Ideals generated by traces in the symplectic reflection algebra $H_{1,{\nu }_1,{\nu }_2}(I_2(2m))$. II

Abstract

The associative algebra of symplectic reflections $\mathcal{H}:=H_{1,{\nu }_1,{\nu }_2}(I_2(2m))$ based on the group generated by the root system $I_2(2m)$ has two parameters, ${\nu }_1$ and ${\nu }_2$. For every value of these parameters, the algebra $\mathcal{H}$ has an $m$-dimensional space of traces. A given trace $\mathrm{tr}$ is called degenerate if the associated bilinear form $B_{\mathrm{tr}}(x,y)=\mathrm{tr}(xy)$ is degenerate. Previously, there were found all values of ${\nu }_1$ and ${\nu }_2$ for which there are degenerate traces in the space of traces, and consequently the algebra $\mathcal{H}$ has a two-sided ideal. We proved earlier that any linear combination of degenerate traces is a degenerate trace. It turns out that for certain values of parameters ${\nu }_1$ and ${\nu }_2$, degenerate traces span a 2-dimensional space. We prove that non-zero traces in this $2d$ space generate three proper ideals of $\mathcal{H}$.

14 pages. arXiv admin note: text overlap with arXiv:1612.00536