Dual representation

If G is a group and ρ is a representation of it over the vector space V, then the dual representation $\bar{\rho}$ is defined over the dual vector space $\bar{V}$ as follows:

$\bar{\rho}(g)$ is the transpose of ρ(g^-1) for all g in G.

$\bar{\rho}$ is also a representation, as you may check explicitly.

If $\mathfrak{g}$ is a Lie algebra and ρ is a representation of it over the vector space V, then the dual representation $\bar{\rho}$ is defined over the dual vector space $\bar{V}$ as follows:

$\bar{\rho}(u)$ is the transpose of -ρ(u) for all u in $\mathfrak{g}$ .

$\bar{\rho}$ is also a representation, as you may check explicitly.

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

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

This mathematics-related article is a stub. You can help Wikipedia by expanding it.

Dual representation

See also: Dual representation, Complex conjugate representation, Dual vector space, Group, Lie algebra, Mathematics, Module, Representation theory, Transpose of a linear map