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[1]:

\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[2]:

\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.

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

[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