nLab diagonal action

Idea

For $G$ a group with action on two objects $X$ and $Y$, the diagonal action is canonically induced action on the product $X \times Y$.

Definition

Consider a group $G$ and a pair of sets $X_1, X_2 \,\in\, Set$, each equipped with a group action

$$\begin{array}{ccc} G \times X_i &\xrightarrow{\phantom{---}}& X_i \\ (g,x_i) &\mapsto& g(x_i) \mathrlap{\,.} \end{array}$$

In components, the diagonal action of $G$ on the Cartesian product $X_1 \times X_2$ is given by

$$\begin{array}{ccc} G \times (X_1 \times X_2) &\xrightarrow{\phantom{-}g\phantom{-}}& X_1 \times X_2 \\ \big( g, (x_1, x_2) \big) &\mapsto& \big( g(x_1) ,\, g(x_2) \big) \mathrlap{\,.} \end{array}$$

More abstractly, regarding the $X_i \in G Set$ as G-sets, the diagonal action is just their Cartesian product as such, formed in the category $G Set$.

Related entries

• trivial action

• adjoint action

Related concepts

• diagonal

• Atiyah Lie groupoid