Supertraces on the algebra of observables of the rational Calogero model based on the classical root system

TL;DR

This paper identifies all supertraces on the algebra of observables for rational Calogero models with harmonic interaction based on classical root systems, extending previous results for type A_N.

Contribution

It provides a complete classification of supertraces for B_N, C_N, and D_N root systems, revealing new algebraic structures in these models.

Findings

Existence of Q independent supertraces for each root system

Q(B_N)=Q(C_N) equals the number of partitions of N

Q(D_N) equals the number of partitions of N into even parts

Abstract

A complete set of supertraces on the algebras of observables of the rational Calogero models with harmonic interaction based on the classical root systems of B_N, C_N and D_N types is found. These results extend the results known for the case A_N. It is shown that there exist Q independent supertraces where Q(B_N)=Q(C_N) is a number of partitions of N into a sum of positive integers and Q(D_N) is a number of partitions of N into a sum of positive integers with even number of even integers.