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

Abstract

We define a complete set of supertraces on the algebra $S{H}_N(\nu )$, 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 $S{H}_N(\nu )$ 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 $S{H}_N(\nu )$ are discussed such as the existence of ideals in associated $W_{\infty }$ - type Lie superalgebras.

22 pages, LATEX

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Cited by in corpus (8)

• Euler-Calogero-Moser system from SU(2) Yang-Mills theory
• Klein operator and the Numbers of independent Traces and Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System
• Traces on the Algebra of Observables of Rational Calogero Model based on the Root System
• The number of independent Traces and Supertraces on Symplectic Reflection Algebras
• Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models
• The number of independent Traces and Supertraces on the Symplectic Reflection Algebra $H_{1,\nu }(\Gamma \wr S_N)$
• Ideals generated by traces or by supertraces in the symplectic reflection algebra $H_{1,\nu }(I_2(2m+1))$
• Supertraces on the algebra of observables of the rational Calogero model based on the classical root system