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S. E. Konstein

Publications and source records attributed to S. E. Konstein.

At least 19 recordsLinked to original sources
arXiv2021

Ideals generated by traces in the symplectic reflection algebra $H_{1,ν_1, ν_2}(I_2(2m))$. II

The associative algebra of symplectic reflections $\mathcal H:= H_{1,ν_1, ν_2}(I_2(2m))$ based on the group generated by the root system $I_2(2m)$ has two parameters, $ν_1$ and $ν_2$. For every value of these parameters, the algebra $\mathcal H$ has an $m$-dimensional space of traces. A given trace ${\rm tr}$ is called degenerate if the associated bilinear form $B_{\rm tr}(x,y)={\rm tr}(xy)$ is degenerate. Previously, there were found all values of $ν_1$ and $ν_2$ for which there are degenerate traces in the space of traces, and consequently the algebra $\mathcal H$ has a two-sided ideal. We proved earlier that any linear combination of degenerate traces is a degenerate trace. It turns out that for certain values of parameters $ν_1$ and $ν_2$, degenerate traces span a 2-dimensional space. We prove that non-zero traces in this $2d$ space generate three proper ideals of $\mathcal H$.

hep-th↗
arXiv2020

Ideals generated by traces or by supertraces in the symplectic reflection algebra $H_{1,ν}(I_2(2m+1))$ II

The algebra $\mathcal H:= H_{1,ν}(I_2(2m+1))$ 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 $ν$ for which either the space of traces contains a~degenerate nonzero trace $tr_ν$ or the space of supertraces contains a~degenerate nonzero supertrace $str_ν$ and, as a~consequence, the algebra $\mathcal H$ has two-sided ideals: one consisting of all vectors in the kernel of the form $B_{tr_ν}(x,y)=tr_ν(xy)$ or another consisting of all vectors in the kernel of the form $B_{str_ν}(x,y)=str_ν(xy)$. We noticed that if $ν=\frac z {2m+1}$, where $z\in \mathbb Z \setminus (2m+1) \mathbb Z$, 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.

math.RT↗
arXiv2019

Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

If $G$ is a finite Coxeter group, then symplectic reflection algebra $H:=H_{1,η}(G)$ 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 $B_i(x,y):=sp_i(x\cdot y)$, where $sp_i$ are (super)traces on $H$, do exist, then $\mathcal I_1=\mathcal I_2$ if and only if $\mathcal I_1 \bigcap H_0=\mathcal I_2 \bigcap H_0$.

math-ph↗
arXiv2019

SO(4)-symmetry of mechanical systems with 3 degrees of freedom

We answered the old question: does there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra o(4) by means of the Poisson brackets? We presented a system which is not centrally symmetric, but has such 6 first integrals. We showed also that not every mechanical system with 3 degrees of freedom possesses such Lie algebra o(4).

physics.class-ph↗
arXiv2017

The number of independent Traces and Supertraces on the Symplectic Reflection Algebra $H_{1,ν}(Γ\wr S_N)$

Symplectic reflection algebra $ H_{1, \,ν}(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 $ν$. In this paper, the values $T(G)$ and $S(G)$ are explicitly calculated for the groups $G= Γ\wr S_N$, where $Γ$ is a finite subgroup of $Sp(2,\mathbb C)$.

math.RT↗
arXiv2017

Klein operator and the Numbers of independent Traces and Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System

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).

math.RT↗
arXiv2016

Ideals generated by traces or by supertraces in the symplectic reflection algebra $H_{1,ν}(I_2(2m+1))$

For each complex number $ν$, an associative symplectic reflection algebra $\mathcal H:= H_{1,ν}(I_2(2m+1))$, 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 $ν$ 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 $H_{1,ν_1, ν_2}(I_2(2m))$ are also presented.

math.RT↗
arXiv2014

The number of independent Traces and Supertraces on Symplectic Reflection Algebras

It is shown that $A:=H_{1,η}(G)$, 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 $[A^L,A^L]/([A^L,A^L]\cap\mathbb C)$ is a simple Lie algebra having at least $T_G$ independent symmetric invariant non-degenerate bilinear forms.

math.RT↗
arXiv2012

Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System

The superalgebra of observables of the rational Calogero model based on the root system R is the associative superalgebra generated by polynomials in N indeterminates, the differential-difference Dunkl's operators and the group algebra of the Coxeter group G generated by the root system R. It is shown that this superalgebra possesses Q_R supertraces, where Q_R is the number of conjugacy classes of the Coxeter group G which have no eigenvalue equal to -1.

math-ph↗
arXiv2012

Traces on the Algebra of Observables of Rational Calogero Model based on the Root System

It is shown that H_R(ν), 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(ν), 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).

math.RT↗
arXiv2010

The deformations of antibracket with even and odd deformation parameters

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.

math-ph↗
arXiv2008

Cohomology of antiPoisson superalgebra

We consider antiPoisson superalgebras realized on the smooth Grassmann-valued functions with compact supports in R^n and with the grading inverse to Grassmanian parity. The lower cohomologies of these superalgebras are found.

hep-th↗
arXiv2006

General form of the deformation of Poisson superbracket on (2,2)-dimensional superspace

Continuous formal deformations of the Poisson superbracket defined on compactly supported smooth functions on n-dimensional space taking values in a Grassmann algebra with m generating elements are described up to an equivalence transformation for the case n=m=2. It is shown that in this case the Poisson superalgebra has an additional deformation comparing with other superdimensions (n,m).

hep-th↗