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Register a new userThe Number of Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System
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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).
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In the Coxeter group W(R) generated by the root system R, let T(R) be the number of conjugacy classes having no eigenvalue 1 and let S(R) be the number of conjugacy classes having no eigenvalue -1. The algebra H{R) of observables of the rational Calogero model based on the root system R possesses T(R) independent traces, the same algebra considered as an associative superalgebra with respect to a certain natural parity possesses S(R) even independent supertraces and no odd trace or supertrace. The numbers T(R) and S(R) are determined for all irreducible root systems (hence for all root systems). It is shown that T(R) =< S(R), and T(R) = S(R) if and only if superalgebra H(R) contains a Klein operator (or, equivalently, W(R) containes -1).
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.
It is shown that H_R( u), the algebra of observables of the rational Calogero model based on the root system R, possesses T(R) independent traces, where T(R) is the number of conjugacy classes of elements without eigenvalue 1 belonging to the Coxeter group W(R) generated by the root system R. Simultaneously, we reproduced an older result: the algebra H_R( u), considered as a superalgebra with a natural parity, possesses ST(R) independent supertraces, where ST(R) is the number of conjugacy classes of elements without eigenvalue -1 belonging to W(R).
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 $mathcal I_1 bigcap H_0=mathcal I_2 bigcap H_0$.
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.
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