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Register a new userConnection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models
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If $G$ is a finite Coxeter group, then symplectic reflection algebra $H:=H_{1,eta}(G)$ has Lie algebra $mathfrak {sl}_2$ of inner derivations and can be decomposed under spin: $H=H_0 oplus H_{1/2} oplus H_{1} oplus H_{3/2} oplus ...$. We show that if the ideals $mathcal I_i$ ($i=1,2$) of all the vectors from the kernel of degenerate bilinear forms $B_i(x,y):=sp_i(xcdot y)$, where $sp_i$ are (super)traces on $H$, do exist, then $mathcal I_1=mathcal I_2$ if and only if $mathcal I_1 bigcap H_0=mathcal I_2 bigcap H_0$.
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In the Coxeter group W(R) generated by the root system R, let Q(R) be the number of conjugacy classes having no eigenvalue -1. The superalgebra of observables of the rational Calogero model based on the root system R possesses Q(R) supertraces. The numbers Q(R) are determined for all irreducible root systems (hence for all root systems).
The associative superalgebra of observables of 3-particle Calogero model giving all wavefunctions of the model via standard Fock procedure has 2 independent supertraces. It is shown here that when the coupling constant u is equal to n+1/3, n-1/3 or n+1/2 for any integer n the existence of 2 independent supertraces leads to existence of nontrivial two-sided ideal in the superalgebra of observables.
It is shown that the superalgebra of observables of the rational Calogero model based on the root system of I_2(n) type possesses [(n+1)/2] supertraces. Model with three-particle interaction based on the root system G_2 belongs to this class of models and its superalgebra of observables has 3 independent supertraces.
For each complex number $ u$, an associative symplectic reflection algebra $mathcal H:= H_{1, u}(I_2(2m+1))$, 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 $ u$ 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 $H_{1, u_1, u_2}(I_2(2m))$ are also presented.
The algebra $mathcal H:= H_{1, u}(I_2(2m+1))$ of observables of the Calogero model based on the root system $I_2(2m+1)$ has an $m$-dimensional space of traces and an $(m+1)$-dimensional space of supertraces. In the preceding paper we found all values of the parameter $ u$ for which either the space of traces contains a~degenerate nonzero trace $tr_{ u}$ or the space of supertraces contains a~degenerate nonzero supertrace $str_{ u}$ and, as a~consequence, the algebra $mathcal H$ has two-sided ideals: one consisting of all vectors in the kernel of the form $B_{tr_{ u}}(x,y)=tr_{ u}(xy)$ or another consisting of all vectors in the kernel of the form $B_{str_{ u}}(x,y)=str_{ u}(xy)$. We noticed that if $ u=frac z {2m+1}$, where $zin mathbb Z setminus (2m+1) mathbb Z$, then there exist both a degenerate trace and a~degenerate supertrace on $mathcal H$. Here we prove that the ideals determined by these degenerate forms coincide.
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