Ideals generated by traces in the algebra of symplectic reflections \({H_{1,{v_{1,}}{v_2}}}\left( {{I_2}\left( {2m} \right)} \right)\)

Abstract

The associative algebra of symplectic reflections \(H: = {H_{1,{v_{1,}}{v_2}}}\left( {{I_2}\left( {2m} \right)} \right)\) based on the group generated by the root system I₂(2m) depends on two parameters, ν₁ and ν₂. For each value of these parameters, the algebra admits an m-dimensional space of traces. A trace tr is said to be degenerate if the corresponding symmetric bilinear form B_tr(x, y) = tr(xy) is degenerate. We find all values of the parameters ν₁ and ν₂ for which the space of traces contains degenerate traces and the algebra H consequently has a two-sided ideal. It turns out that a linear combination of degenerate traces is also a degenerate trace. For the ν₁ and ν₂ values corresponding to degenerate traces, we find the dimensions of the space of degenerate traces.