# Frobenius-Schur indicators for semisimple Lie algebras

## TL;DR

This paper provides a closed formula for Frobenius-Schur indicators of finite-dimensional representations of complex semisimple Lie algebras, showing they are integers and relate to zero weight space dimensions for large m.

## Contribution

It introduces a representation-theoretic formula for Frobenius-Schur indicators and establishes their integer values and asymptotic behavior for classical Lie algebras.

## Key findings

- Indicators are integers for all m>1.
- For large m, indicators equal the dimension of the zero weight space.
- Specific bounds for m where this equality holds in classical Lie algebras.

## Abstract

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.