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

Abstract

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