Dual representation

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:
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:
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" representation formula_1 of a group formula_2, and let us work in an orthonormal basis. Thus, formula_1 maps formula_2 into the group of unitary matrices. Then the abstract transpose in the definition of the dual representation may be identified with the ordinary matrix transpose. Since the adjoint of a matrix is the complex conjugate of the transpose, the transpose is the conjugate of the adjoint. Thus, formula_5 is the complex conjugate of the adjoint of the inverse of formula_6. But since formula_6 is assumed to be unitary, the adjoint of the inverse of formula_6 is just formula_6.
The upshot of this discussion is that when working with unitary representations in an orthonormal basis, formula_5 is just the complex conjugate of formula_6.
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), the dual of the irreducible representation with label formula_12 is the irreducible representation with label formula_13. In particular, the standard three-dimensional representation of SU(3) (with highest weight formula_14) is not isomorphic to its dual. In the theory of quarks in the physics literature, the standard representation and its dual are called "formula_15" and "formula_16."
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. (See the figure.) Now, for a given Lie algebra, if it should happen that operator formula_17 is an element of the Weyl group, then the weights of every representation are automatically invariant under the map formula_18. For such Lie algebras, "every" irreducible representation will be isomorphic to its dual. (This is the situation for SU(2), where the Weyl group is formula_19.) Lie algebras with this property include the odd orthogonal Lie algebras formula_20 (type formula_21) and the symplectic Lie algebras formula_22 (type formula_23). 
If, for a given Lie algebra, formula_17 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 formula_25 mapping the negative of the fundamental Weyl chamber to the fundamental Weyl chamber. Then if we have an irreducible representation with highest weight formula_26, the "lowest" weight of the dual representation will be formula_27. It then follows that the "highest" weight of the dual representation will be formula_28. Since we are assuming formula_17 is not in the Weyl group, formula_25 cannot be formula_17, which means that the map formula_32 is not the identity. Of course, it may still happen that for certain special choices of formula_26, we might have formula_34. The adjoint representation, for example, is always isomorphic to its dual.
In the case of SU(3) (or its complexified Lie algebra, formula_35), we may choose a base consisting of two roots formula_36 at an angle of 120 degrees, so that the third positive root is formula_37. In this case, the element formula_25 is the reflection about the line perpendicular to formula_39. Then the map formula_32 is the reflection about the line "through" formula_39. The self-dual representations are then the ones that lie along the line through formula_39. These are the representations with labels of the form formula_43, 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,
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.
Consider the group formula_49 of complex numbers of absolute value 1. The irreducible representations are all one dimensional, as a consequence of Schur's lemma. The irreducible representations are parameterized by integers formula_50 and given explicitly as
The dual representation to formula_52 is then the inverse of the transpose of this one-by-one matrix, that is,
That is to say, the dual of the representation formula_52 is formula_55.
Generalization.
A general ring module does not admit a dual representation. Modules of Hopf algebras do, however.