Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

TL;DR

This paper investigates the relationship between ideals generated by traces and supertraces in superalgebras associated with Calogero models, revealing conditions under which these ideals coincide based on their intersections with a specific subalgebra.

Contribution

It establishes a connection between ideals generated by traces and supertraces in superalgebras of Calogero models, providing criteria for their equality based on subalgebra intersections.

Findings

Ideals generated by traces and supertraces are equal if their intersections with the zero-grade subalgebra are equal.

The structure of the superalgebra decomposes under spin, influencing the ideals generated by (super)traces.

Conditions for the equality of ideals are characterized by their intersections with the kernel of certain bilinear forms.

Abstract

If $G$ is a finite Coxeter group, then symplectic reflection algebra $H:=H_{1,\eta}(G)$ has Lie algebra $\mathfrak {sl}_2$ of inner derivations and can be decomposed under spin: $H=H_0 \oplus H_{1/2} \oplus H_{1} \oplus H_{3/2} \oplus ...$. We show that if the ideals $\mathcal I_i$ ($i=1,2$) of all the vectors from the kernel of degenerate bilinear forms $B_i(x,y):=sp_i(x\cdot y)$, where $sp_i$ are (super)traces on $H$, do exist, then $\mathcal I_1=\mathcal I_2$ if and only if $\mathcal I_1 \bigcap H_0=\mathcal I_2 \bigcap H_0$.