Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models
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If G is a finite Coxeter group, then symplectic reflection algebra H := H1,η (G) has Lie algebra 𝔰𝔩2 of inner derivations and can be decomposed under spin: H = H0 ⊕ H1/2 ⊕ H1 ⊕ H3/2 ⊕ ... 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 · y), where spi are (super)traces on H, do exist, then ℐ1 = ℐ2 if and only if ℐ1 ∩ H0 = ℐ2 ∩H0.
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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.
A linear function str on 𝒜 is called a supertrace if
A linear function tr on 𝒜 is called a trace if
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 = ℝN be endowed with a positive definite symmetric bilinear form (·,·). For any nonzero , define the reflections r as follows:
A finite set of non-zero vectors ℛ ⊂ V is said to be a root system and any vector is called a root if the following conditions hold:
ℛ is -invariant for any ,
if , are proportional to each other, then either or .
The Coxeter group G ⊂ O(N,ℝ) ⊂ End(V) generated by all reflections with 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., if and belong to one conjugacy class of G.
Observe an important property of the superalgebra H: the Lie (super)algebra of its inner derivations contains the Lie subalgebra 𝔰𝔩2 generated by operators
These operators satisfy the following relations:
It follows from Eq. (3.3) that the operators D00, D11 and D01 = D10 constitute an 𝔰𝔩2-triple:
The polynomials Tαβ commute with ℂ[G], i.e., ]= 0, and act on the as on vectors of the irreducible 2-dimensional 𝔰𝔩2-modules:
We will denote this 𝔰𝔩2 thus realized by the symbol SL2.
Introduce also the subspaces , which is the direct sum of all irreducible SL2-modules 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
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 Tαβ are even elements, each two-sided ideal ℐ ⊂ H can be decomposed in an analogous way: ℐ = ℐ0 ⊕ ℐ1/2 ⊕ ....
Since Tαβ are even elements of the superalgebra H, we have sp(Dαβ f)= 0 for any (super)trace sp on H, and hence the following proposition takes placec:
sp(f)= sp(f0) for any f ∈ H and any (super)trace sp on H.
If s ≠ 0, then the elements of the form Dαβf, where α, β = 0, 1, and , f ≠ 0, span the irreducible SL2-module . 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,η(M) for all finite Coxeter groups M, see . In particular, there is an m-dimensional space of traces and an (m + 1)-dimensional space of supertraces on H1,η(I2(2m + 1)).
Every (super)trace sp(·) on any associative (super)algebra 𝒜 generates the following bilinear form on 𝒜:
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 ∈ 𝒜 such that Bsp(f, g)= 0 for any g ∈ 𝒜) is the two-sided ideal ℐsp ⊂ 𝒜.
Theorem 9.1 from  may be shortened to the following theorem:
Let m ∈ , where m ⩾ 1 and n = 2m + 1. Then
The associative algebra H1,η(I2(n)) has nonzero traces trη such that the symmetric invariant bilinear form Btrη (x,y)= trη(x · y) is degenerate if and only if , where z ∈ \ n. For each such z, all nonzero degenerate traces on H1,z/n(I2(n)) are proportional to each other.
The associative superalgebra H1,η(I2(n)) has nonzero supertraces strη such that the supersymmetric invariant bilinear form Bstrη (x,y)= strη(x · y) is degenerate if , where z ∈ \ n. For each such z, all nonzero degenerate supertraces on H1,z/n(I2(n)) are proportional to each other.
The associative superalgebra H1,η(I2(n)) has nonzero supertraces strη such that the supersymmetric invariant bilinear form Bstrη (x,y)= strη(x · y) is degenerate if , where z ∈ . For each such z, all nonzero degenerate supertraces on H1,z+1/2(I2(n)) are proportional to each other.
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 trz generating the ideal ℐtrz consisting of the kernel of the degenerate form Btrz (f, g)= trz(f · g), and simultaneously the degenerate supertrace strz generating the ideal ℐstrz consisting of the kernel of the degenerate form Bstrz (f, g) = strz(f · 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:
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.
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.
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.
This algebra has a faithful representation via Dunkl differential-difference operators Di, see , acting on the space of G-invariant smooth functions on V, namely , see [1,14]. The Hamiltonian of the Calogero model based on the root system [2–4, 13] is the operator defined in (3.2) (see ). The wave functions are obtained in this model via the standard Fock procedure with the Fock vacuum |0〉 such that for all i by acting on |0〉 with G-invariant polynomials of the .
This elementary fact is known for a long time, see, eg, .
Cite this article
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 -