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^[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.

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 you may check explicitly.

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.

See also

• Complex conjugate representation
• Kirillov Character Formula

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