Knowlpedia

Dual (contragredient) representation

Given a representation on , the induced representation on is ; infinitesimally, .

Let VV be a finite-dimensional vector space.

Lie group version

If ρ:G→GL(V)\rho:G\to GL(V) is a GG, the dual (contragredient) representation on V∗V^* is

ρ∗:G→GL(V∗),(ρ∗(g)λ)(v):=λ(ρ(g−1)v).\rho^*:G\to GL(V^*),\qquad (\rho^*(g)\lambda)(v) := \lambda\bigl(\rho(g^{-1})v\bigr).

Equivalently, in a chosen basis, ρ∗(g)\rho^*(g) is represented by (ρ(g−1))T(\rho(g^{-1}))^{\mathsf T}.

Lie algebra version

If π:g→gl(V)\pi:\mathfrak g\to \mathfrak{gl}(V) is a g\mathfrak g, its dual representation π∗:g→gl(V∗)\pi^*:\mathfrak g\to\mathfrak{gl}(V^*) is defined by

(π∗(X)λ)(v):=− λ(π(X)v).(\pi^*(X)\lambda)(v) := -\,\lambda\bigl(\pi(X)v\bigr).

In matrix form, π∗(X)=− π(X)T\pi^*(X) = -\,\pi(X)^{\mathsf T}.

These definitions are compatible under differentiation: if π=dρe\pi=d\rho_e (see ), then d(ρ∗)e=π∗d(\rho^*)_e=\pi^*.

Remarks

Duals interact naturally with other constructions such as . In highest-weight theory, dualizing typically negates weights (compare and ).