Ideals generated by traces in the~algebra of symplectic reflections $H_{1,\nu_1,\nu_2}(I_2(2m))$

S. E. Konstein ; I. V. Tyutin

Teoretičeskaâ i matematičeskaâ fizika, Tome 187 (2016) no. 2, pp. 297-309

Résumé

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 $I_2(2m)$ depends on two parameters, $\nu_1$ and $\nu_2$. For each value of these parameters, the algebra admits an $m$-dimensional space of traces. A trace $\operatorname{tr}$ is said to be degenerate if the corresponding symmetric bilinear form $B_{\operatorname{tr}}(x,y)=\operatorname{tr}(xy)$ is degenerate. We find all values of the parameters $\nu_1$ and $\nu_2$ for which the space of traces contains degenerate traces and the algebra $\mathcal H$ consequently has a two-sided ideal. It turns out that a linear combination of degenerate traces is also a degenerate trace. For the $\nu_1$ and $\nu_2$ values corresponding to degenerate traces, we find the dimensions of the space of degenerate traces.

Détail

Keywords: algebra of symplectic reflections, ideal, trace, group algebra.
Mots-clés : supertrace, Coxeter group