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

S.E. Konstein; I.V. Tyutin

doi:10.1080/14029251.2020.1684005

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Volume 27, Issue 1, October 2019, Pages 7 - 11

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

Authors

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,konstein@lpi.ru,tyutin@lpi.ru

^*Corresponding author

Corresponding Authors

S.E. Konstein, I.V. Tyutin

Received 19 August 2019, Accepted 30 August 2019, Available Online 25 October 2019.

DOI: 10.1080/14029251.2020.1684005 How to use a DOI?
Abstract: If G is a finite Coxeter group, then symplectic reflection algebra H := H_1,_η (G) has Lie algebra 𝔰𝔩₂ of inner derivations and can be decomposed under spin: H = H₀ ⊕ H_1/2 ⊕ H₁ ⊕ H_3/2 ⊕ ... We show that if the ideals ℐ_i (i = 1,2) of all the vectors from the kernel of degenerate bilinear forms B_i(x,y) := sp_i(x · y), where sp_i are (super)traces on H, do exist, then ℐ₁ = ℐ₂ if and only if ℐ₁ ∩ H₀ = ℐ₂ ∩H₀.
Copyright: © 2020 The Authors. Published by Atlantis and Taylor & Francis
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. Preliminaries and notation

Let 𝒜 be an associative superalgebra with parity π. All expressions of linear algebra are given for homogenous elements only and are supposed to be extended to inhomogeneous elements via linearity.

Definition 1.1.

A linear function str on 𝒜 is called a supertrace if

str(f⋅g)=(−1)π(f)π(g)str(g⋅f) for all f,g∈𝒜.

Definition 1.2.

A linear function tr on 𝒜 is called a trace if

tr(f⋅g)=tr(g⋅f) for all f,g∈𝒜.

We will use the notation “sp” and the term “(super)trace” to denote both cases, traces and super-traces, simultaneously.

2. The superalgebra of observables

Let V = ℝ^N be endowed with a positive definite symmetric bilinear form (·,·). For any nonzero v→∈V, define the reflections r_rv→ as follows:

rv→:x→↦x→−2(x→,v→)(v→,v→)v→ for any x→∈V.(2.1)

A finite set of non-zero vectors ℛ ⊂ V is said to be a root system and any vector v→∈ℛ is called a root if the following conditions hold:

i)
ℛ is rw→-invariant for any w→∈ℛ,
ii)
if v→1,v→2∈ℛ are proportional to each other, then either v→1=v→2 or v→1=−v→2.

The Coxeter group G ⊂ O(N,ℝ) ⊂ End(V) generated by all reflections rv→ with v→∈ℛ is finite.

We do not apply any conditions on the scalar products of the roots because we want to consider both crystallographic and non-crystallographic root systems, e.g., I₂(n) (see Theorem 4.1).

Let η be a complex-valued G-invariant function on ℛ, i.e., η(v→)=η(w→) if rv→ and rw→ belong to one conjugacy class of G.

We consider here the Symplectic Reflection (Super)algebra over complex numbers (see [6]) H := H_1,η(G) and call it the superalgebra of observables of Calogero model based on root system ℛ.^a

This algebra consists of noncommuting polynomials in 2N indeterminates aiα, where α = 0,1 and i = 1, ..., N, with coefficients in ℂ[G] satisfying the relations (see [6] Eq. (1.15))^b

[aiα,ajβ]=εαβ(δij+∑v→∈ℛη(v→)vivj(v→,v→)rv→),(2.2)

and

rv→aiα=∑j=1N(δij−2vivj(v→,v→))ajαrv→.(2.3)

Here ε^αβ is the antisymmetric tensor such that ε⁰¹ = 1, and v_i (i = 1,...,N) are the coordinates of the vector v→. The commutation relations (2.2), (2.3) suggest to define the parity π by setting:

π(aiα)=1 for any α, i; π(rv→)=0 for any v→∈ℛ.(2.4)

and we can consider the algebra H as a superalgebra as well.

3. 𝔰𝔩₂

Observe an important property of the superalgebra H: the Lie (super)algebra of its inner derivations contains the Lie subalgebra 𝔰𝔩₂ generated by operators

Dαβ:f↦Dαβf=[Tαβ,f],(3.1)

where α,β = 0,1, and f ∈ H, and polynomials T^αβ are defined as follows:

Tαβ:=12∑i=1N(aiαaiβ+aiβaiα).(3.2)

These operators satisfy the following relations:

[Dαβ,Dγδ]=εαγDβδ+εαδDβγ+εβγDαδ+εβδDαγ,(3.3)

since

[Tαβ,Tγδ]=εαγTβδ+εαδTβγ+εβγTαδ+εβδTαγ.

It follows from Eq. (3.3) that the operators D⁰⁰, D¹¹ and D⁰¹ = D¹⁰ constitute an 𝔰𝔩₂-triple:

[D01,D11]=2D11, [D01,D00]=−2D00, [D11,D00]=−4D01.

The polynomials T^αβ commute with ℂ[G], i.e., [Tαβ,rv→]=0]= 0, and act on the aiα as on vectors of the irreducible 2-dimensional 𝔰𝔩₂-modules:

Dαβaiγ=[Tαβ,aiγ]=εαγaiβ+εβγaiα, where i=1,…,N.(3.4)

We will denote this 𝔰𝔩₂ thus realized by the symbol SL2.

The subalgebra

H0:={f∈H|Dαβf=0 for any α,β}⊂H(3.5)

is called the subalgebra of singlets.

Introduce also the subspaces Hs:=⊕is=1∞Hsis, which is the direct sum of all irreducible SL2-modules Hsis of spin s, for s = 0, 1/2, 1,.... It is clear that H₀ is the defined above subalgebra of singlets.

The (super)algebra H can be decomposed in the following way

H=H0⊕Hrest, where Hrest:H1/2⊕H1⊕H3/2⊕….

Then each element f ∈ H can be represented in the form f = f₀ + f_rest, where f₀ ∈ H₀ and f_rest ∈ H_rest.

Note, that since SL2 is generated by inner derivations and T^αβ are even elements, each two-sided ideal ℐ ⊂ H can be decomposed in an analogous way: ℐ = ℐ₀ ⊕ ℐ_1/2 ⊕ ....

Since T^αβ are even elements of the superalgebra H, we have sp(D^αβ f)= 0 for any (super)trace sp on H, and hence the following proposition takes place^c:

Proposition 3.1.

sp(f)= sp(f₀) for any f ∈ H and any (super)trace sp on H.

Proof.

If s ≠ 0, then the elements of the form D^αβf, where α, β = 0, 1, and f∈Hsis, f ≠ 0, span the irreducible SL2-module Hsis. This implies sp f = 0 for any (super)trace on H and any f ∈ H_rest.

4. The (super)traces on H

It is shown in [9, 10, 12] that the algebra H has a multitude of independent (super)traces. For the list of dimensions of the spaces of the (super)traces on H_1,η(M) for all finite Coxeter groups M, see [8]. In particular, there is an m-dimensional space of traces and an (m + 1)-dimensional space of supertraces on H_1,η(I₂(2m + 1)).

Every (super)trace sp(·) on any associative (super)algebra 𝒜 generates the following bilinear form on 𝒜:

Bsp(f,g)=sp(f⋅g) for any f,g∈𝒜.(4.1)

It is obvious that if such a bilinear form B_sp is degenerate, then the kernel of this form (i.e., the set of all vectors f ∈ 𝒜 such that B_sp(f, g)= 0 for any g ∈ 𝒜) is the two-sided ideal ℐ^sp ⊂ 𝒜.

The ideals of this sort are found, for example, in [11, Theorem 9.1] (generalizing the results of [15, 16] and [7] for the two- and three-particle Calogero models).

Theorem 9.1 from [11] may be shortened to the following theorem:

Theorem 4.1.

Let m ∈ 𝕑, where m ⩾ 1 and n = 2m + 1. Then

1)
The associative algebra H_1,η(I₂(n)) has nonzero traces tr_η such that the symmetric invariant bilinear form B_trη (x,y)= tr_η(x · y) is degenerate if and only if η=zn, where z ∈ 𝕑 \ n𝕑. For each such z, all nonzero degenerate traces on H_1,_z_/_n(I₂(n)) are proportional to each other.
2)
The associative superalgebra H_1,η(I₂(n)) has nonzero supertraces str_η such that the supersymmetric invariant bilinear form B_strη (x,y)= str_η(x · y) is degenerate if η=zn, where z ∈ 𝕑 \ n𝕑. For each such z, all nonzero degenerate supertraces on H_1,_z_/_n(I₂(n)) are proportional to each other.
3)
The associative superalgebra H_1,η(I₂(n)) has nonzero supertraces str_η such that the supersymmetric invariant bilinear form B_str_η (x,y)= str_η(x · y) is degenerate if η=z+12, where z ∈ 𝕑. For each such z, all nonzero degenerate supertraces on H_1,_z_+1/2(I₂(n)) are proportional to each other.
4)
For all other values of η, all nonzero traces and supertraces are nondegenerate.

Theorem 4.1 implies that if z ∈ 𝕑 \ n𝕑, then there exists the degenerate trace tr_z generating the ideal ℐ^{tr^z} consisting of the kernel of the degenerate form B_{tr_z} (f, g)= tr_z(f · g), and simultaneously the degenerate supertrace str_z generating the ideal ℐ^{str^z} consisting of the kernel of the degenerate form B_{str_z} (f, g) = str_z(f · g).

A question arises: is it true that ℐ^{tr^z} = ℐ^{str^z}?

Answer to this and other similar questions can be considerably simplified by considering only the singlet parts of these ideals.

The following theorem justifies this method:

Theorem 4.2.

Let sp₁ and sp₂ be degenerate (super)traces on H. They generate the two-sided ideals ℐ₁ and ℐ₂ consisting of the kernels of bilinear forms B₁(f, g)= sp₁(f ·g) and B₂(f, g)= sp₂(f ·g), respectively.

Then ℐ₁ = ℐ₂ if and only if ℐ₁∩H₀ = ℐ₂∩H₀.

Proof.

It suffices to prove that if I₁∩H₀ = I₂∩H₀, then ℐ₁ = ℐ₂.

Consider any non-zero element f ∈ ℐ₁. For any g ∈ H, we have sp₁(f · g)= 0, f · g ∈ ℐ₁ and (f · g)₀ ∈ ℐ₁. So (f · g)₀ ∈ ℐ₁∩H₀. Due to hypotheses of this Theorem, (f · g)₀ ∈ ℐ₂ ∩H₀, and hence sp₂((f · g)₀)= 0. Proposition 3.1 gives sp₂(f ·g)= sp₂((f ·g)₀) which implies sp₂(f ·g)= 0.

Therefore, f ∈ ℐ₂.

Acknowledgments

The authors (S.K. and I.T.) are grateful to Russian Fund for Basic Research (grant No. 17-02-00317) for partial support of this work.

Footnotes

This algebra has a faithful representation via Dunkl differential-difference operators D_i, see [5], acting on the space of G-invariant smooth functions on V, namely a^iα=12(xi+(−1)αDi), see [1,14]. The Hamiltonian of the Calogero model based on the root system [2–4, 13] is the operator T^01 defined in (3.2) (see [1]). The wave functions are obtained in this model via the standard Fock procedure with the Fock vacuum |0〉 such that a^i0|0〉=0 for all i by acting on |0〉 with G-invariant polynomials of the a^i1.

The sign and coefficient of the sum in the rhs of Eq. (2.2) is chosen for obtaining the Calogero model in the form [1], Eq. (1), Eq. (5), Eq. (9), Eq. (10) when ℛ is of type A_N₋₁.

This elementary fact is known for a long time, see, eg, [12].

References

[1]L. Brink, H. Hansson, and M.A. Vasiliev, Explicit solution to the N-body Calogero problem, Phys. Lett, No. B286, 1992, pp. 109-111.

[2]F. Calogero, Solution of a three-body problem in one dimension, J. Math. Phys, Vol. 10, 1969, pp. 2191-2196.

[3]F. Calogero, Ground state of a one dimensional N-body problem, J. Math. Phys, Vol. 10, 1969, pp. 2197-2200.

[4]F. Calogero, Solution of the One-Dimensional N-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials, J. Math. Phys, Vol. 12, 1971, pp. 419.

[5]C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc, Vol. 311, No. 1, 1989, pp. 167-183.

[6]P. Etingof and V. Ginqzburg, Symplectic reflection algebras, Calogero–Moser space, and deformed Harish–Chandra homomorphism, Inv. Math, Vol. 147, 2002, pp. 243-348.

[7]S.E. Konstein, 3-particle Calogero Model: Supertraces and Ideals on the Algebra of Observables, Teor. Mat. Fiz, Vol. 116, 1998, pp. 122. arXiv:hep-th/9803213

[8]S.E. Konstein and R. Stekolshchik, Klein operator and the Number of Traces and Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System, Journal of Nonlinear Mathematical Physics, Vol. 20, No. 2, 2013, pp. 295-308.

[9]S.E. Konstein and I.V. Tyutin, Traces on the Algebra of Observables of the Rational Calogero Model Based on the Root System, Journal of Nonlinear Mathematical Physics, Vol. 20, No. 2, 2013, pp. 271-294. arXiv:1211.6600

[10]S.E. Konstein and I.V. Tyutin, The number of independent traces and supertraces on symplectic reflection algebras, Journal of Nonlinear Mathematical Physics, Vol. 21, No. 3, 2014, pp. 308-335. arXiv:1308.3190

[11]S.E. Konstein and I.V. Tyutin, Ideals generated by traces or by supertraces in the symplectic reflection algebra H1,ν(I2(2m + 1)), Journal of Nonlinear Mathematical Physics, Vol. 24, No. 3, 2017, pp. 405-425. arXiv:1612.00536

[12]S.E. Konstein and M.A. Vasiliev, Supertraces on the Algebras of Observables of the Rational Calogero Model with Harmonic Potential, J. Math. Phys, Vol. 37, 1996, pp. 2872.

[13]M.A. Olshanetsky and A.M. Perelomov, Quantum integrable systems related to lie algebras, Phys. Rep, Vol. 94, No. 6, 1983, pp. 313-404.

[14]A. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett, Vol. 69, 1992, pp. 703-705.

[15]M.A. Vasiliev, Quantization on sphere and high-spin superalgebras, JETP Letters, Vol. 50, 1989, pp. 377-379.

[16]M.A. Vasiliev, Higher spin algebras and quantization on the sphere and hyperboloid, Int. J. Mod. Phys, Vol. A6, No. 07, 1991, pp. 1115-1135.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 27 - 1
Pages: 7 - 11
Publication Date: 2019/10/25
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2020.1684005 How to use a DOI?
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/).

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TY  - JOUR
AU  - S.E. Konstein
AU  - I.V. Tyutin
PY  - 2019
DA  - 2019/10/25
TI  - Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models
JO  - Journal of Nonlinear Mathematical Physics
SP  - 7
EP  - 11
VL  - 27
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2020.1684005
DO  - 10.1080/14029251.2020.1684005
ID  - Konstein2019
ER  -

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Journal of Nonlinear Mathematical Physics

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

1. Preliminaries and notation

Definition 1.1.

Definition 1.2.

2. The superalgebra of observables

3. 𝔰𝔩2

Proposition 3.1.

Proof.

4. The (super)traces on H

Theorem 4.1.

Theorem 4.2.

Proof.

Acknowledgments

Footnotes

References

Cite this article

3. 𝔰𝔩₂