Frobenius-Schur indicators for semisimple Lie algebras

Details Frobenius-Schur indicators for semisimple Lie algebras Frobenius-Schur indicators for semisimple Lie algebras

2007-04-02

arxiv.org/abs/0704.0165v1

Mathematics

Quantum Algebra

12 pages, to appear in Journal of Algebra

Scientific paper

Let g be a finite dimensional complex semisimple Lie algebra, and let V be a finite dimensional represenation of g. We give a closed formula for the mth Frobenius-Schur indicator, m>1, of V in representation-theoretic terms. We deduce that the indicators take integer values, and that for a large enough m, the mth indicator of V equals the dimension of the zero weight space of V. For the classical Lie algebras sl(n), so(2n), so(2n+1) and sp(2n), this is the case for m greater or equal to 2n-1, 4n-5, 4n-3 and 2n+1, respectively.

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