Let $a \otimes b, a'\otimes b' \in \mathbb{C}_q[X] \otimes \mathbb{C}_q[Y]$, where $X, Y$ are two algebraic varieties. Suppose that algebraic group $T$ acts on $X, Y$. Then there are coactions $\delta: \mathbb{C}_q[Y] \to \mathbb{C}_q[T] \otimes \mathbb{C}_q[Y]$, $\delta: \mathbb{C}_q[X] \to \mathbb{C}_q[T] \otimes \mathbb{C}_q[X]$. There is a formula: \begin{align} (a \otimes b) (a'\otimes b') = a \cdot b'_{(-1)}(a') \otimes b'_{(0)} b, \end{align} where $b'_{(-1)} \otimes b'_{(0)} = \delta(b')$ (here we use the Sweedler notation). Are there some reference about this formula? Thank you very much.
Edit: the formula should be \begin{align} (a \otimes b) (a'\otimes b') = a (b_{(-1)}.a') \otimes b_{(0)} b, \end{align} $\delta(b) = b_{(-1)} \otimes b_{(0)} \in \mathbb{C}_q[T] \otimes \mathbb{C}_q[Y]$. We also require that $\mathbb{C}_q[T]$ acts on $\mathbb{C}_q[X]$.
The map $\delta$ gives what is called a Yetter--Drinfeld-module. The quantized coordinate rings are algebra objects within this category, and the tensor product is the braided tensor product in the category of Yetter--Drinfeld modules over $\mathbb{C}_q[T]$. This means that the product structure is defined as follows: $$m_{A\otimes B}:=(m_A\otimes m_B)(Id_A\otimes \Psi_{A,B}\otimes Id_B)$$ Here, both $A,B$ are object in this category of Yetter--Drinfeld modules, and $\Psi$ is the braiding in the category, while $m_A$ denotes multiplication on $A$.
The Yetter--Drinfeld structure on the quantum coordinate rings is probably induced by the dual quasitriangular structure on $\mathbb{C}_q[T]$.
As this is a reference request: The fact that the category of Yetter--Drinfeld modules is braided is standard. It can be found in textbooks such as Majid's Foundations of Quantum Group Theory. I am not sure where to find the other details.