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

is defined over the dual vector space as follows^[1]:

is the transpose of ρ(g⁻¹)

for all g in G. Then is also a representation, as may be checked explicitly. The dual representation is also known as the contragredient representation.

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^[2]:

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

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.

[edit] Generalization

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

[edit] See also

• Complex conjugate representation
• Kirillov Character Formula

[edit] References

1. ^ Lecture 1 of William Fulton & Joe Harris, Representation Theory. A First Course, Graduate Texts in Mathematics, 129, Springer-Verlag, 1991. ISBN: 0-387-97527-6; 0-387-97495-4
2. ^ Lecture 8 of William Fulton & Joe Harris, Representation Theory. A First Course, Graduate Texts in Mathematics, 129, Springer-Verlag, 1991. ISBN: 0-387-97527-6; 0-387-97495-4