Klein operator and the Numbers of independent Traces and Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System

TL;DR

This paper determines the number of independent traces and supertraces on the algebra of observables in the rational Calogero model based on root systems, revealing conditions for the existence of a Klein operator.

Contribution

It provides explicit formulas for T(R) and S(R) for all irreducible root systems and characterizes when T(R) equals S(R) based on the presence of a Klein operator.

Findings

T(R) and S(R) are explicitly calculated for all irreducible root systems.

T(R) is less than or equal to S(R), with equality if and only if a Klein operator exists.

The algebra's structure depends on the root system's properties and the Coxeter group.

Abstract

In the Coxeter group W(R) generated by the root system R, let T(R) be the number of conjugacy classes having no eigenvalue 1 and let S(R) be the number of conjugacy classes having no eigenvalue -1. The algebra H{R) of observables of the rational Calogero model based on the root system R possesses T(R) independent traces, the same algebra considered as an associative superalgebra with respect to a certain natural parity possesses S(R) even independent supertraces and no odd trace or supertrace. The numbers T(R) and S(R) are determined for all irreducible root systems (hence for all root systems). It is shown that T(R) =< S(R), and T(R) = S(R) if and only if superalgebra H(R) contains a Klein operator (or, equivalently, W(R) containes -1).