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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** Π is defined over the complex conjugate vector space V as follows:

- Π(
*g*) is the conjugate of Π(*g*) for all*g*in*G*.

Π 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 π is defined over the conjugate vector space *V* as follows:

- π(
*X*) is the conjugate of π(*X*) for all*X*in**g**.^{[1]}

π 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 is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket),

- π(
*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.

## See also

## Notes

**^**This is the mathematicians' convention. Physicists use a different convention where the Lie bracket of two real vectors is an imaginary vector. In the physicist's convention, insert a minus in the definition.