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

is the transpose of ρ(g^-1) for all g in G.

is also a representation, as you may check explicitly.

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

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

is also a representation, as you may check explicitly.

Unfortunately, a general ring module does not admit a dual representation.

See also complex conjugate representation

For a unitary representation, the conjugate representation and the dual representation coincides.