Dual representation

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This article is about a mathematical concept. For the psychological concept, see Dual representation (psychology).

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), that is, ρ*(g) = ρ(g−1)T for all gG.

The dual representation is also known as the contragredient representation.

If g is a Lie algebra and π is a representation of it on the vector space V, then the dual representation π* is defined over the dual vector space V* as follows:[2]

π*(X) = −π(X)T for all Xg.

In both cases, the dual representation is a representation in the usual sense.

Motivation[edit]

In representation theory, both vectors in V and linear functionals in V* are considered as column vectors so that the representation can act (by matrix multiplication) from the left. Given a basis for V and the dual basis for V*, the action of a linear functional φ on v, φ(v) can be expressed by matrix multiplication,

\langle\varphi, v\rangle \equiv \varphi(v) = \varphi^Tv,

where the superscript T is matrix transpose. Consistency requires

\langle{\rho}^*(g)\varphi, \rho(g)v\rangle = \langle\varphi, v\rangle.[3]

With the definition given,

\langle{\rho}^*(g)\varphi, \rho(g)v\rangle = \langle\rho(g^{-1})^T\varphi, \rho(g)v\rangle = (\rho(g^{-1})^T\varphi)^T \rho(g)v = \varphi^T\rho(g^{-1})\rho(g)v = \varphi^Tv = \langle\varphi, v\rangle.

For the Lie algebra representation one chooses consistency with a possible group representation. Generally, if Π is a representation of a Lie group, then π given by

\pi(X) = \frac{d}{dt}\Pi(e^{tX})|_{t = 0}.

is a representation of its Lie algebra. If Π* is dual to Π, then its corresponding Lie algebra representation π* is given by

\pi^*(X) = \frac{d}{dt}\Pi^*(e^{tX})|_{t = 0} = \frac{d}{dt}\Pi(e^{-tX})^T|_{t = 0} = -\pi(X)^T..[4]

Generalization[edit]

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

See also[edit]

References[edit]

  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, MR 1153249, 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, MR 1153249, ISBN 978-0-387-97527-6 
  3. ^ Lecture 1, page 4 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, MR 1153249, ISBN 978-0-387-97527-6 
  4. ^ Lecture 8, page 111 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, MR 1153249, ISBN 978-0-387-97527-6