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تسجيل مستخدم جديدIdeals generated by traces or by supertraces in the symplectic reflection algebra $H_{1, u}(I_2(2m+1))$ II
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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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اقرأ أيضاً
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 supertra
ces. 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.
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 $Sp(2,mathbb C)$.
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.
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 $mathcal I_1 bigcap H_0=mathcal I_2 bigcap H_0$.
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).
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