Ideals generated by traces or by supertraces in the symplectic reflection algebra $H_{1,\nu}(I_2(2m+1))$ II

TL;DR

This paper investigates the ideals generated by traces and supertraces in a specific symplectic reflection algebra related to the Calogero model, showing that under certain conditions, the ideals coincide.

Contribution

It proves that the ideals generated by degenerate traces and supertraces in the algebra $H_{1, u}(I_2(2m+1))$ are identical when both degenerate forms exist.

Findings

Degenerate trace and supertrace ideals coincide under specific parameter conditions.

Identifies parameter values where both degenerate forms exist.

Provides a detailed analysis of the algebra's ideal structure.

Abstract

The algebra $\mathcal H:= H_{1,\nu}(I_2(2m+1))$ of observables of the Calogero model based on the root system $I_2(2m+1)$ has an $m$-dimensional space of traces and an $(m+1)$-dimensional space of supertraces. In the preceding paper we found all values of the parameter $\nu$ for which either the space of traces contains a~degenerate nonzero trace $tr_{\nu}$ or the space of supertraces contains a~degenerate nonzero supertrace $str_{\nu}$ and, as a~consequence, the algebra $\mathcal H$ has two-sided ideals: one consisting of all vectors in the kernel of the form $B_{tr_{\nu}}(x,y)=tr_{\nu}(xy)$ or another consisting of all vectors in the kernel of the form $B_{str_{\nu}}(x,y)=str_{\nu}(xy)$. We noticed that if $\nu=\frac z {2m+1}$, where $z\in \mathbb Z \setminus (2m+1) \mathbb Z$, then there exist both a degenerate trace and a~degenerate supertrace on $\mathcal H$. Here we prove that the ideals determined by these degenerate forms coincide.