Journal of Nonlinear Mathematical Physics

Volume 28, Issue 1, March 2021, Pages 123 - 133

Ideals Generated by Traces or by Supertraces in the Symplectic Reflection Algebra H1,V(I2(2m + 1)) II

Authors
I.A. Batalin, S.E. Konstein*, I.V. Tyutin
I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, RAS 119991, Leninsky Prosp., 53, Moscow, Russia
*Corresponding author. Email: konstein@lpi.ru
Corresponding Author
S.E. Konstein
Received 21 August 2020, Accepted 24 August 2020, Available Online 10 December 2020.
DOI
https://doi.org/10.2991/jnmp.k.200922.012How to use a DOI?
Keywords
symplectic reflection algebra, trace, supertrace, ideal, dihedral group
Abstract

The algebra 𝒣≔H1,ν(I2(2m+1)) of observables of the Calogero model based on the root system I2(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 trν or the space of supertraces contains a degenerate nonzero supertrace strν and, as a consequence, the algebra 𝒣 has two-sided ideals: one consisting of all vectors in the kernel of the form Btrν(x,y)=trν(xy) or another consisting of all vectors in the kernel of the form Bstrν(x,y)=strν(xy) . We noticed that if ν=z2m+1 , where z∈𝕑\(2m+1)𝕑 , then there exist both a degenerate trace and a degenerate supertrace on 𝒣 . Here we prove that the ideals determined by these degenerate forms coincide.

Copyright
© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

This paper is a continuation of [5]; we advise the reader to recall [5].

1.1. Definitions

Let 𝒜 be an associative 𝕑2 -graded algebra with unit; let ε denote its parity. All expressions of linear algebra are given for homogenous elements only and are supposed to be extended to inhomogeneous elements via linearity.

A linear complex-valued function tr on 𝒜 is called a trace if tr(fg − gf) = 0 for all f,g∈𝒜 . A linear complex-valued function str on 𝒜 is called a supertrace if str(fg − (−1)ε(f)ε(g)gf) = 0 for all f,g∈𝒜 . These two definitions can be unified as follows.

Let ϰ=±1 . A linear complex-valued function spϰ on 𝒜 is called ϰ -trace if spϰ(fg-ϰɛ(f)ɛ(g)gf)=0 for all f,g∈𝒜 .

Each nonzero ϰ -trace spϰ defines the nonzero symmetric1 bilinear form Bspϰ(f,g)≔spϰ(fg) .

If Bspϰ is degenerate, then the set of the vectors of its kernel is a proper ideal in 𝒜 . We say that the ϰ -trace spϰ is degenerate if the bilinear form Bspϰ is degenerate.

1.2. The Goal and Structure of the Paper

The simplicity (or, alternatively, existence of ideals) of Symplectic Reflection Algebras or, briefly, SRA (for definition, see [3]) was investigated in a number of papers, see, e.g., [2,9]. In particular, it is shown that all SRA H1,ν(G) with ν = 0 are simple (see [2,10]).

It follows from [4] and [7] that an associative algebra of observables of the Calogero model with harmonic term in the potential and with coupling constant ν based on the root system I2(2m + 1) (this algebra is SRA denoted H1,ν(I2(2m + 1))) has an m-dimensional space of traces and an (m + 1)-dimensional space of supertraces.

We say that the parameter ν is singular, if the algebra H1,ν(I2(n)) has a degenerate trace or a degenerate supertrace.

In [5], we found all singular values of ν for the algebras 𝒣≔H1,ν(I2(n)) in the case of n odd (n = 2m + 1) and found the corresponding degenerate traces and supertraces; the result is formulated in Theorem 10.1.

We noticed that if ν=z2m+1 , where z∈𝕑\(2m+1)𝕑 , then there exist both a degenerate trace and a degenerate supertrace on H.

Denote this degenerate trace by trz and the degenerate supertrace by strz

Theorem 10.1 proved in [5] implies that if z∈𝕑\n𝕑 , then

  1. (i)

    the trace given by the formula (10.1) in [5] is degenerate and generates the ideal 𝒤trz consisting of all the vectors in the kernel of the degenerate form Btrz(x,y)=trz(xy) ,

  2. (ii)

    the supertrace (10.2) is degenerate and generates the ideal 𝒤strz consisting of all the vectors in the kernel of the degenerate form Bstrz(x,y)=strz(xy) .

The goal of this paper is Theorem 13.1, which proves

Conjecture 1.1.

([5, Conjecture 9.1]) 𝒤trz=𝒤strz .

In Sections 2–10 we recall the necessary definitions and preliminary facts.

2. THE GROUP I2(2m + 1)

Hereafter in this paper, n = 2m + 1.

Definition 2.1.

The group I2(n) is a finite subgroup of the orthogonal group O(2,𝕉) generated by the root system I2(n).

The group I2(n) is the symmetry group of a flat regular n-gon; I2(n) consists of n reflections Rk and n rotations Sk, where k = 0, 1, …, 2m. We consider the indices k as integers modulo n.

These elements (Rk and Sk for all k) satisfy the relations

RkRl=Sk-l,  SkSl=Sk+l,  RkSl=Rk-l,  SkRl=Rk+l.

The element S0 is the unit in the group I2(n). Obviously, since n is odd, all the reflections Rk are in the same conjugacy class.

The rotations Sk and Sl constitute a conjugacy class if k + l = n.

Let

λ≔exp(2πin).

Let

G≔𝔺[I2(n)] (2.1)

be the group algebra of the group I2(n). In G, it is convenient to introduce the following basis

Lp≔1n∑k=0n-1λkpRk,  Qp≔1n∑k=0n-1λ-kpSk.

3. SYMPLECTIC REFLECTION ALGEBRA H1,v(I2(2m + 1))

Definition 3.1.

The symplectic reflection algebra 𝒣≔H1,ν(I2(2m+1)) is the associative algebra of polynomials in the noncommuting elements aα and bα, where α = 0, 1, with coefficients in G [see Eq. (2.1)], satisfying the relations

Lpaα=-bαLp+1,  Lpbα=-aαLp-1,Qpaα=aαQp+1,  Qpbα=bαQp-1,LkLl=δk+lQl,  LkQl=δk-lLl,QkLl=δk+lLl,  QkQl=δk-lQl, (3.1)
[aα,bβ]=ɛαβ(1+μL0),[aα,aβ]=ɛαβμL1,[bα,bβ]=ɛαβμL-1,
where δk ≔ δk0, and εαβ is the skew-symmetric tensor normalized so that ε01 = 1, and
μ≔nν.

Defining the parity in 𝒣 by setting

ɛ(aα)=ɛ(bα)=1,   ɛ(Rk)=ɛ(Sk)=0,

we turn this algebra into a superalgebra.

The algebra H1,ν(I2(2m + 1)) depends on one complex parameter ν.

4. SUBALGEBRA OF SINGLETS

Consider the elements2 Tαβ≔12({aα,bβ}+{bα,aβ}) of the algebra 𝒣 , and the inner derivations of 𝒣 they generate:

Dαβ:  f↦[f,Tαβ]        for any f∈𝒣.

It is easy to verify that the linear span of these derivations is a Lie algebra isomorphic to sl2.

Definition 4.1.

A singlet is any element f∈𝒣 such that [f, Tαβ] = 0 for all α, β. The subalgebra H0⊂𝒣 consisting of all singlets of the algebra 𝒣 is called the subalgebra of singlets.

One can consider the algebra 𝒣 as an sl2-module and decompose it into the direct sum of irreducible submodules.

Observe, that any ϰ-trace is identically zero on all irreducible sl2-submodules of 𝒣 , except for singlets.

Let the skew-symmetric tensor εαβ be normalized so that ε01 = 1 and so ∑αɛαβɛαγ=δβγ . We set

𝔰≔14i∑α,β=0,1({aα,bβ}-{bα,aβ})ɛαβ.

Proposition 4.1.

([5, Proposition 4.2]) The subalgebra of singlets H0 is the algebra of polynomials in the element 𝔰 with coefficients in the group algebra 𝔺[I2(2m+1)] .

The commutation relations of the singlet 𝔰 with generators of the algebra 𝒣 have the form:

[𝔰,Qp]=[𝔰,Sk]=[Tαβ,𝔰]=0,𝔰Lp=-Lp𝔰,  𝔰Rk=-Rk𝔰,(𝔰-iμL0)aα=aα(𝔰+i+iμL0).

5. THE FORM OF IDEALS IN 𝒣 AND IN H0

Theorem 5.1.

([5, Theorem 4.3]) Let 𝒤 be a proper ideal in the algebra 𝒣 , and 𝒤0≔𝒤∩H0 . Then, there exist nonzero polynomials ϕk0∈𝔺[𝔰] , where k = 0, ..., n − 1, such that 𝒤0 is the span over 𝔺[𝔰] of the elements

ϕk0(𝔰)Qk,   ϕn-k0Lk,  where k=0,...,n-1 and ϕn0≔ϕ00.

Proposition 5.1.

([5, Proposition 4.4]) If 𝒤⊂𝒣 is a proper ideal, then 𝒤0=𝒤∩H0 is a proper ideal in H0.

Definition 5.1.

For each p = 0, …, 2m, we define the ideals 𝒥p and 𝒥p in the algebra 𝔺[𝔰] by setting

𝒥p≔{f∈𝔺[𝔰]| f(𝔰)Qp∈𝒤},  𝒥p≔{f∈𝔺[𝔰]| f(𝔰)Lp∈𝒤}.

Proposition 5.2.

([5, Proposition 4.7]) We have 𝒥p=𝒥-p .

Proposition 5.3.

([5, Proposition 4.8]). We have 𝒥p≠0 for any p = 0, …, 2m.

Since 𝔺[𝔰] is a principal ideal ring, we have the following statement:

Corollary 5.1.

For any p = 0, …, 2m, there exists a nonzero polynomial ϕp0∈𝔺[𝔰] such that 𝒥p=ϕp0𝔺[𝔰] .

Theorem 5.1 evidently follows from Corollary 5.1.

6. GENERATING FUNCTIONS OF Ï°-TRACES

For each ϰ-trace ϰ on spϰ 𝒣 , one can define the following set of generating functions which allow one to calculate the ϰ-trace of arbitrary element in H0 via finding the values of the derivatives of these functions with respect to parameter t at zero:

Fpspϰ(t)≔spϰ(exp(t(𝔰-iμL0))Qp),Ψpspϰ(t)≔spϰ(exp(t𝔰)Lp),    where p = 0,...,2m. (6.1)

Since L0Qp = 0 for any p ≠ 0, it follows from the definition (6.1) that

Fpspϰ(t)=spϰ(exp(t𝔰)Qp)    if p≠0,F0spϰ(t)=spϰ(exp(t(𝔰-iμL0))Q0).

It is easy to find Ψpspϰ for  p≠0. Since 𝔰Lq=-Lq𝔰 for any q = 0, …, 2m, we have

Ψqspϰ(t)=spϰ(exp(t𝔰)Lq)=spϰ(Lq). (6.2)

Next, since spϰ(Rk) does not depend on k, we have spϰ(Lp) = 0 for any p ≠ 0 and

Ψpspϰ(t)≡0      for any p≠0. (6.3)

The value of spÏ°(L0) will be calculated later, in Section 9.

We consider also the functions

Φpspϰ(t)≔spϰ(exp(t(𝔰+iμL0))Qp).

It is easily verified, by expanding the exponential in a series, that these functions are related with the functions FpspÏ° by the formula

Φpspϰ(t)=Fpspϰ(t)+2iΔpspϰ(t),       whereΔp(t)spϰ=δpsin(μt)spϰ(L0).

The form of generating functions is related with (non)degeneracy of the form Bspκ as described in Proposition 7.1 below.

7. DEGENERACY CONDITIONS FOR THE Ï°-TRACE

Proposition 7.1.

([5, Proposition 6.1]). The ϰ-trace on the algebra 𝒣 is degenerate if and only if the generating functions Fpspϰ defined by formula (6.1) have the following form

Fpspϰ(t)=∑j=1jpexp(tωj,p)φj,p(t), (7.1)
where ωj,p∈𝔺 and φj,p∈𝔺[t] might depend on ϰ.

8. EQUATIONS FOR THE GENERATING FUNCTIONS FpspÏ°

In [5, Eq. (7.1)], the following system of differential equations for the generating functions is obtained:

ddtFpspϰ-ϰeitddtFp+1spϰ=iFpspϰ+ϰieitFp+1spϰ+2ϰiddt(eitΔp+1spϰ). (8.1)

The initial conditions for this system are:

FpspÏ°(0)=spÏ°(Qp).

To solve the system (8.1), we consider its Fourier transform. Let

λ≔e2πi/(2m+1),Gkspϰ≔∑p=02mλkpFpspϰ,                                  where k=0,...,2m,Δ˜kspϰ≔∑p=02mλkpΔp+1spϰ=λ-k(sin(μt)spϰ(L0)), where k=0,...,2m. (8.2)

For the functions GkspÏ° , we then obtain the equations

ddtGkspϰ=iλk+ϰeitλk-ϰeitGkspϰ+2iϰλkλk-ϰeitddt(eitΔ˜kspϰ) (8.3)

with the initial conditions

GkspÏ°(0)=spÏ°(Sk). (8.4)

We choose the following form of the solution of the system (8.3):

Gkspϰ(t)=ϰeit(ϰeit-λk)2λkgkspϰ(t), (8.5)

where

gkspϰ(t)=(2μ(cos(tμ)-1)+2iλ-k(λk-ϰeit)sin(tμ))spϰ(L0)+ϰλ-k(ϰ-λk)2spϰ(Sk). (8.6)

Evidently, this solution satisfies the initial condition (8.4) for each Ï° and k, except for the case where Ï° = +1 and k = 0.

If ϰ = +1 and k = 0, then the expression (8.5) for G0tr has a removable singularity at t = 0. In this case, instead of the condition G0tr(0)=tr(S0) we consider the condition limt→0G0tr(t)=tr(S0) .

When Ï° = +1 the solution (8.5) and (8.6) gives

G0tr(t)=eit(eit-1)2(2μ(cos(tμ)-1)+2i(1-eit)sin(tμ))tr(L0),
and one can easily see that
limt→0G0tr(t)=-μtr(L0).

It is shown in Subsection 9.1 that if Ï° = +1, then

tr(S0)=-μtr(L0)

for any trace tr on 𝒣 .

So, G0tr(t) satisfies the initial conditions (8.4) also.

In the case where ϰ = −1, the ϰ-trace is a supertrace (see [4]). In this case, the m + 1 values str(Sk) = str(S2m+1−k) for k = 0, …, m completely define the supertrace on 𝒣 (see [7]).

In the case where ϰ = +1, the ϰ-trace is a trace (see [4]). In this case, the m values tr(Sk) = tr(S2m+1−k) for k = 1, …, m completely define the trace on 𝒣 (see [7]). The value tr(S0) linearly depends on parameters tr(Sk), where k = 1, …, m, and this value is found in Subsection 9.1 (see Eqs. (9.4 and 9.5)).

9. VALUES OF THE ϰ-TRACE ON 𝔺[I2(2m+1)]

From [5] we have

spϰ(Rk)=-2μ2m+1(1+ϰ2Xtr+1-ϰ2Ystr), (9.1)

where

Xtr≔∑r=12msin2(πr2m+1)tr(Sr), (9.2)

Ystr≔∑r=02mcos2(πr2m+1)str(Sr). (9.3)

Below we consider these values for the traces and supertraces separately.

9.1. Values of the Traces (ϰ = +1) on 𝔺[I2(2m+1)]

The group I2(2m + 1) has m conjugacy classes without the eigenvalue +1 in the spectrum: {Sp, S2m+1−p}, where p = 1, ..., m.

By Theorem 2.3 in [4], the values of the trace on these conjugacy classes

sk≔tr(Sk),      where s2m+1-k=sk,  k=1,...,m,
are arbitrary and completely define the trace on the algebra 𝒣 . Therefore, the dimension of the space of traces is equal to m.

Further, the group I2(2m + 1) has one conjugacy class with one eigenvalue +1 in its spectrum: {R1, ..., R2m+1}. The value of tr(Rk) is expressed via sk by formula (9.1).

Besides, the group I2(2m + 1) has one conjugacy class with two eigenvalues +1 in its spectrum: {S0}.

The traces on conjugacy classes with two eigenvalues +1 in the spectrum is calculated in [5] using Ground Level Conditions (for their definition, see [4]):

tr(S0)=2ν2(2m+1)Xtr. (9.4)

We also note that

tr(L0)=-2μ2m+1Xtr, tr(Lp)=0 for p≠0,    tr(S0)=-μtr(L0). (9.5)

9.2. Values of the Supertraces (ϰ = −1) on 𝔺[I2(2m+1)]

The group I2(2m + 1) has m + 1 conjugacy classes without the eigenvalue −1 in the spectrum:

{S0},{Sp,S2m+1-p}, where p=1,...,m.

By [4, Theorem 2.3], the values of the supertrace on these conjugacy classes

uk≔str(Sk)=str(S2m+1-k), where k=0,...,m,

are arbitrary parameters that completely define the supertrace str on the algebra 𝒣 , and therefore the dimension of the space of supertraces is equal to m + 1.

Besides, the group I2(2m + 1) has one conjugacy class with one eigenvalue −1 in the spectrum: {R1, ..., R2m+1}.

The supertraces of the conjugacy class with eigenvalue −1 in its spectrum are given by Eq. (9.1): str(Rk) = −2νYstr, where k = 0, 1, …, 2m, and where Ystr is defined by Eq (9.3).

10. SINGULAR VALUES OF THE PARAMETER μ

The solution Eq. (8.5) and (8.6) determines the generation functions of traces and supertraces on H0 for any trace and any supertrace on 𝒣 . Generally speaking, Gkspϰ is a meromorphic function on t, but if µ and spϰ are such that the form Bspϰ is degenerate, then Gkspϰ is an integer function on t for each k. The complete list of such pairs of µ and spϰ is given in Theorem 10.1. For these values of µ and spϰ, the functions Gkspϰ are Laurent polynomials in exp(it).

Theorem 10.1.

([5, Theorem 9.1]). Let m∈𝕑 , where m ≥ 1, and n = 2m + 1. Then

  1. (1)

    The associative algebra H1,ν(I2(n)) has a one-parameter set of nonzero traces trz such that the symmetric invariant bilinear form Btrz(x,y)=trz(xy) is degenerate if and only if ν=zn , where z∈𝕑\n𝕑 . These traces are completely defined by their values at Sk for k = 1, …, m:

    trz(Sk)=τnsin2πkn(1-cos2πkzn),    where τ∈𝔺, τ≠0. (10.1)

    Here τ is an arbitrary parameter specifying the trace in one-dimensional space of traces.

  2. (2)

    The associative superalgebra H1,ν(I2(n)) has a one-parameter set of nonzero supertraces strz such that the symmetric invariant bilinear form Bstrz(x,y)=strz(xy) is degenerate if ν=zn , where z∈𝕑\n𝕑 . These supertraces are completely defined by their values at Sk for k = 0, …, m:

    strz(Sk)=τncos2πkn(1-(-1)zcos2πkzn),   where τ∈𝔺, τ≠0. (10.2)

    Here τ is an arbitrary parameter specifying the supertrace in one-dimensional space of supertraces.

  3. (3)

    The associative superalgebra H1,ν(I2(n)) has a one-parameter set of nonzero supertraces str1/2 such that the symmetric invariant bilinear form Bstr1/2(x,y)=str1/2(xy) is degenerate if ν=z+12 , where z∈𝕑 . These supertraces are completely defined by their values at Sk for k = 0, …, m:

    str1/2(Sk)=τncos2πkn,   where τ∈𝔺, τ≠0.

    Here τ is an arbitrary parameter specifying the supertrace in one-dimensional space of supertraces.

  4. (4)

    For all other values of ν, all nonzero traces and supertraces are nondegenerate.

11. GENERATING FUNCTIONS FpspÏ° FOR THE DEGENERATE Ï°-TRACE

Let μ∈𝕑\n𝕑 . Substitute the solutions (10.1) for the case ϰ = +1 and (10.2) for the case ϰ = −1 to Eqs. (8.5) and (8.6). We obtain the formula for both values of ϰ

gkspϰ=-4τn[cos(tμ)+iμλ-k(λk-ϰeit)sin(tμ)-ϰμcos2πkμn]. (11.1)

Introducing the new variable y instead of t

y≔ϰeit (11.2)

we can rewrite Eq. (11.1) in the form

gkspϰ=-2τnϰμ[(yμ+y-μ)+μλ-k(λk-y)(yμ-y-μ)-2cos2πkμn]

and Eq. (8.5) in the form

Gkspϰ=λky(y-λk)2gkspϰ. (11.3)

Now we see that Gkspϰ are the Laurent polynomials in y with the highest degree ≤|μ| and the lowest degree ≥1−|μ|.

Note, that the expressions (11.3) are even functions of the parameter µ, so we can assume that µ is a positive integer.

Let µ > 0 in what follows.

Thus, GkspÏ° can be expressed in the form

Gkspϰ=ϰμ∑𝓁=μ1-μβ𝓁ky𝓁, (11.4)
where the β𝓁k are constants not depending on ϰ and not all of them equal to zero.

Equation (11.4) implies that

βμk=2τμn. (11.5)

Further, Eq. (8.2) implies

Fpspϰ=1n∑k=02mλ-kpGkspϰ
and the generating functions FpspÏ° have the form
Fpspϰ=ϰμ∑𝓁=μ-μα𝓁py𝓁, (11.6)
where the α𝓁k are constants not depending on ϰ . Observe that Fpspϰ can be equal to zero for some p ≠ 0 (e.g., if µ = 1, then Fpspϰ=0 for each p ≠ 0), but F0spϰ≠0 since Eq. (11.5) implies αμ0=2τμn≠0 . Equation (11.5) implies also that α-μ0=0 .

12. THE GENERATING FUNCTION 𝒡spϰ = spϰ(exp(t𝔰)Q0) FOR THE DEGENERATE ϰ-TRACE

Let μ∈𝕑\n𝕑 and ϰ-trace be defined by Eq. (10.1) in the case ϰ = +1 and by Eq. (10.2) in the case ϰ = −1.

In this section we introduce the function

𝒡spϰ≔spϰ(exp(ts)Q0)
and express it via F0spÏ° .

Proposition 12.1.

𝒡spϰ is an even function of t:

𝒡spϰ=spϰ(cosh(t𝔰)Q0. (12.1)

Indeed, 𝒡spϰ=spϰ(cosh(t𝔰)Q0+sinh(t𝔰)Q0) and spϰ(sinh(t𝔰)Q0)=0 since

spϰ(sinh(t𝔰)Q0)=spϰ((sinh(t𝔰)L0)L0)=spϰ(L0(sinh(t𝔰)L0))==spϰ((L0sinh(t𝔰))L0)=spϰ((-(sinh(t𝔰)L0))L0)=spϰ(-sinh(t𝔰)Q0)

Now, decompose F0spÏ° :

F0spϰ=spϰ(et(𝔰-iμL0)Q0)=Feven+Fodd                     whereFeven=spϰ(∑s=0∞1(2s)!(t(𝔰-iμL0))2sQ0)=spϰ(∑s=0∞1(2s)!t2s(𝔰2-μ2)sQ0), (12.2)
Fodd=spϰ(∑s=0∞1(2s+1)!(t(𝔰-iμL0))2s+1Q0)     =spϰ(∑s=0∞1(2s+1)!t2s+1(𝔰2-μ2)s(𝔰-iμL0)Q0)     =spϰ(∑s=0∞1(2s+1)!t2s+1(𝔰2-μ2)s(-iμL0)Q0)     =∑s=0∞1(2s+1)!t2s+1(-μ2)s(-iμ)spϰL0     =sinh(-iμ)spϰL0=-ϰμ2(yμ-y-μ)spϰL0. (12.3)
Equation (11.6) implies that
Fodd=ϰμ2(∑𝓁=μ-μα𝓁0y𝓁-∑𝓁=μ-μα-𝓁0y𝓁). (12.4)

Comparing Eq. (12.4) with Eq. (12.3) implies

α𝓁0=α-𝓁0,  if 𝓁≠μ,𝓁≠-μ,αμ0-α-μ0=-spϰL0, (12.5)
and
Feven=ϰμ2αμ0(yμ+y-μ)+ϰμ2∑𝓁=0μ-1α𝓁0(y𝓁+y-𝓁)=αμ0cosh(itμ)+ϰμ∑𝓁=0μ-1ϰ𝓁α𝓁0cosh(it𝓁). (12.6)

Proposition 12.2.

𝒡spϰ(t)=αμ0+ϰμ∑𝓁=0μ-1ϰ𝓁α𝓁0cosh(tμ2-𝓁2).

Proof. Taking Proposition 12.1 into account let us decompose Eq. (12.1) into the Taylor series:

𝒡spϰ(t)=∑s=0∞a2st2s(2s)!,
where a2s≔spϰ(𝔰2sQ0) for s = 0, 1, 2, ….

Equation (12.2) implies

a2s=(d2dt2+μ2)sFeven|t=0,
and Eq. (12.6) implies
a2s={aμ0+ϰμ∑𝓁=0μ-1ϰ𝓁α𝓁0          if s=0ϰμ∑𝓁=0μ-1ϰ𝓁α𝓁0(-𝓁2+μ2)s   if s≠0.

So

𝒡spϰ(t)=∑s=0∞a2st2s(2s)!=αμ0+ϰμ∑𝓁=0μ-1ϰ𝓁α𝓁0cosh(tμ2-𝓁2).

13. IDEALS GENERATED BY DEGENERATE Ï°-TRACES

Let μ=z∈𝕑\n𝕑 and the ϰ-trace be defined by Eq. (10.1) for the case ϰ = +1 and by Eq. (10.2) for the case ϰ = −1.

These degenerate Ï°-traces are denoted in Theorem 10.1 by trz and strz.

Denote the ideals generated by these traces spϰ by 𝒤ϰ ; in H0, consider the ideals 𝒤0ϰ≔𝒤ϰ∩H0 .

Now we can prove Conjecture 1.1 ([5, Conjecture 9.1]):

Theorem 13.1.

𝒤+1=𝒤-1 .

To prove Theorem 13.1 we use Theorem 4.2 from [6] which in our case implies

Theorem 13.2.

([6, Theorem 4.2]) 𝒤+1=𝒤-1 if and only if 𝒤+1∩H0=𝒤-1∩H0 .

So, Theorem 13.1 follows from

Theorem 13.3.

𝒤+1∩H0=𝒤-1∩H0 .

Proof. For degenerate spÏ°, we established the following facts:

Fpspϰ(t)=spϰ(et𝔰Qp)=ϰμ∑𝓁=μ-μα𝓁pϰ𝓁eit𝓁                               for p=1,2,...,n-1,𝒡spϰ(t)=spϰ(et𝔰Q0)=αμ0+ϰμ∑𝓁=0μ-1ϰ𝓁α𝓁0cosh(tμ2-𝓁2)  where  αμ0≠0,
and where the α-s do not depend on ϰ.

For any p = 1, …, n, it is easy to find the lowest degree polynomial differential operators with constant coefficients Dpϰ(d/dt) such that Dpϰ(d/dt)Fpspϰ(t)=0 :

Dpϰ(ddt)=(∏𝓁=-μ: α𝓁p≠0μ(ddt-i𝓁)  if Fpspϰ≠0,1  if Fpspϰ=0,
and D0ϰ(d/dt) such that D0ϰ(d/dt)𝒡spϰ(t)=0 :
D0ϰ(ddt)=ddt∏𝓁=0: α𝓁0≠0μ-1(d2dt2-μ2+𝓁2).

Further, it is a simple exercise to prove that

Dpϰ(𝔰)Qp, Dpϰ(𝔰)L-p∈𝒤0ϰ   for any   p=0,...n-1,

namely,

Bspϰ(Dpϰ(𝔰)Qp,f)=Bspϰ(Dpϰ(𝔰)L-p,f)=0   for any f∈H0 and p = 0,...,n-1.

Consider, for example, Bspϰ(D0ϰ(𝔰)Q0,f) for f=g(𝔰)Qp and f=g(𝔰)Lp :

spϰ(D0ϰ(𝔰)Q0g(𝔰)Qp)=spϰ(D0ϰ(𝔰)Q0g(𝔰)Lp)=0  for  p≠0,

since Q0Qp = Q0Lp = 0 for p ≠ 0,

spϰ(D0ϰ(𝔰)Q0g(𝔰)Q0)=spϰ(D0ϰ(𝔰)g(𝔰)Q0)=D0ϰ(ddt)g(ddt)spϰ(et𝔰Q0)|t=0=g(ddt)D0ϰ(ddt)𝒡spϰ(t)|t=0=0,spϰ(D0ϰ(𝔰)Q0g(𝔰)L0)=spϰ(D0ϰ(𝔰)g(𝔰)L0)=D0ϰ(ddt)g(ddt)spϰ(et𝔰L0)|t=0=g(ddt)D0ϰ(ddt)spϰ(L0)=0

due to Eq. (6.2) and since the operator D0Ï°(ddt) contains the factor ddt .

Further, it is easy to see that for each of the ideals 𝒤0ϰ , where ϰ=±1 , the polynomials ϕp0∈𝔺[𝔰] defined in Corollary 5.1 satisfy the relations ϕp0(𝔰)=Dpϰ(𝔰) for p = 0, …, n − 1.

So, Theorem 5.1 implies that the 𝔺[𝔰] -span of the Dpϰ(𝔰)Qp and Dpϰ(𝔰)L-p for p = 0, …, n − 1 is 𝒤0ϰ .

Since Dp+1=Dp-1 , we have 𝒤0+1=𝒤0-1 , and as result, 𝒤+1=𝒤-1 .

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

ACKNOWLEDGMENTS

The authors are grateful to Russian Fund for Basic Research (grant No. 20-02-00193) for partial support of this work.

Footnotes

1

Initially, we used the term “(super)symmetric bilinear form” currently used by many, e.g., in the paper [1], even in its title. However, in a recent preprint [8], it is explained that the supersymmetry B(ν, w) = (−1)p(ν)p(w) B(w, ν) is related with the isomorphism V ⊗ W ≃ W ⊗ V; of superspaces and has nothing to do with the (anti)symmetry of the bilinear form B on V = W.

2

Here the brackets {.,.} denote anticommutator.

REFERENCES

[10]DS Passman, Infinite crossed products, Pure and Applied Mathematics, vol. 135, Academic Press, San Diego, 1989.
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
28 - 1
Pages
123 - 133
Publication Date
2020/12
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.2991/jnmp.k.200922.012How to use a DOI?
Copyright
© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - I.A. Batalin
AU  - S.E. Konstein
AU  - I.V. Tyutin
PY  - 2020
DA  - 2020/12
TI  - Ideals Generated by Traces or by Supertraces in the Symplectic Reflection Algebra H₁,V(I₂(2m + 1)) II
JO  - Journal of Nonlinear Mathematical Physics
SP  - 123
EP  - 133
VL  - 28
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.k.200922.012
DO  - https://doi.org/10.2991/jnmp.k.200922.012
ID  - Batalin2020
ER  -