# 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 Authors
S.E. Konstein, I.V. Tyutin
Received 19 August 2019, Accepted 30 August 2019, Available Online 25 October 2019.
DOI
10.1080/14029251.2020.1684005How to use a DOI?
Abstract

If G is a finite Coxeter group, then symplectic reflection algebra H := H1,Ξ· (G) has Lie algebra π°π©2 of inner derivations and can be decomposed under spin: H = H0 β H1/2 β H1 β H3/2 β ... We show that if the ideals βi (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 β1 = β2 if and only if β1 β© H0 = β2 β©H0.

Open Access

## 1. Preliminaries and notation

Let π 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 is called a supertrace if

str(fβg)=(β1)Ο(f)Ο(g)str(gβf)βββforββallββf,gβπ.

### Definition 1.2.

A linear function tr on π is called a trace if

tr(fβg)=tr(gβf)ββforββallββf,gβπ.

We will use the notation βspβ and the term β(super)traceβ to denote both cases, traces and super-traces, simultaneously.

## 2. The superalgebra of observables

Let V = βN be endowed with a positive definite symmetric bilinear form (Β·,Β·). For any nonzero vββV, define the reflections rrvβ as follows:

rvβ:xββ¦xββ2(xβ,vβ)(vβ,vβ)vβββββforββanyββxββV.(2.1)

A finite set of non-zero vectors β β V is said to be a root system and any vector vβββ is called a root if the following conditions hold:

1. i)

β is rwβ-invariant for any wβββ,

2. ii)

if vβ1,vβ2ββ are proportional to each other, then either vβ1=vβ2 or vβ1=βvβ2.

The Coxeter group G β O(N,β) β End(V) generated by all reflections rvβ with vβββ 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 Ξ· be a complex-valued G-invariant function on β, i.e., Ξ·(vβ)=Ξ·(wβ) if rvβ and rwβ 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]=Ξ΅Ξ±Ξ²(Ξ΄ij+βvβββΞ·(vβ)vivj(vβ,vβ)rvβ),(2.2)
and
rvβaiΞ±=βj=1N(Ξ΄ijβ2vivj(vβ,vβ))ajΞ±rvβ.(2.3)

Here Ξ΅Ξ±Ξ² 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±)=1βββforββanyββΞ±,ββi;ββββΟ(rvβ)=0βββforββanyββvβββ.(2.4)
and we can consider the algebra H as a superalgebra as well.

Observe an important property of the superalgebra H: the Lie (super)algebra of its inner derivations contains the Lie subalgebra π°π©2 generated by operators

DΞ±Ξ²:fβ¦DΞ±Ξ²f=[TΞ±Ξ²,f],(3.1)
where Ξ±,Ξ² = 0,1, and f β H, and polynomials TΞ±Ξ² are defined as follows:
TΞ±Ξ²:=12βi=1N(aiΞ±aiΞ²+aiΞ²aiΞ±).(3.2)

These operators satisfy the following relations:

[DΞ±Ξ²,DΞ³Ξ΄]=Ξ΅Ξ±Ξ³DΞ²Ξ΄+Ξ΅Ξ±Ξ΄DΞ²Ξ³+Ξ΅Ξ²Ξ³DΞ±Ξ΄+Ξ΅Ξ²Ξ΄DΞ±Ξ³,(3.3)
since
[TΞ±Ξ²,TΞ³Ξ΄]=Ξ΅Ξ±Ξ³TΞ²Ξ΄+Ξ΅Ξ±Ξ΄TΞ²Ξ³+Ξ΅Ξ²Ξ³TΞ±Ξ΄+Ξ΅Ξ²Ξ΄TΞ±Ξ³.

It follows from Eq. (3.3) that the operators D00, D11 and D01 = D10 constitute an π°π©2-triple:

[D01,D11]=2D11,ββββ[D01,D00]=β2D00,ββββ[D11,D00]=β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 π°π©2-modules:

DΞ±Ξ²aiΞ³=[TΞ±Ξ²,aiΞ³]=Ξ΅Ξ±Ξ³aiΞ²+Ξ΅Ξ²Ξ³aiΞ±,ββββwhereββi=1,β¦,N.(3.4)

We will denote this π°π©2 thus realized by the symbol SL2.

The subalgebra

H0:={fβH|DΞ±Ξ²f=0ββforββanyββΞ±,Ξ²}βH(3.5)
is called the subalgebra of singlets.

Introduce also the subspaces Hs:=βis=1β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=H0βHrest,ββββwhereββββHrest:H1/2βH1βH3/2ββ¦.

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

Note, that since SL2 is generated by inner derivations and TΞ±Ξ² are even elements, each two-sided ideal β β H can be decomposed in an analogous way: β = β0 β β1/2 β ....

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 β 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 fβHsis, f β  0, span the irreducible SL2-module Hsis. This implies sp f = 0 for any (super)trace on H and any f β 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 generates the following bilinear form on π:

Bsp(f,g)=sp(fβg)ββforββanyββf,gβπ.(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 β π such that Bsp(f, g)= 0 for any g β π) is the two-sided ideal βsp β π.

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 β π, 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 β π \ nπ. 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 β π \ nπ. 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 β π. 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 β π \ nπ, then there exists the degenerate trace trz generating the ideal βtrz consisting of the kernel of the degenerate form Btrz (f, g)= trz(f Β· g), and simultaneously the degenerate supertrace strz generating the ideal βstrz consisting of the kernel of the degenerate form Bstrz (f, g) = strz(f Β· g).

A question arises: is it true that βtrz = βstrz?

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 β1 and β2 consisting of the kernels of bilinear forms B1(f, g)= sp1(f Β·g) and B2(f, g)= sp2(f Β·g), respectively.

### Proof.

It suffices to prove that if I1β©H0 = I2β©H0, then β1 = β2.

Consider any non-zero element f β β1. For any g β H, we have sp1(f Β· g)= 0, f Β· g β β1 and (f Β· g)0 β β1. So (f Β· g)0 β β1β©H0. Due to hypotheses of this Theorem, (f Β· g)0 β β2 β©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 β β2.

## 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+(β1)Ξ±Di), see [1,14]. The Hamiltonian of the Calogero model based on the root system [2β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 |0βͺ such that a^i0|0βͺ=0 for all i by acting on |0βͺ 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 ANβ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
2019/10/25
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2020.1684005How to use a DOI?
Open Access

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  -