Is the representation $mathbfNoplusbarmathbfN$ of $SU(N)$ real or pseudo-real?

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Let $N geq 3$ be an integer. Let us call $mathbfN$ its fundamental representation, and let $barmathbfN$ be its complex conjugated representation.



Is the direct sum $mathbfNoplusbarmathbfN$ real or pseudo-real ?




representation-theory lie-groups lie-algebras




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  • The similarity transformation interchanging 1-2 spaces suggests just pseudoreal. If you took N = 3, you could convince yourself by explicit calculation that this rep is pseudoreal, not real.
    – Cosmas Zachos
    Jul 31 at 15:22














up vote
0
down vote

favorite












Let $N geq 3$ be an integer. Let us call $mathbfN$ its fundamental representation, and let $barmathbfN$ be its complex conjugated representation.



Is the direct sum $mathbfNoplusbarmathbfN$ real or pseudo-real ?




representation-theory lie-groups lie-algebras




share|cite|improve this question



















  • The similarity transformation interchanging 1-2 spaces suggests just pseudoreal. If you took N = 3, you could convince yourself by explicit calculation that this rep is pseudoreal, not real.
    – Cosmas Zachos
    Jul 31 at 15:22












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $N geq 3$ be an integer. Let us call $mathbfN$ its fundamental representation, and let $barmathbfN$ be its complex conjugated representation.



Is the direct sum $mathbfNoplusbarmathbfN$ real or pseudo-real ?




representation-theory lie-groups lie-algebras




share|cite|improve this question











Let $N geq 3$ be an integer. Let us call $mathbfN$ its fundamental representation, and let $barmathbfN$ be its complex conjugated representation.



Is the direct sum $mathbfNoplusbarmathbfN$ real or pseudo-real ?





representation-theory lie-groups lie-algebras





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 20 at 12:45









Antoine

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  • The similarity transformation interchanging 1-2 spaces suggests just pseudoreal. If you took N = 3, you could convince yourself by explicit calculation that this rep is pseudoreal, not real.
    – Cosmas Zachos
    Jul 31 at 15:22
















  • The similarity transformation interchanging 1-2 spaces suggests just pseudoreal. If you took N = 3, you could convince yourself by explicit calculation that this rep is pseudoreal, not real.
    – Cosmas Zachos
    Jul 31 at 15:22















The similarity transformation interchanging 1-2 spaces suggests just pseudoreal. If you took N = 3, you could convince yourself by explicit calculation that this rep is pseudoreal, not real.
– Cosmas Zachos
Jul 31 at 15:22




The similarity transformation interchanging 1-2 spaces suggests just pseudoreal. If you took N = 3, you could convince yourself by explicit calculation that this rep is pseudoreal, not real.
– Cosmas Zachos
Jul 31 at 15:22















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