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:
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ρ*(g) is the transpose of ρ(g−1), that is, ρ*(g) = ρ(g−1)T for all g ∈ G.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:
π*(X) = −π(X)T for all X ∈ g.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,
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
Generalization
A general ring module does not admit a dual representation. Modules of Hopf algebras do, however.