Dual representation

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In mathematics, if G is a group and ρ is a linear representation of it on 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. Then $\bar{\rho}$ is also a representation, as may be checked explicitly. The dual representation is also known as the contragredient representation.

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: