Traces on the Algebra of Observables of Rational Calogero Model based on the Root System

TL;DR

This paper investigates the algebra of observables in the rational Calogero model based on root systems, revealing the number of independent traces and supertraces related to conjugacy classes in the associated Coxeter group.

Contribution

It establishes the existence and counts of independent traces and supertraces in the algebra of observables for the Calogero model based on root systems, linking algebraic properties to Coxeter group conjugacy classes.

Findings

Number of independent traces equals the count of conjugacy classes without eigenvalue 1.

Number of independent supertraces equals the count of conjugacy classes without eigenvalue -1.

Reproduction of an older result regarding supertraces in the algebra as a superalgebra.

Abstract

It is shown that H_R(\nu), the algebra of observables of the rational Calogero model based on the root system R, possesses T(R) independent traces, where T(R) is the number of conjugacy classes of elements without eigenvalue 1 belonging to the Coxeter group W(R) generated by the root system R. Simultaneously, we reproduced an older result: the algebra H_R(\nu), considered as a superalgebra with a natural parity, possesses ST(R) independent supertraces, where ST(R) is the number of conjugacy classes of elements without eigenvalue -1 belonging to W(R).