On Dunkl angular momenta algebra

We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincare-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl(N) version of the subalgebra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.


I. INTRODUCTION
The differential-difference operators is shown to define a superintegrable system on N − 1 dimensional hypersphere ( [12], see also [15]), and a way to represent conserved charges was developed in [11]. We note that identifying Liouville charges is still an open problem. The operators (1.2) were already used in [13] as building blocks for the intertwining operators between the angular Hamiltonians with coupling constants different by an integer.
In this paper we firstly deal with the subalgebra of the rational Cherednik algebra of type A N −1 generated by the elements M ij and by the permutations s ij . A close algebra already appeared in the work of V. Kuznetsov [16] in connection with the Calogero-Moser system. We explain that the angular Calogero-Moser Hamiltonian can be realised as an element in this algebra. Furthermore, we show that it generates the centre of the algebra.
This operator can be considered as a deformation of the usual quadratic Casimir invariant of so(N), which corresponds to the angular momentum square.
We also describe all the relations which generators M ij satisfy. The commutation relations are simple extensions of the usual so(N) relations with metrics replaced by pairwise particle permutations. There are additional "crossing" relations which are quadratic in the generators too. We describe a basis in this algebra and show that it is a non-homogeneous quadratic algebra of Poincaré-Birkhoff-Witt (PBW) type in the sense of [17] (see also [18]).
These results are generalised for any Coxeter root system R in the final section.
We also consider the gl(N) version of the subalgebra of the rational Cherednik algebra which is generated by the elements x k D l rather than their combinations M kl , and by the group algebra. A closely related algebra in type A N −1 was considered in [19]. We describe relations in this algebra too and it gives another example of a non-homogeneous quadratic algebra of PBW type.

II. ANGULAR CALOGERO-MOSER HAMILTONIAN
We will work with the gauged Dunkl operators defined by They correspond to (1.1) via the transformation ψ → i<j (x i − x j ) g ψ of the wavefunctions.
Applying them instead of usual momenta operators for the free-particle system, we arrive at the modified Hamiltonian [4] Let now Res(A) ≡ Res + (A) be the restriction of an S N -invariant operator A to the space of symmetric functions, and let Res − (A) be its restriction to the antisymmetric functions. Then where H ± is the Calogero-Moser Hamiltonian [5]: Apart from g = 0 case, when the algebra generated by ∇ i , x j reduces to the Heisenberg algebra, the algebra formed by the coordinates and Dunkl operators for g = 0 is not closed: where for the later convenience the pairwise permutation operators are rescaled and the new notation S ij is introduced: 1 + g k =i s ik = 1 − k =i S ik , for i = j. (2.6) Introduce spherical coordinates and let H Ω , H Ω,± be the corresponding angular Calogero-Moser Hamiltonians obtained by the separation of radial and angular variables:

Define Dunkl angular momentum operators
where we define the Dunkl angular momentum square and the symmetric group algebra invariant operators, respectively, as Proof. The proposition is a consequence of the following relation where we use the usual vector and scalar product notations.
Indeed, using x 2 = r 2 and substituting x · ∇ = x · ∂ + S = r∂ r + S into (2.14), we obtain In order to complete the proof, we have to justify the relation (2.14). Substituting (2.9) into (2.12) and using the commutation relations (2.5), we have: The last term can be simplified further by rewriting the definition (2.6) of S ij in the gener- Its application leads to the following identity: By substituting (2.17) into (2.15) we get the desired expression (2.14). This completes the proof.
We mention that a related statement in the case of Dunkl operators associated with the group Z N 2 is contained in [20].

III. DUNKL ANGULAR MOMENTA ALGEBRA
Recall that the creation-annihilation operators a + i , a − i ≡ a i for the Calogero-Moser rational model in the harmonic confinement [7], [6] are given by They satisfy the relations and are hermitian conjugate to each other. This coincides with the relations (2.5) for the The commutation relations between creation-annihilation operators and group elements have the form Recall also that where differently denoted indexes are assumed again to have different values, and it follows that The deformed angular momentum generators (2.9) are expressed in terms of the deformed creation-annihilation operators (3.1) as They inherit the standard (anti-)commutation relations with the permutations operators: provided that differently denoted indexes have different values.
When the deformation parameter g vanishes, (3.7) corresponds to the standard representation of the generators of the algebra so(N) in terms of bosonic creation-annihilation operators or vector fields. In this case, the corresponding operators M 0 ij satisfy the well known commutation relation For nontrivial values of the deformation parameter, these generators, like a ± i themselves, do not form a Lie algebra any more, since their commutators include the coefficients dependent on the permutation operators.
We are interested in the Dunkl angular momenta algebra H so(N ) g which is generated by the operators M kl and by the group algebra CS N . It can be considered as a subalgebra of the rational Cherednik algebra H g (S N ) via the homomorphism which maps generators To get the commutators between the operators M ij , one just replaces the Euclidean metric δ ij in (3.9) by the permutation operator S ij . This matches the deformation rules in the commutators given by (2.5) or (3.2). More precisely, the following statement takes place whose version in another gauge is contained in [16] (Section 8, arXiv version only).
Proposition 2. (cf. [16]) Operators (3.7) satisfy the following commutation relations: Proof. In order to prove (3.10), firstly, we verify the commutators which is a simple consequence of the relations (3.2). Then using them together with the definition (3.7), we get (a + i S jk a l − a + l S jk a i ), (3.11) where in order to shorten the formulae, we introduce the operator, which antisymmetrizes over the pairs of indexes included in square brackets: The simplest case is when the values of all four indices differ. Then permutation operators commute with the annihilation-creation operators in all terms, and we arrive at the desired commutation relation (3.10).
It remains to consider the less trivial case when only three indexes differ. It is sufficient to check k = j case only. Then (3.11) acquires the following form: Then moving permutations to the right in order to get the desirable result, the additional commutators appear in contrast to the previous case. However, such unwanted terms are canceled out. Indeed, the only nontrivial commutations between permutations and creationannihilation operators are given by 14) as can be deduced from the definition (2.6). Due to (3.14), the appeared commutators eliminate each other: Therefore, the commutation relation (3.10) holds for arbitrary index values, and we have finished its proof.
The relation (3.10) is reduced to the usual commutation (3.9) for so(N) generators in the nondeformed limit g = 0.
The order of operators S ij and M kl in the right-hand side of the equation (3.10) must be respected in general. Using the relations (3.8), some permutations in the right-hand side of the commutator (3.10) can be moved to the left of the Dunkl angular momenta generators.
In particular, if all the indexes differ then the operators commute but, in general case, the obtained in this way relations will be more complicate. However, it is easy to verify that all permutations can be moved to the left simultaneously: The above relation follows also from the invariance of the subalgebra H so(N ) g under the Hermitian conjugation, which is inherited from the bigger algebra H g (S N ).
We also note the equivariance of the relations (3.10), (3.16) under the symmetric group action. It is easy to verify using the relations (3.8), (3.5), and (3.6) that any permutation σ ∈ S N , when acting on the commutator (3.10), just permutes all its indexes: i → σ(i).

IV. CROSSING RELATION AND PBW PROPERTY
Let us establish another quadratic relation, which the generators of the deformed angular momentum algebra H so(N ) g satisfy. We call it the crossing relation.
Proof. Note that in both sides of the relation (4.1) the sum is taken over the cyclic permutations of the first three indexes i, j, k, so that it can be rewritten as where for convenience the cyclic permutation sum operator is introduced: Note that the right-hand side of the equality (4.1) contains the terms, which appear also in the right-hand side of the commutation relation (3.10). It is easy to verify that if the values of any two of the indexes i, j, k, l coincide, then the relation (4.1) either reduces to the commutation relation, or its both side vanish trivially. Therefore, it is enough to consider the case when the values of all four indexes differ.
Define the usual normal order N () with the creation operators placed on the left hand side and annihilation operators on the right. For instance, N (a + i a j a + k a l ) = a + i a + k a j a l . The operators commute under the normal order: N ([M ij , M kl ]) = 0. Since the classical momenta obey the crossing relations, the same relation holds also for the normal order: Using the definition (3.7), one can express each product in terms of the normal order as Therefore, the right hand side of the crossing relation equals Canceling out and grouping together similar terms, we arrive at the desired expression Note that all the permutation operators in the right-hand side of the crossing relations can be moved in front of the angular momentum operators: In the the case of two equal indexes the relations (4.2) and (4.5) are reduced to the commutation relations (3.10) and (3.16) respectively, whose equivalence is already established.

FIG. 1. Graphical representation for the deformed angular momentum and permutation operators.
Applying the hermitian conjugate to the crossing relations (4.1) and using (3.17), we obtain the equivalent relations Like for the commutators, the system of crossing relations (4.6) is invariant with respect to the permutation group.
The sum of the equations (4.2) and (4.5) gives rise to a simpler crossing relation written in terms of anticommutators with vanishing right-hand side: This is the analogue of the crossing relations in quantum case, which possesses the same symmetry as the classical one. Its direct consequence is the vanishing of antisymmetrized product asym [ijkl] M ij M kl = 0. (4.8) For even values of N, this results in vanishing of the Pfaffian which corresponds to a peculiar Casimir elements of so(N) Lie algebra. Here ε i 1 ...i N is the Levi-Civita anti-symmetric tensor.
The crossing relation has a clear graphical interpretation. We represent the Dunkl angular momentum tensor M ij by an arrow with tensor indexes at endpoints as shown on Figure 1.  Consider the monomials composed from deformed angular momenta and pairwise permutations. Then using the commutation relations (3.8), the permutations can be moved to the right giving rise to The induction on n = k s=1 n s and the commutation relation (3.10) ensure a rearrangement of the elements M isjs in (4.10) according to some predefined order. Thus we choose the indexes to be ordered first by the values of i s , then, if they equal, by the values of j s : represented on Figure 2. Therefore, we arrive at the set of monomials, which do not have intersecting semicircles, that is they satisfy the condition  Proof. It is easy to see in the representation (2.9) that monomials (4.10) with the restrictions (4.11) and (4.12), are linearly independent for g = 0. Moreover their classical version, when M ij is replaced with the classical angular momenta are linearly independent too (see e.g. [11]). By taking the highest symbols, considered as elements of the smash product algebra C[x, p] # S N , it follows that these monomials are linearly independent for any g.
Theorem 4 and its proof show that H so(N ) g is a flat family of nonhomogeneous quadratic algebras over CS N of Poincaré-Birkhoff-Witt type in the terminology of [17] (see also [8], [18]).
Note that the constructed non-intersection monomial basis is similar to the well-known overcomplete valence-bond basis among the singlet states for usual quantum spins, introduced by Temperley and Lieb and subjecting to the similar crossing relations (with the trivial right-hand side) [21].

V. THE CENTRE
Consider the square of the Dunkl angular momentum operator M 2 given by (2.12), which is an analogue of the second order Casimir element of the usual so(N) algebra. Evidently, it is invariant with respect to the permutations S ij . However, it commutes with all the elements M ij only in the limit g = 0. In order to obtain a true central element, the operator (2.12) must be supplemented by an element from the symmetric group algebra. Such an operator appeared in [16]. We show that this operator generates the whole centre. Based on Proposition 1 this element can be identified with the angular Calogero-Moser Hamiltonian.
As a consequence, we obtain the relation (5.2) using also the commutations (2.5): where we have used the index antisymmetrization (3.12) for the convenience.
Then we note that the Dunkl angular momentum operators (2.9), (2.1) depend on angular coordinates only. This becomes evident after the application of the involutive antiautomorphism of the Cherednik algebra with the mutual exchange of x i and ∇ i , to the equation Then the commutation relations (3.5) and (3.8) between the generators take the form and their cyclic permutations.
The commutation relations (3.10) are equivalent to and other relations are obtained by the cyclic permutations of three indexes. They can be rewritten in a more symmetric form, The Casimir element, which is proportional to the angular Hamiltonian, has the following form in terms of introduced operators: It extends the spin square operator.
Note that for so(3) case the crossing relations are either trivial or reduce to the commutation relations.

VII. gl(N ) CASE
The algebra H so(N ) g can be included into a bigger subalgebra H gl(N ) g of the rational Cherednik algebra. We define it to be generated by the operators and by the group algebra CS N (1 ≤ k, l ≤ N). The operators can be considered acting on meromorphic functions. We also have M kl = E kl − E lk .
The algebra can be realised as a subalgebra of the rational Cherednik algebra H g (S N ) via a conjugation which maps ∇ k → D k . Further, there are more general but equivalent choices of the generators E kl = (αx k + β∇ k )(γx l + δ∇ l ), where parameters α, β, γ, δ are such that αδ − βγ = 0. They define isomorphic algebras. Indeed, the operators αx k + β∇ k with different indecies k pairwise commute. Therefore these generators correspond to different choices of generators of the rational Cherednik algebra. In particular, the algebra H gl(N ) g is isomorphic to the subalgebra of the rational Cherednik algebra H g (S N ) which is generated by the elements x k D l and by the group algebra CS N .
Generators E kl obey the standard commutation relations with the permutation operators: where all indexes are pairwise different.
As in the case of the usual gl(N), the rising and lowering generators are Hermitian conjugate of each other: We start from the analogue of the crossing relation (4.1), which in this case takes the following form: For different values of i, j, k this relation follows immediately from (7.1) and (3.2), (3.4).
The nontrivial case is when j = k = l, for which we have: where in the second equality above the relation (3.14) was applied.
Taking the Hermitian conjugate of (7.4), we get an equivalent relation, with antisymmetrization over the first indexes of E ij generators. Both relations can be written down in the following compact form: Their combination with suitably chosen indexes results in the commutation relation among the deformed gl(N) generators (7.6) which holds for any index values. Its antisymmetrisation over i, j and k, l immediately gives the commutation relations (3.10) for deformed angular momenta. When all indexes differ, the commutator in the right-hand side of (7.6) disappears, and we arrive at a natural extension of gl(N) commutation relations with metrics tensor δ ij replaced by the pairwise permutation operator S ij , as for the so(N) case considered above. Some other commutation relations are less straightfowrward. Below we write down all particular cases of the commutation relation (7.6) provided that the values of all four indexes differ pairwise.
and the equalities from the third line in (7.7) must also be supplemented by their conjugate relations. As for H so(N ) g case, considered above, any monomial in E ij and S ij can be expressed in the following form by moving all permutations to the right hand side: Using the Wicks's theorem, one can decompose (7.8) as was described in Section IV. In the example below, only the highest-order term in this normal ordering decomposition is shown E n 1 11 E n 2 12 E n 3 22 E n 4 32 E n 5 33 E n 6 34 = (a + 1 ) n 1 +n 2 (a + 2 ) n 3 (a + 3 ) n 4 +n 5 +n 6 a n 1 1 a n 2 +n 3 +n 4 2 a n 5 3 a n 6 4 + . . . .
Here the lower-order terms in a ± i are indicated by the dots. The monomials (7.8) with the restriction (7.9) are linearly independent, since it is easy to see that there is a one-to-one correspondence between them and their "highest symbols" a + n 1 i 1 . . . a + n k i k a n 1 j 1 . . . a n k j k , and because of the PBW theorem for the rational Cherednik algebra [8].
Let V be a vector space of dimension N with the basis e 1 , . . . , e N . Consider the tensor algebra T (V ⊗V ) and its smash product with the symmetric group algebra CS N where group elements act on V and hence on V ⊗ V by permuting the basis elements. We view operators E ij as elements of this smash product T (V ⊗ V ) # CS N via the mapping E ij → (e i ⊗ e j )e, where e ∈ S N is the identity element.
Similarly to the case of H From the commutation relations (7.6) it is easy to deduce that the element . It describes the Calogero Hamiltonian (2.11) in confined oscillator potential [2], which can be seen from (2.2) and (3.1) [7]: extended out of the space of identical particles.
Finally we note that ρ and constants generate the centre of H gl(N ) g . This follows from the following lemma similarly to the proof of Theorem 5.

Lemma 1. Consider the classical Poisson algebra generated by
Proof. Let f (x, p) ∈ Z. Since the Poisson bracket it follows for i = j that f is a function of the variables y i = x i p i (1 ≤ i ≤ N). In these variables, the relation (7.12) implies that (∂ y i −∂ y j )f = 0 so that f is a function of C = y i as stated.

VIII. COXETER GROUPS GENERALISATIONS
The previous results can be generalised to the case when the symmetric group S N is replaced with a finite reflection group.
Let R be a Coxeter root system in the Euclidean space V ∼ = R N with the inner product denoted as (·, ·) (see [22]). Let e 1 , . . . , e N be the standard basis. Denote by s α the orthogonal reflection with respect to the hyperplane (α, x) = 0: Let W be the corresponding finite Coxeter group generated by reflections s α , α ∈ R. Let g : R → C be a W -invariant function, denote g α = g(α).

The Dunkl operators take the form
where ξ ∈ R N , and R + is a positive half of the root system. Given ξ, η ∈ R N we define the element S ξη of the group algebra CW by This element comes from the commutation Define the Dunkl angular momentum operator Note that for any w ∈ W and ξ, η ∈ R N the following relations hold: The Dunkl angular momenta algebra H so(N ) g (W ) is defined to be generated by the operators M ξη , and by the group algebra CW . It can be considered as a subalgebra of the rational
The proposition can be checked straightforwardly by making use of the formulas Similarly to Theorem 4 for W = S N , the algebra H Similarly to the case W = S N considered in Section II, let Res A be the restriction of a W -invariant element of the algebra H so(N ) g (W ) to the space of W -invariant functions ψ: Then the Calogero-Moser Hamiltonian for a general Coxeter group W is Generalisation of Theorem 5 holds for any W and it has the following form. where the elements S ij = S e i e j ∈ CW are given by (8.2). The set of monomials (7.8), (7.9) with σ ∈ W gives a basis of the algebra for any W -invariant multiplicity function g, and this is also a quadratic algebra over CW of PBW type. The centre is generated by constants In the representation of the algebra E ξη → E ξη = 1 2 ((x, ξ) − ∇ ξ )((x, η) + ∇ η ) the central generator takes the form of the Calogero-Moser operator in the harmonic confinement associated with W : where H is given by (8.7). A sheaf of algebras was associated to varieties with finite group actions in [24]. In the case of the projective space with the natural action of the Coxeter group W the algebra of global sections is isomorphic to a quotient of the algebra H gl(N ) g (W ) over the ideal generated by a central element [24] (see also [25]). Affinity of this sheaf of algebras was investigated in [25]. Another interesting question is about the extension of these results to the algebra H so(N ) g (W ) and the sheaf of algebras associated with the subspace of isotropic lines in the projective space.
We also note that integrability of the angular Calogero-Moser systems deserves further analysis as, in particular, a complete set of Liouville charges remains unknown. The problem of Liouville integrability was one of the main motivations for this work and we hope that the work is a useful step towards its solution. There are a few reasons why the problem is difficult and remains unsolved which we try to explain (see also [11]). Firstly, in comparison with the case of the Calogero-Moser systems in the linear space, commutativity of homogeneous quantum integrals for the latter problem is quite straightforward and can be derived from the property that the highest term of any quantum integral should be polynomial [26] .

Secondly, one way to try to establish Liouville integrability for the angular Hamiltonian
H Ω is to consider the natural embedding of the universal enveloping algebras U(so (2)) ⊂ U(so (3) Nonetheless, in the gl(N) case a version of this construction exists at least for classical Coxeter groups W . In this case a correction E ′ ii of the operator E ii by group algebra elements is known so that [E ′ ii , E ′ jj ] = 0 [27]. It would be interesting to try to extend this construction to other finite reflection groups W and to try to modify Casimir elements M 2 to a commuting family. It is not plausible though that modification purely by group algebra elements is possible in the so(N) case. A more general algebra at g = 0 one can alternatively start with is the quantum shift of argument algebra (see [28]) in which case one allows operators of order higher than 2.
In the case of gl(N) and classical Coxeter groups W it follows from [6] (for W = A N −1 ) and [29] (for W = B N , D N ) that the algebra H gl(N ) g (W ) contains a complete set of Liouville quantum integrals for the Calogero-Moser Hamiltonian in harmonic confinement. Indeed, it is shown that a complete set of commuting quantum integrals is expressed via the combinations a + i a i . This suggests that the symmetry algebra H so(N ) g (W ) might also be the right one for the angular Calogero-Moser Hamiltonian. We also note that the algebra and the above description of its centre has been recently explored in the study of Calogero-Moser deformation of the Coulomb problem and, in particular, for the construction of the generalisation of the Runge-Lenz vector [30].