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

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.