adjoint action in nLab

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.

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Contents

Idea

An adjoint action is an action by conjugation .

Definition

Of a group on itself

The adjoint action of a group GG on itself is the action Ad:G×G→GAd : G \times G \to G given by

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

Of a Lie group on its Lie algebra

The adjoint action ad:G×𝔤→𝔤ad : G \times \mathfrak{g} \to \mathfrak{g} of a Lie group GG on its Lie algebra 𝔤\mathfrak{g} is for each g∈Gg \in G the derivative dAd(g):T eG→T eGd Ad(g) : T_e G \to T_e G of this action in the second argument at the neutral element of GG

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

This is often written as ad(g)(x)=g −1xgad(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 GG it is: in this case both gg as well as xx 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:𝔤×𝔤⟶𝔤 ad \,\colon\, \mathfrak{g} \times \mathfrak{g} \longrightarrow \mathfrak{g}

which is simply the Lie bracket

ad x:y↦[x,y]. ad_x \,\colon\, y \mapsto [x,y] \,.

Of a Hopf algebra on itself

Let kk be a commutative unital ring and H=(H,m,η,Δ,ϵ,S)H = (H,m,\eta,\Delta,\epsilon, S) be a Hopf kk-algebra with multiplication mm, unit map η\eta, comultiplication Δ\Delta, counit ϵ\epsilon and the antipode map S:H→H opS: H\to H^{op}. We can use Sweedler notation Δ(h)=∑h (1)⊗ kh (2)\Delta(h) = \sum h_{(1)}\otimes_k h_{(2)}. The adjoint action of HH on HH is given by

h▹g=∑h (1)gS(h (2)) h\triangleright g = \sum h_{(1)} g S(h_{(2)})

and it makes HH not only an HH-module, but in fact a monoid in the monoidal category of HH-modules (usually called HH-module algebra).

Of a simplicial group on itself

Let

• 𝒢\mathcal{G} be a simplicial group,

and write

• 𝒢Actions(sSet)\mathcal{G}\Actions(sSet) for the category of 𝒢\mathcal{G}-action objects internal to SimplicialSetsl

• W𝒢∈𝒢Actions(sSet)W \mathcal{G} \in \mathcal{G}Actions(sSet) for its universal principal simplicial complex;

• W¯𝒢=W𝒢𝒢∈sSet\overline{W}\mathcal{G} \,=\, \frac{W \mathcal{G}}{\mathcal{G}} \in sSet for the simplicial classifying space;

• 𝒢 ad∈𝒢Actions(sSet)\mathcal{G}_{ad} \in \mathcal{G}Actions(sSet) for the adjoint action of 𝒢\mathcal{G} on itself:

  (1)𝒢 ad×𝒢 → 𝒢 ad (g k,h k) ↦ h k⋅g k⋅h k −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 𝒢→diag𝒢×𝒢\mathcal{G} \xrightarrow{diag} \mathcal{G} \times \mathcal{G} of the following action of the direct product group:

  𝒢 ad×(𝒢×𝒢) → 𝒢 ad (g k,(h′ k,h k)) ↦ h′ k⋅g k⋅h k −1. \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{\,.} }

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

• trivial action, diagonal action

• 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