Dual (contragredient) representation

Let $V$ be a finite-dimensional vector space.

Lie group version

If $\rho:G\to GL(V)$ is a representation of a Lie group $G$, the dual (contragredient) representation on $V^*$ is

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

Lie algebra version

If $\pi:\mathfrak g\to \mathfrak{gl}(V)$ is a representation of a Lie algebra $\mathfrak g$, its dual representation $\pi^*:\mathfrak g\to\mathfrak{gl}(V^*)$ is defined by

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

These definitions are compatible under differentiation: if $\pi=d\rho_e$ (see differentiation of a group homomorphism ), then $d(\rho^*)_e=\pi^*$.

Context

Duals interact naturally with other constructions such as tensor products . In highest-weight theory, dualizing typically negates weights (compare weights and weights in the dual Cartan ).