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

TL;DR

This paper characterizes the parameter values in symplectic reflection algebras where traces or supertraces become degenerate, leading to the existence of two-sided ideals, thus revealing structural properties of these algebras.

Contribution

It determines all parameter values causing degeneracy of traces or supertraces in symplectic reflection algebras associated with dihedral groups.

Findings

Identifies parameter values with degenerate traces or supertraces.

Establishes existence of two-sided ideals of null-vectors at these parameters.

Provides analogous results for related algebra $H_{1, u_1, u_2}(I_2(2m))$.

Abstract

For each complex number $\nu$, an associative symplectic reflection algebra $\mathcal H:= H_{1,\nu}(I_2(2m+1))$, based on the group generated by root system $I_2(2m+1)$, has an $m$-dimensional space of traces and an $(m+1)$-dimensional space of supertraces. A (super)trace $sp$ is said to be degenerate if the corresponding bilinear (super)symmetric form $B_{sp}(x,y)=sp(xy)$ is degenerate. We find all values of the parameter $\nu$ for which either the space of traces contains a degenerate nonzero trace or the space of supertraces contains a degenerate nonzero supertrace and, as a consequence, the algebra $\mathcal H$ has a two-sided ideal of null-vectors. The analogous results for the algebra $H_{1,\nu_1, \nu_2}(I_2(2m))$ are also presented.