathbb{C}[G] \otimes \mathbb{C}[Y]$ be a coaction given by $\delta_2(g) = \sum g_{(-1)} \otimes g_{(0)}$. Are there some natural coaction $\delta$ of $\mathbb{C}[G]$ on $\mathbb{C}[X] \otimes \mathbb{C}[Y]$ induced by $\delta_1, \delta_2$? What is $\delta(f \otimes g)$? Thank you very much.

Answer 1

If $C$ is a bialgebra over a field ${\mathbb k}$, then the category $C\text{-CoMod}$ of $C$-comodules admits a natural monoidal structure by defining, for $(X,\eta_X),(Y,\eta_Y)\in C\text{-CoMod}$, the coaction of $C$ on $X\otimes_{\mathbb k} Y$ as $$X\otimes_{\mathbb k} Y\xrightarrow{\eta_X\otimes\eta_Y} (C\otimes_{\mathbb k} X)\otimes_{\mathbb k} (C\otimes_{\mathbb k} Y)\cong (C\otimes_{\mathbb k} C)\otimes_{\mathbb k}(X\otimes_{\mathbb k} Y)\xrightarrow{\mu_C\otimes\text{id}_{X\otimes_{{\mathbb k}} Y}} C\otimes_{\mathbb k} (X\otimes_{\mathbb k} Y).$$ For an affine group $G/{\mathbb k}$ a $G$-representation on some ${\mathbb k}$-vector space $V$ is the same as a ${\mathbb k}[G]$-comodule structure on $V$, and since ${\mathbb k}[G]$ is even a bialgebra, $G\text{-Rep}$ therefore inherits a monoidal structure.