# Dual representation

In mathematics, if G is a group and ρ is a linear representation of it on the real vector space V, then the dual representation ρ is defined over the dual vector space V as follows:[1]

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. For a complex vector space, take $\overline{\rho}(g) := \rho(g)^{\dagger}$

If $\mathfrak{g}$ 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:[2]

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

## Generalization

A general ring module does not admit a dual representation. Modules of Hopf algebras do, however.

## References

1. ^ Lecture 1 of Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR1153249, ISBN 978-0-387-97527-6
2. ^ Lecture 8 of Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR1153249, ISBN 978-0-387-97527-6