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Given a non-acyclic quiver without loops with Kac's root system associated. When do we know there are infinitely many real roots?

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  • $\begingroup$ This is a good example of a bad title... $\endgroup$
    – Tom De Medts
    Commented Apr 1, 2015 at 9:05

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Always. If the quiver has a cycle, then the root system is not of finite type (since the quivers corresponding to finite type root systems are trees), and any root system not of finite type has an infinite number of real roots (more or less by definition).

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