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Topological states of matter are characterized by topological invariants, which are physical quantities whose values are quantized and do not depend on details of the measured system. Among them the electrical Hall conductance, which is expressed in units of e2/h (e the electron charge, h the Planck constant), is easiest to probe. In the fractional quantum Hall effect (FQHE) regime, fractional quantized values of the electrical Hall conductance attest to topologically ordered states, which are states that carry quasi-particles with fractional charge and (expected) anyonic statistics. Another topological invariant, which is much harder to measure, is the thermal Hall conductance, KT, expressed in units of κ0T=(π2kB2/3h)T (kB the Boltzmann constant, T the temperature). Remarkably, in one-dimensional transport it does not depend on the particles charge, particles exchange statistics, and is even insensitive to the interaction strength among the particles. A fractional value of the quantized thermal Hall conductance shows that the probed state of matter is non-abelian. Quasiparticles in nonabelian states lead to a ground state degeneracy and perform topological unitary transformations among ground states when braided. As such, they may be useful for topological quantum computation. In this talk, I will report our measurements of the thermal Hall conductance in several quantum Hall states in the lowest [1] and first

excited [2] Landau levels. Remarkably, we find the thermal Hall conductance of the v=5/2 state to be fractional, with a value K=2.5κ0, implying non-abelian nature of the state.