This entry is about conjugation in the sense of adjoint actions, as in forming conjugacy classes. For conjugation in the sense of anti-involutions on star algebras see at complex conjugation.
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).
Let
and write
$\mathcal{G}\Actions(sSet)$ for the category of $\mathcal{G}$-action objects internal to SimplicialSetsl
$W \mathcal{G} \in \mathcal{G}Actions(sSet)$ for its universal principal simplicial complex;
$\overline{W}\mathcal{G} \,=\, \frac{W \mathcal{G}}{\mathcal{G}} \in sSet$ for the simplicial classifying space;
$\mathcal{G}_{ad} \in \mathcal{G}Actions(sSet)$ for the adjoint action of $\mathcal{G}$ on itself:
which we may understand as the restriction along the diagonal morphism $\mathcal{G} \xrightarrow{diag} \mathcal{G} \times \mathcal{G}$ of the following action of the direct product group:
The free loop space object of the simplicial classifying space $\overline{W} \mathcal{G}$ is isomorphic in the classical homotopy category to the Borel construction of the adjoint action (1):
For proof and more background see at free loop space of classifying space.
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces
Eckhard Meinrenken, Clifford algebras and Lie theory, Springer
Last revised on July 4, 2021 at 12:58:28. See the history of this page for a list of all contributions to it.