Dual 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:

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:

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

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,

where the superscript $T$ is matrix transpose. Consistency requires

With the definition given,

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

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

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

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