nLab adjoint action

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.

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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

$$ad \,\colon\, \mathfrak{g} \times \mathfrak{g} \longrightarrow \mathfrak{g}$$

which is simply the Lie bracket

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

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

• $\mathcal{G}$ be a simplicial group,

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.

Related concepts

• coadjoint action

• adjoint representation, adjoint bundle

• free loop space of classifying space

• conjugation action

• conjugacy class

• twisted ad-equivariant K-theory

References

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

• Eckhard Meinrenken, Clifford algebras and Lie theory, Springer

• eom: adjoint representation of a Lie group, adjoint group