Supriya Ghosh (Editor)

Dual representation

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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:

Contents

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,

⟨ φ , v ⟩ ≡ φ ( v ) = φ T v ,

where the superscript T is matrix transpose. Consistency requires

⟨ ρ ∗ ( g ) φ , ρ ( g ) v ⟩ = ⟨ φ , v ⟩ .

With the definition given,

⟨ ρ ∗ ( g ) φ , ρ ( g ) v ⟩ = ⟨ ρ ( g − 1 ) T φ , ρ ( g ) v ⟩ = ( ρ ( g − 1 ) T φ ) T ρ ( g ) v = φ T ρ ( g − 1 ) ρ ( g ) v = φ T v = ⟨ φ , v ⟩ .

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

π ( X ) = d d t Π ( e t X ) | 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

π ∗ ( X ) = d d t Π ∗ ( e t X ) | t = 0 = d d t Π ( e − t X ) T | t = 0 = − π ( X ) T . .

Generalization

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

References

Dual representation Wikipedia
(Text) CC BY-SA