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Register a new userThe number of independent Traces and Supertraces on Symplectic Reflection Algebras
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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 symplectic 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.
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Symplectic reflection algebra $ H_{1, , u}(G)$ has a $T(G)$-dimensional space of traces whereas, when considered as a superalgebra with a natural parity, it has an $S(G)$-dimensional space of supertraces. The values of $T(G)$ and $S(G)$ depend on the symplectic reflection group $G$ and do not depend on the parameter $ u$. In this paper, the values $T(G)$ and $S(G)$ are explicitly calculated for the groups $G= Gamma wr S_N$, where $Gamma$ is a finite subgroup of $Sp(2,mathbb C)$.
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.
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).
The symplectic structures on $3$-Lie algebras and metric symplectic $3$-Lie algebras are studied. For arbitrary $3$-Lie algebra $L$, infinite many metric symplectic $3$-Lie algebras are constructed. It is proved that a metric $3$-Lie algebra $(A, B)$ is a metric symplectic $3$-Lie algebra if and only if there exists an invertible derivation $D$ such that $Din Der_B(A)$, and is also proved that every metric symplectic $3$-Lie algebra $(tilde{A}, tilde{B}, tilde{omega})$ is a $T^*_{theta}$-extension of a metric symplectic $3$-Lie algebra $(A, B, omega)$. Finally, we construct a metric symplectic double extension of a metric symplectic $3$-Lie algebra by means of a special derivation.
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