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 $t{r}_{\nu }$ or the space of supertraces contains a~degenerate nonzero supertrace $st{r}_{\nu }$ and, as a~consequence, the algebra $\mathcal{H}$ has two-sided ideals: one consisting of all vectors in the kernel of the form $B_{t{r}_{\nu }}(x,y)=t{r}_{\nu }(xy)$ or another consisting of all vectors in the kernel of the form $B_{st{r}_{\nu }}(x,y)=st{r}_{\nu }(xy)$. 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.

13 pages. arXiv admin note: substantial text overlap with arXiv:1612.00536

References in corpus (2)

• Nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2
• Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models