Dual 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.

is the complex conjugate of the adjoint of the inverse of

is assumed to be unitary, the adjoint of the inverse of

The upshot of this discussion is that when working with unitary representations in an orthonormal basis,

[5] In particular, the standard three-dimensional representation of SU(3) (with highest weight

In the theory of quarks in the physics literature, the standard representation and its dual are called "

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.

Now, for a given Lie algebra, if it should happen that operator

is an element of the Weyl group, then the weights of every representation are automatically invariant under the map

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

To understand how this works, we note that there is always a unique Weyl group element

Then if we have an irreducible representation with highest weight

, the lowest weight of the dual representation will be

It then follows that the highest weight of the dual representation will be

The adjoint representation, for example, is always isomorphic to its dual.

In the case of SU(3) (or its complexified Lie algebra,

), we may choose a base consisting of two roots

at an angle of 120 degrees, so that the third positive root is

[8] The self-dual representations are then the ones that lie along the line through

, which are the representations whose weight diagrams are regular hexagons.

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.

The irreducible representations are all one dimensional, as a consequence of Schur's lemma.

The irreducible representations are parameterized by integers

is then the inverse of the transpose of this one-by-one matrix, that is, That is to say, the dual of the representation

A general ring module does not admit a dual representation.