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:

ρ(g) is the transpose of ρ(g−1)

for all g in G. Then ρ is also a representation, as may be checked explicitly. The dual representation is also known as the contragredient representation.

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

ρ(u) is the transpose of −ρ(u) for all u in .

ρ is also a representation, as can be explicitly checked.

For a unitary representation, the conjugate representation and the dual representation coincide, up to equivalence of representations.