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Complex conjugate representation
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, if <i>G</i> is a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> and Π is a <a href="/facts/Representation_theory/3ydLtaGH">representation</a> of it over the <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> <i>V</i>, then the complex conjugate representation Π is defined over the <a href="/facts/Complex_conjugate_vector_space/trWp8y4A">complex conjugate vector space</a> V as follows: </p> Π(<i>g</i>) is the <a href="/facts/Conjugate_linear_map/trWp8y4A">conjugate</a> of Π(<i>g</i>) for all <i>g</i> in <i>G</i>. <p>Π is also a representation, as one may check explicitly. </p><p>If g is a <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> and π is a representation of it over the vector space <i>V</i>, then the conjugate representation π is defined over the conjugate vector space <i>V</i> as follows: </p> π(<i>X</i>) is the conjugate of π(<i>X</i>) for all <i>X</i> in g.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> <p>π is also a representation, as one may check explicitly. </p><p>If two real Lie algebras have the same <a href="/facts/Complexification/fDfFLccF">complexification</a>, and we have a complex representation of the complexified Lie algebra, their conjugate representations are still going to be different. See <a href="/facts/Spinor/sKFNnEwP">spinor</a> for some examples associated with spinor representations of the <a href="/facts/Spin_group/tfvR9cqM">spin groups</a> Spin(<i>p</i> + <i>q</i>) and Spin(<i>p</i>, <i>q</i>). </p><p>If g {\displaystyle {\mathfrak {g}}} is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), </p> π(<i>X</i>) is the conjugate of −π(<i>X</i>*) for all <i>X</i> in g <p>For a finite-dimensional <a href="/facts/Unitary_representation/W1XRatG8">unitary representation</a>, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations. </p>