Complex conjugate representation
title: "Complex conjugate representation" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["representation-theory-of-groups"] topic_path: "general/representation-theory-of-groups" source: "https://en.wikipedia.org/wiki/Complex_conjugate_representation" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
In mathematics, if G is a group and Π is a representation of it over the complex vector space V, then the complex conjugate representation is defined over the complex conjugate vector space as follows:
:(g) is the conjugate of Π(g) for all g in G.
is also a representation, as one may check explicitly.
If g is a real Lie algebra and π is a representation of it over the vector space V, then the conjugate representation is defined over the conjugate vector space as follows:
:(X) is the conjugate of π(X) for all X in g.
is also a representation, as one may check explicitly.
If two real Lie algebras have the same complexification, and we have a complex representation of the complexified Lie algebra, their conjugate representations are still going to be different. See spinor for some examples associated with spinor representations of the spin groups Spin(p + q) and Spin(p, q).
If \mathfrak{g} is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket),
:(X) is the conjugate of −π(X*) for all X in g
For a finite-dimensional unitary representation, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations.
Notes
References
- This is the mathematicians' convention. Physicists use a different convention where the [[Lie bracket of vector fields. Lie bracket]] of two real vectors is an imaginary vector. In the physicist's convention, insert a minus in the definition.
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