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adjoint action

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×GGAd : G \times G \to G given by

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

Of a Lie group on its Lie algebra

The adjoint action ad:G×𝔤𝔤ad : G \times \mathfrak{g} \to \mathfrak{g} of a Lie group GG on its Lie algebra 𝔤\mathfrak{g} is for each gGg \in G the derivative dAd(g):T eGT 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×𝔤𝔤 ad : \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}

which is simply the Lie bracket

ad x:y[x,y]. ad_x : 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:HH 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

hg=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).

Revised on February 8, 2013 12:21:55 by Urs Schreiber (89.204.138.214)