symmetric monoidal (∞,1)-category of spectra
An adjoint action is an action by conjugation .
The adjoint action of a group $G$ on itself is the action $Ad : G \times G \to G$ given by
The adjoint action $ad : G \times \mathfrak{g} \to \mathfrak{g}$ of a Lie group $G$ on its Lie algebra $\mathfrak{g}$ is for each $g \in G$ the derivative $d Ad(g) : T_e G \to T_e G$ of this action in the second argument at the neutral element of $G$
This is often written as $ad(g)(x) = g^{-1} x g$ even though for a general Lie group the expression on the right is not the product of three factors in any way. But for a matrix Lie group $G$ it is: in this case both $g$ as well as $x$ are canonically identified with matrices and the expression on the right is the product of these matrices.
Since this is a linear action, it is called the adjoint representation of a Lie group. The associated bundles with respect to this representation are called adjoint bundles.
Differentiating the above example also in the second argument, yields the adjoint action of a Lie algebra on itself
which is simply the Lie bracket
Let $k$ be a commutative unital ring and $H = (H,m,\eta,\Delta,\epsilon, S)$ be a Hopf $k$-algebra with multiplication $m$, unit map $\eta$, comultiplication $\Delta$, counit $\epsilon$ and the antipode map $S: H\to H^{op}$. We can use Sweedler notation $\Delta(h) = \sum h_{(1)}\otimes_k h_{(2)}$. The adjoint action of $H$ on $H$ is given by
and it makes $H$ not only an $H$-module, but in fact a monoid in the monoidal category of $H$-modules (usually called $H$-module algebra).