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

TL;DR

This paper characterizes all supertraces on the algebra of observables for the rational Calogero model with harmonic potential, revealing a rich structure of multiple supertraces for arbitrary particle number.

Contribution

It extends the classification of supertraces from small cases to general N, showing the algebra admits multiple independent supertraces related to partitions of N.

Findings

Number of supertraces equals the number of partitions of N into odd parts.

Existence of multiple supertraces implies ideals in related Lie superalgebras.

Results generalize previous specific cases to arbitrary N.

Abstract

We define a complete set of supertraces on the algebra $SH_N(\nu)$, 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 $SH_N(\nu)$ 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 $N\ge 3$. Some consequences of the existence of several independent supertraces of $SH_N (\nu )$ are discussed such as the existence of ideals in associated $W_{\infty}$ - type Lie superalgebras.