In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows:

1. is the transpose of , that is, = for all .

The dual representation is also known as the contragredient representation.

If is a Lie algebra and is a representation of it on the vector space , then the dual representation is defined over the dual vector space as follows:

1. = for all .

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.

Properties

Irreducibility and second dual

If a (finite-dimensional) representation is irreducible, then the dual representation is also irreducible—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.

Unitary representations

Consider a unitary, and let us work in an orthonormal basis into the group of unitary matrices.

.

The SU(2) and SU(3) cases

In the representation theory of SU(2), the dual of each irreducible representation does turn out to be isomorphic to the representation. But for the representations of SU(3).) is not isomorphic to its dual. In the theory of quarks."

General semisimple Lie algebras

More generally, in the representation theory of semisimple Lie algebras (or the closely related representation theory of compact Lie groups), the weights of the dual representation are the negatives of the weights of the original representation. is an element of the Weyl group. For such Lie algebras, every).

is not in the Weyl group, then the dual of an irreducible representation will generically not be isomorphic to the original representation. To understand how this works, we note that there is always a unique Weyl group element, the lowest. It then follows that the highest.. The adjoint representation, for example, is always isomorphic to its dual.

is the reflection about the line through., which are the representations whose weight diagrams are regular hexagons.

Motivation

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

1. ,

where the superscript is matrix transpose. Consistency requires

With the definition given,

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

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

Example

of complex numbers of absolute value 1. The irreducible representations are all one dimensional, as a consequence of Schur's lemma and given explicitly as

is then the inverse of the transpose of this one-by-one matrix, that is,

.

Generalization

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

See also

• Complex conjugate representation

• Tensor product of representations

• Kirillov Character Formula

References

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