It is shown that $A:=H_{1,eta}(G)$, the Sympectic Reflection Algebra, has $T_G$ independent traces, where $T_G$ is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group $G$ generated by the system of symplecti
c reflections. Simultaneously, we show that the algebra $A$, considered as a superalgebra with a natural parity, has $S_G$ independent supertraces, where $S_G$ is the number of conjugacy classes of elements without eigenvalue -1 belonging to $G$. We consider also $A$ as a Lie algebra $A^L$ and as a Lie superalgebra $A^S$. It is shown that if $A$ is a simple associative algebra, then the supercommutant $[A^{S},A^{S}]$ is a simple Lie superalgebra having at least $S_G$ independent supersymmetric invariant non-degenerate bilinear forms, and the quotient $[A^L,A^L]/([A^L,A^L]capmathbb C)$ is a simple Lie algebra having at least $T_G$ independent symmetric invariant non-degenerate bilinear forms.
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 Calo
gero 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).
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 n
umbers Q(R) are determined for all irreducible root systems (hence for all root systems).
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).
We consider antiPoisson superalgebra realized on the smooth Grassmann-valued functions of the form xi f_0(x)+f_1(x), where f_0 has compact support on R, and with the parity opposite to that of the Grassmann superalgebra realized on these functions. T
he deformations with even and odd deformation parameters of this superalgebra are found.
We consider antibracket superalgebras realized on the smooth Grassmann-valued functions with compact supports in n-dimensional space and with the grading inverse to Grassmanian parity. The deformations with even and odd deformation parameters of these superalgebras are presented for arbitrary n.
