# Complex representation

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{{distinguish|Complex envelope}}
In [mathematics](/source/mathematics), a '''complex representation''' is a [representation](/source/group_representation) of a [group](/source/Group_(mathematics)) (or [that](/source/Lie_algebra_representation) of [Lie algebra](/source/Lie_algebra)) on a complex vector space. Sometimes (for example in physics), the term '''complex representation''' is reserved for a representation on a complex vector space that is neither [real](/source/real_representation) nor [pseudoreal](/source/pseudoreal_representation) (quaternionic). In other words, the group elements are expressed as complex matrices, and the complex conjugate of a complex representation is a different, non-equivalent representation. For compact groups, the [Frobenius-Schur indicator](/source/Frobenius-Schur_indicator) can be used to tell whether a representation is real, complex, or pseudo-real. 

For example, the N-dimensional [fundamental representation](/source/fundamental_representation) of SU(N) for N greater than two is a complex representation whose complex conjugate is often called the [antifundamental representation](/source/antifundamental_representation).

== References ==
*{{Fulton-Harris}}

Category:Representation theory of groups

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Adapted from the Wikipedia article [Complex representation](https://en.wikipedia.org/wiki/Complex_representation) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Complex_representation?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.