Journal of Nonlinear Mathematical Physics

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 Author
S.E. Konstein
Received 19 August 2019, Accepted 30 August 2019, Available Online 25 October 2019.
DOI
https://doi.org/10.1080/14029251.2020.1684005How to use a DOI?
Abstract

If G is a finite Coxeter group, then symplectic reflection algebra H := H1,Ξ· (G) has Lie algebra 𝔰𝔩2 of inner derivations and can be decomposed under spin: H = H0 βŠ• H1/2 βŠ• H1 βŠ• H3/2 βŠ• ... We show that if the ideals ℐi (i = 1,2) of all the vectors from the kernel of degenerate bilinear forms Bi(x,y) := spi(x Β· y), where spi are (super)traces on H, do exist, then ℐ1 = ℐ2 if and only if ℐ1 ∩ H0 = ℐ2 ∩H0.

Copyright
Β© 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 π’œ be an associative superalgebra with parity Ο€. 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 π’œ is called a supertrace if

str(fβ‹…g)=(βˆ’1)Ο€(f)Ο€(g)str(gβ‹…f)   for  all  f,gβˆˆπ’œ.

Definition 1.2.

A linear function tr on π’œ is called a trace if

tr(fβ‹…g)=tr(gβ‹…f)  for  all  f,gβˆˆπ’œ.

We will use the notation β€œsp” and the term β€œ(super)trace” to denote both cases, traces and super-traces, simultaneously.

2. The superalgebra of observables

Let V = ℝN be endowed with a positive definite symmetric bilinear form (Β·,Β·). For any nonzero vβ†’βˆˆV, define the reflections rrvβ†’ as follows:

rvβ†’:x→↦xβ†’βˆ’2(xβ†’,vβ†’)(vβ†’,vβ†’)v→    for  any  xβ†’βˆˆV.(2.1)

A finite set of non-zero vectors β„› βŠ‚ V is said to be a root system and any vector vβ†’βˆˆβ„› is called a root if the following conditions hold:

  1. i)

    β„› is rwβ†’-invariant for any wβ†’βˆˆβ„›,

  2. ii)

    if vβ†’1,vβ†’2βˆˆβ„› are proportional to each other, then either vβ†’1=vβ†’2 or vβ†’1=βˆ’vβ†’2.

The Coxeter group G βŠ‚ O(N,ℝ) βŠ‚ End(V) generated by all reflections rvβ†’ with vβ†’βˆˆβ„› 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 := H1,Ξ·(G) and call it the superalgebra of observables of Calogero model based on root system β„›.a

This algebra consists of noncommuting polynomials in 2N indeterminates aiΞ±, where Ξ± = 0,1 and i = 1, ..., N, with coefficients in β„‚[G] satisfying the relations (see [6] Eq. (1.15))b

[aiΞ±,ajΞ²]=Ραβ(Ξ΄ij+βˆ‘vβ†’βˆˆβ„›Ξ·(vβ†’)vivj(vβ†’,vβ†’)rvβ†’),(2.2)
and
rvβ†’aiΞ±=βˆ‘j=1N(Ξ΄ijβˆ’2vivj(vβ†’,vβ†’))ajΞ±rvβ†’.(2.3)

Here Ραβ is the antisymmetric tensor such that Ξ΅01 = 1, and vi (i = 1,...,N) are the coordinates of the vector vβ†’. The commutation relations (2.2), (2.3) suggest to define the parity Ο€ by setting:

Ο€(aiΞ±)=1   for  any  α,  i;    π(rvβ†’)=0   for  any  vβ†’βˆˆβ„›.(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

DΞ±Ξ²:f↦DΞ±Ξ²f=[TΞ±Ξ²,f],(3.1)
where α,β = 0,1, and f ∈ H, and polynomials Tαβ are defined as follows:
TΞ±Ξ²:=12βˆ‘i=1N(aiΞ±aiΞ²+aiΞ²aiΞ±).(3.2)

These operators satisfy the following relations:

[Dαβ,Dγδ]=ΡαγDβδ+ΡαδDβγ+ΡβγDαδ+ΡβδDαγ,(3.3)
since
[Tαβ,Tγδ]=ΡαγTβδ+ΡαδTβγ+ΡβγTαδ+ΡβδTαγ.

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

[D01,D11]=2D11,    [D01,D00]=βˆ’2D00,    [D11,D00]=βˆ’4D01.

The polynomials TΞ±Ξ² commute with β„‚[G], i.e., [TΞ±Ξ²,rvβ†’]=0]= 0, and act on the aiΞ± as on vectors of the irreducible 2-dimensional 𝔰𝔩2-modules:

DΞ±Ξ²aiΞ³=[TΞ±Ξ²,aiΞ³]=ΡαγaiΞ²+ΡβγaiΞ±,    where  i=1,…,N.(3.4)

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

The subalgebra

H0:={f∈H|DΞ±Ξ²f=0  for  any  α,Ξ²}βŠ‚H(3.5)
is called the subalgebra of singlets.

Introduce also the subspaces Hs:=βŠ•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=H0βŠ•Hrest,    where    Hrest:H1/2βŠ•H1βŠ•H3/2βŠ•β€¦.

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

Note, that since SL2 is generated by inner derivations and TΞ±Ξ² are even elements, each two-sided ideal ℐ βŠ‚ H can be decomposed in an analogous way: ℐ = ℐ0 βŠ• ℐ1/2 βŠ• ....

Since TΞ±Ξ² are even elements of the superalgebra H, we have sp(DΞ±Ξ² f)= 0 for any (super)trace sp on H, and hence the following proposition takes placec:

Proposition 3.1.

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

Proof.

If s β‰  0, then the elements of the form DΞ±Ξ²f, where Ξ±, Ξ² = 0, 1, and f∈Hsis, f β‰  0, span the irreducible SL2-module Hsis. This implies sp f = 0 for any (super)trace on H and any f ∈ Hrest.

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 H1,Ξ·(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 H1,Ξ·(I2(2m + 1)).

Every (super)trace sp(Β·) on any associative (super)algebra π’œ generates the following bilinear form on π’œ:

Bsp(f,g)=sp(fβ‹…g)  for  any  f,gβˆˆπ’œ.(4.1)

It is obvious that if such a bilinear form Bsp is degenerate, then the kernel of this form (i.e., the set of all vectors f ∈ π’œ such that Bsp(f, g)= 0 for any g ∈ π’œ) is the two-sided ideal ℐsp βŠ‚ π’œ.

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 β©Ύ 1 and n = 2m + 1. Then

  1. 1)

    The associative algebra H1,Ξ·(I2(n)) has nonzero traces trΞ· such that the symmetric invariant bilinear form BtrΞ· (x,y)= trΞ·(x Β· y) is degenerate if and only if Ξ·=zn, where z ∈ 𝕑 \ n𝕑. For each such z, all nonzero degenerate traces on H1,z/n(I2(n)) are proportional to each other.

  2. 2)

    The associative superalgebra H1,Ξ·(I2(n)) has nonzero supertraces strΞ· such that the supersymmetric invariant bilinear form BstrΞ· (x,y)= strΞ·(x Β· y) is degenerate if Ξ·=zn, where z ∈ 𝕑 \ n𝕑. For each such z, all nonzero degenerate supertraces on H1,z/n(I2(n)) are proportional to each other.

  3. 3)

    The associative superalgebra H1,Ξ·(I2(n)) has nonzero supertraces strΞ· such that the supersymmetric invariant bilinear form BstrΞ· (x,y)= strΞ·(x Β· y) is degenerate if Ξ·=z+12, where z ∈ 𝕑. For each such z, all nonzero degenerate supertraces on H1,z+1/2(I2(n)) are proportional to each other.

  4. 4)

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

Theorem 4.1 implies that if z ∈ 𝕑 \ n𝕑, then there exists the degenerate trace trz generating the ideal ℐtrz consisting of the kernel of the degenerate form Btrz (f, g)= trz(f Β· g), and simultaneously the degenerate supertrace strz generating the ideal ℐstrz consisting of the kernel of the degenerate form Bstrz (f, g) = strz(f Β· g).

A question arises: is it true that ℐtrz = ℐstrz?

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 Β·g) and B2(f, g)= sp2(f Β·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 Β· g)= 0, f Β· g ∈ ℐ1 and (f Β· g)0 ∈ ℐ1. So (f Β· g)0 ∈ ℐ1∩H0. Due to hypotheses of this Theorem, (f Β· g)0 ∈ ℐ2 ∩H0, and hence sp2((f Β· g)0)= 0. Proposition 3.1 gives sp2(f Β·g)= sp2((f Β·g)0) which implies sp2(f Β·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 Di, see [5], acting on the space of G-invariant smooth functions on V, namely a^iΞ±=12(xi+(βˆ’1)Ξ±Di), 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 |0βŒͺ such that a^i0|0βŒͺ=0 for all i by acting on |0βŒͺ 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 β„› is of type ANβˆ’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.
[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
[15]M.A. Vasiliev, Quantization on sphere and high-spin superalgebras, JETP Letters, Vol. 50, 1989, pp. 377-379.
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
27 - 1
Pages
7 - 11
Publication Date
2019/10
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1080/14029251.2020.1684005How to use a DOI?
Copyright
Β© 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/).

Cite this article

TY  - JOUR
AU  - S.E. Konstein
AU  - I.V. Tyutin
PY  - 2019
DA  - 2019/10
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  - https://doi.org/10.1080/14029251.2020.1684005
ID  - Konstein2019
ER  -