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

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.