Adjoint action notations $\operatorname{ad}(X)(Y)$, $\operatorname{ad}(X)$ and $\operatorname{ad}_x(Y)$ are equivalent??

Question

As the title says I'm a bit confused with these notations of adjoint action of Lie algebra on itself. Are these notations ($\operatorname{ad}(X)(Y)$, $\operatorname{ad}(X)$ and $\operatorname{ad}_x(Y)$) equivalent?

Answer 1

No, they are not. The linear map $ad: L\rightarrow \mathfrak{gl}(V)$ is a Lie algebra representation by the Jacobi identity. It is defined by $x\mapsto ad(x)$ with $ad(x)(y)=[x,y]$. Some people write $ad_x(y)$ for $ad(x)(y)$. So $ad(x)$ is an endomorphism of the vector space of $L$, while $ad(x)(y)$ is an element of $L$ itself.

Answer 2

They are not all quite the same, but the first and last are equivalent. The key thing here is to keep track of the "data types" involved, i.e. what the inputs and outputs of the functions involved are. $\DeclareMathOperator{\ad}{ad}$ $\newcommand{\g}{\mathfrak g}$ $\newcommand{\gl}{\mathfrak {gl}}$

We begin with a function $\ad : \g \to \gl(\g)$. That is, for any $X \in \g$, $\ad(X)$ is an element of $\gl(\g)$, which is to say that $\ad(X)$ is a (not necessarily invertible) linear map from $\g$ to $\g$. That is, we have $\ad(X):\g \to \g$, so that $\ad(X)$ maps an element $Y \in \g$ to $\ad(X)(Y)$.

To make the notation easier, sometimes we define $\ad_X = \ad(X)$. Now, we have $\ad_X: \g \to \g$, so that $\ad_X$ maps $Y$ to $\ad_X(Y)$.

So, in particular, given $X,Y \in \g$: $\ad(X) = \ad_X \in \gl(\g)$, while $\ad(X)(Y) = \ad_X(Y) \in \g$