Supertraces on the Algebras of Observables of the Rational Calogero Model with Harmonic Potential

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We define a complete set of supertraces on the algebra SHₙ(ν), the algebra of observables of the N-body rational Calogero model with harmonic interaction. This result extends the previously known results for the simplest cases of N=1 and N=2 to arbitrary N. It is shown that SHₙ(ν) admits q(N) independent supertraces where q(N) is a number of partitions of N into a sum of odd positive integers, so that q(N)>1 for N≥ 3. Some consequences of the existence of several independent supertraces of SHₙ (ν) are discussed such as the existence of ideals in associated W∞ - type Lie superalgebras.