# Complex conjugate representation

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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 you 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 you 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 ${\displaystyle {\mathfrak {g}}}$ 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 unitary representation, the dual representation and the conjugate representation coincide.