Ideals generated by traces in the symplectic reflection algebra $H_{1,\nu_1, \nu_2}(I_2(2m))$. II

TL;DR

This paper investigates the structure of traces in the symplectic reflection algebra associated with the dihedral group, identifying conditions under which degenerate traces form a 2-dimensional space and generate specific ideals.

Contribution

It characterizes the space of degenerate traces in the algebra and shows how they generate three proper ideals for certain parameter values.

Findings

Degenerate traces form a 2-dimensional space for specific parameters.

Non-zero traces in this space generate three proper ideals.

The structure of these ideals depends on the parameters $ u_1$ and $ u_2$.

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 ${\rm tr}$ is called degenerate if the associated bilinear form $B_{\rm tr}(x,y)={\rm tr}(xy)$ 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 $\mathcal H$.