# Dual representation

(Redirected from Contragredient representation)

In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representation ρ* is defined over the dual vector space V* as follows:[1]

ρ*(g) is the transpose of ρ(g−1), that is, ρ*(g) = ρ(g−1)T for all gG.

The dual representation is also known as the contragredient representation.

If g is a Lie algebra and π is a representation of it on the vector space V, then the dual representation π* is defined over the dual vector space V* as follows:[2]

π*(X) = −π(X)T for all Xg.

In both cases, the dual representation is a representation in the usual sense.

## Motivation

In representation theory, both vectors in V and linear functionals in V* are considered as column vectors so that the representation can act (by matrix multiplication) from the left. Given a basis for V and the dual basis for V*, the action of a linear functional φ on v, φ(v) can be expressed by matrix multiplication,

${\displaystyle \langle \varphi ,v\rangle \equiv \varphi (v)=\varphi ^{T}v}$,

where the superscript T is matrix transpose. Consistency requires

${\displaystyle \langle {\rho }^{*}(g)\varphi ,\rho (g)v\rangle =\langle \varphi ,v\rangle .}$[3]

With the definition given,

${\displaystyle \langle {\rho }^{*}(g)\varphi ,\rho (g)v\rangle =\langle \rho (g^{-1})^{T}\varphi ,\rho (g)v\rangle =(\rho (g^{-1})^{T}\varphi )^{T}\rho (g)v=\varphi ^{T}\rho (g^{-1})\rho (g)v=\varphi ^{T}v=\langle \varphi ,v\rangle }$.

For the Lie algebra representation one chooses consistency with a possible group representation. Generally, if Π is a representation of a Lie group, then π given by

${\displaystyle \pi (X)={\frac {d}{dt}}\Pi (e^{tX})|_{t=0}.}$

is a representation of its Lie algebra. If Π* is dual to Π, then its corresponding Lie algebra representation π* is given by

${\displaystyle \pi ^{*}(X)={\frac {d}{dt}}\Pi ^{*}(e^{tX})|_{t=0}={\frac {d}{dt}}\Pi (e^{-tX})^{T}|_{t=0}=-\pi (X)^{T}.}$.[4]

## Generalization

A general ring module does not admit a dual representation. Modules of Hopf algebras do, however.