Ideals Generated by Traces or by Supertraces in the Symplectic Reflection Algebra H1,V(I2(2m + 1)) II
- DOI
- 10.2991/jnmp.k.200922.012How to use a DOI?
- Keywords
- symplectic reflection algebra; trace; supertrace; ideal; dihedral group
- Abstract
The algebra
- 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 linear complex-valued function tr on
Let
Each nonzero
If
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
We noticed that if
Denote this degenerate trace by trz and the degenerate supertrace by strz
Theorem 10.1 proved in [5] implies that if
- (i)
- (ii)
the supertrace (10.2) is degenerate and generates the ideal
The goal of this paper is Theorem 13.1, which proves
Conjecture 1.1.
([5, Conjecture 9.1])
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
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
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
Let
be the group algebra of the group I2(n). In G, it is convenient to introduce the following basis
3. SYMPLECTIC REFLECTION ALGEBRA H1,v(I2(2m + 1))
Definition 3.1.
The symplectic reflection algebra
Defining the parity in
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
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
One can consider the algebra
Observe, that any ϰ-trace is identically zero on all irreducible sl2-submodules of
Let the skew-symmetric tensor εαβ be normalized so that ε01 = 1 and so
Proposition 4.1.
([5, Proposition 4.2]) The subalgebra of singlets H0 is the algebra of polynomials in the element
The commutation relations of the singlet
5. THE FORM OF IDEALS IN
Theorem 5.1.
([5, Theorem 4.3]) Let
Proposition 5.1.
([5, Proposition 4.4]) If
Definition 5.1.
For each p = 0, …, 2m, we define the ideals
Proposition 5.2.
([5, Proposition 4.7]) We have
Proposition 5.3.
([5, Proposition 4.8]). We have
Since
Corollary 5.1.
For any p = 0, …, 2m, there exists a nonzero polynomial
Theorem 5.1 evidently follows from Corollary 5.1.
6. GENERATING FUNCTIONS OF ϰ-TRACES
For each ϰ-trace ϰ on spϰ
Since L0Qp = 0 for any p ≠ 0, it follows from the definition (6.1) that
It is easy to find
Next, since spϰ(Rk) does not depend on k, we have spϰ(Lp) = 0 for any p ≠ 0 and
The value of spϰ(L0) will be calculated later, in Section 9.
We consider also the functions
It is easily verified, by expanding the exponential in a series, that these functions are related with the functions
The form of generating functions is related with (non)degeneracy of the form
7. DEGENERACY CONDITIONS FOR THE ϰ-TRACE
8. EQUATIONS FOR THE GENERATING FUNCTIONS
F p s p ϰ
In [5, Eq. (7.1)], the following system of differential equations for the generating functions is obtained:
The initial conditions for this system are:
To solve the system (8.1), we consider its Fourier transform. Let
For the functions
with the initial conditions
We choose the following form of the solution of the system (8.3):
where
Evidently, this solution satisfies the initial condition (8.4) for each ϰ and k, except for the case where ϰ = +1 and k = 0.
If ϰ = +1 and k = 0, then the expression (8.5) for
When ϰ = +1 the solution (8.5) and (8.6) gives
It is shown in Subsection 9.1 that if ϰ = +1, then
for any trace tr on
So,
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
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
9. VALUES OF THE ϰ-TRACE ON
[ I 2 ( 2 m + 1 ) ]
From [5] we have
where
Below we consider these values for the traces and supertraces separately.
9.1. Values of the Traces (ϰ = +1) on
[ I 2 ( 2 m + 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
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]):
We also note that
9.2. Values of the Supertraces (ϰ = −1) on
[ I 2 ( 2 m + 1 ) ]
The group I2(2m + 1) has m + 1 conjugacy classes without the eigenvalue −1 in the spectrum:
By [4, Theorem 2.3], the values of the supertrace on these conjugacy classes
are arbitrary parameters that completely define the supertrace str on the algebra
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
Theorem 10.1.
([5, Theorem 9.1]). Let
- (1)
The associative algebra H1,ν(I2(n)) has a one-parameter set of nonzero traces trz such that the symmetric invariant bilinear form
Here τ is an arbitrary parameter specifying the trace in one-dimensional space of traces.
- (2)
The associative superalgebra H1,ν(I2(n)) has a one-parameter set of nonzero supertraces strz such that the symmetric invariant bilinear form
Here τ is an arbitrary parameter specifying the supertrace in one-dimensional space of supertraces.
- (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
Here τ is an arbitrary parameter specifying the supertrace in one-dimensional space of supertraces.
- (4)
For all other values of ν, all nonzero traces and supertraces are nondegenerate.
11. GENERATING FUNCTIONS
F p s p ϰ
Let
Introducing the new variable y instead of t
we can rewrite Eq. (11.1) in the form
and Eq. (8.5) in the form
Now we see that
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,
Equation (11.4) implies that
Further, Eq. (8.2) implies
12. THE GENERATING FUNCTION
s p ϰ s p ϰ ( exp ( t 𝔰 ) Q 0 )
Let
In this section we introduce the function
Proposition 12.1.
Indeed,
Now, decompose
Comparing Eq. (12.4) with Eq. (12.3) implies
Proposition 12.2.
Proof. Taking Proposition 12.1 into account let us decompose Eq. (12.1) into the Taylor series:
Equation (12.2) implies
So
13. IDEALS GENERATED BY DEGENERATE ϰ-TRACES
Let
These degenerate ϰ-traces are denoted in Theorem 10.1 by trz and strz.
Denote the ideals generated by these traces spϰ by
Now we can prove Conjecture 1.1 ([5, Conjecture 9.1]):
Theorem 13.2.
([6, Theorem 4.2])
So, Theorem 13.1 follows from
Theorem 13.3.
Proof. For degenerate spϰ, we established the following facts:
For any p = 1, …, n, it is easy to find the lowest degree polynomial differential operators with constant coefficients
Further, it is a simple exercise to prove that
namely,
Consider, for example,
since Q0Qp = Q0Lp = 0 for p ≠ 0,
due to Eq. (6.2) and since the operator
Further, it is easy to see that for each of the ideals
So, Theorem 5.1 implies that the
Since
CONFLICTS OF INTEREST
The authors declare they have no conflicts of interest.
ACKNOWLEDGMENTS
The authors are grateful to
Footnotes
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.
Here the brackets {.,.} denote anticommutator.
REFERENCES
Cite this article
TY - JOUR AU - I.A. Batalin AU - S.E. Konstein AU - I.V. Tyutin PY - 2020 DA - 2020/12/10 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 - 10.2991/jnmp.k.200922.012 ID - Batalin2020 ER -