Contents

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.

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

An adjoint action is an action by conjugation .

## Definition

### Of a group on itself

The adjoint action of a group $G$ on itself is the action $Ad : G \times G \to G$ given by

$Ad : (g,h) \mapsto g^{-1} \cdot h \cdot g \,.$

### Of a Lie group on its Lie algebra

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 $G$

$ad : (g,x) \mapsto Ad(g)_*(x) \,.$

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.

### Of a Lie algebra on itself

Differentiating the above example also in the second argument, yields the adjoint action of a Lie algebra on itself

$ad : \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$

which is simply the Lie bracket

$ad_x : y \mapsto [x,y] \,.$

### Of a Hopf algebra on itself

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

$h\triangleright g = \sum h_{(1)} g S(h_{(2)})$

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).

### Of a simplicial group on itself

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:

(1)$\array{ \mathcal{G}_{ad} \times \mathcal{G} &\xrightarrow{\;\;\;}& \mathcal{G}_{ad} \\ (g_k,h_k) &\mapsto& h_k \cdot g_k \cdot h_k^{-1} }$

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:

$\array{ \mathcal{G}_{ad} \times (\mathcal{G} \times \mathcal{G}) &\xrightarrow{\;\;\;}& \mathcal{G}_{ad} \\ (g_k, (h'_k, h_k)) &\mapsto& h'_k \cdot g_k \cdot h^{-1}_k \mathrlap{\,.} }$

###### Proposition

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):

$\mathcal{L} \big( \overline{W}\mathcal{G} \big) \;\; \simeq \;\; \mathcal{G}_{ad} \sslash \mathcal{G} \;\;\;\;\;\; \in \;\; Ho\big( sSet_{Qu} \big)$

For proof and more background see at free loop space of classifying space.

## References

• Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces

• Eckhard Meinrenken, Clifford algebras and Lie theory, Springer

Last revised on July 4, 2021 at 08:58:28. See the history of this page for a list of all contributions to it.