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:[1][2]

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:[3]

The motivation for this definition is that Lie algebra representation associated to the dual of a Lie group representation is computed by the above formula. But the definition of the dual of a Lie algebra representation makes sense even if it does not come from a Lie group representation.

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

If a (finite-dimensional) representation is irreducible, then the dual representation is also irreducible[4]—but not necessarily isomorphic to the original representation. On the other hand, the dual of the dual of any representation is isomorphic to the original representation.

Two nonisomorphic dual representations of SU(3), with highest weights (1,2) and (2,1)

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,

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.