Symmetry, Integrability and Geometry: Methods and Applications Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle

A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space $E$, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the Dunkl operators, originally introduced as part of a program to generalize harmonic analysis in Euclidean spaces. This gives us a new, geometric way of viewing the Dunkl operators. In particular, we present a new proof of the commutativity of these operators among themselves as a consequence of a geometric property, namely, that the connection has curvature zero.


Coxeter groups -a bit of Algebra
We consider R N with its standard inner product: x, y = N j=1 x j y j For 0 = α ∈ R N , let σ α be the orthogonal reflection in the hyperplane H α orthogonal to α. So, with the action on the right, we have: • xσ α = x for all x ∈ H α , • ασ α = −α. G is finite and so we call it a finite Coxeter group.
Take j ∈ {1, 2, . . . , N } and κ as above. Define to be the Dunkl operator for the j-th coordinate.
These are not local operators for non-zero κ.
One says Dunkl operators are deformations of directional and partial derivatives, not perturbations. They break the symmetry group from the orthogonal group O(R N ) (κ ≡ 0) down to the Coexter group G (non-zero κ).
Theorem 0.1 The family of operators This theorem is a bit of a surprise. It was proved by Dunkl in his original paper on this subject (1989).

Dunkl miscelanea -Analysis
What we can do with this theory: 1. Generalized exponential function or Dunkl kernel.
(Simultaneous eigenfunction of all the T j,κ .) 2. Dunkl version of the Fourier transform (also known as the Dunkl transform).
3. Dunkl heat equation ∂u ∂t = 1 2 ∆ κ u and its associated Dunkl heat kernel. Quantum Principal Bundles (Non-commutative Geometry) All algebras are over the field of complex numbers C, are associative and have an identity element 1.
A quantum principal bundle (QPB) P = (B, A, F ) consists of the following objects.
• A * -algebra B. ('Functions' on the total space.) • A quantum group A which is a * -algebra.
(Compact Matrix Pseudogroups, Woronowicz, 1987.) ('Functions' on the model fiber space.) • A right co-action of the quantum group A on the total space B, i.e., F : B → B ⊗A. F is also a unital * -homomorphism. (Unital means that F (1) = 1 ⊗ 1 and * -homomorphism means that Furthermore, we require that these objects satisfy one more property, namely that the map β : (The right invariant 'functions' on the total space, i.e., the 'functions' on the base space.) Exterior Algebras on a Quantum Principal Bundle (Non-commutative Geometry) An (N-graded) differential calculus on a quantum principal bundle P = (B, A, F ) is given by these objects: 1. A graded differential * -algebra (Ω(P ), d P ) over B.

A bicovariant and
3. An extension of the right co-action F : B → B⊗A of the QPB to a right co-actionF : Ω(P ) → Ω(P )⊗Γ of Ω(P ) overΓ, whereΓ is the enveloping differential calculus of the fodc (Γ, d) andF is a differential, unital * -homomorphism.
Moreover, we have these properties: 1. Ω(P ) is generated as a graded differential calculus by Ω 0 (P ) = B, the elements of degree zero. TheF -invariant elements of Ω(P ) is a differential * -subalgebra denoted by Ω(M ). This is the differential calculus for the base space. We do not necessarily have that Ω(M ) is generated by V.

Connections and Covariant Derivatives on a Quantum Principal Bundle (Non-commutative Geometry)
Consider a QPB with a given differential calculus.
Here κ : A → A is the antipode of the quantum group. (Not to be confused with the multiplicity function.)

A specific QPB
A specific example of a QPB with a specific connection will now give us the Dunkl operators. First: the QPB.
The spaces are classical, but the differential calculi (DC) are not classical for two of the spaces: • Total space: P = R N \ (∪ α∈R H α ). Quantum DC.
The standard differential calculus on P is denoted hor(P ) and will indeed turn out to be the horizontal forms on the total space P .
As a graded * -algebra Ω(P ) = hor(P ) ⊗ Γ ∧ inv . The product, the * -operation and the differential on Ω(P ) are defined in the paper; Γ ∧ inv is defined as the quotient of the tensor algebra over Γ inv by dividing out the quadratic relations g 1 g 2 =h [g 1 ] ⊗ [g 2 ] where g 1 , g 2 ∈ S and e = h / ∈ S. Also, [g] = π(g) with g ∈ S is the canonical basis of Γ inv .
• Group: Coxeter group G associated with root system R (which acts freely on P ). Quantum DC Γ ∧ .
• Base space: M = P/G ∼ = any connected component of P with DC taken to be the classical DC on M .
The DC on the quantum group A is completely quantum. The classical calculus of the zero dimensional compact Hausdorff differential manifold G is worthless! Following Gelfand theory, we take the quantum group to be the finite dimensional vector space of all complex valued (continuous) functions on G.

The Quantum DC for A
A is commutative. All one-sided ideals are two-sided and given uniquely by By the theory of fodc, we must consider all the ideals I ⊂ ker where : A → C is the co-unit given by where e ∈ G is the identity element. So, I ⊂ ker if and only if e ∈ S.
To get a * -invariant fodc is equivalent to κ(I) * = I which in turn is equivalent to S −1 = S.
To get a bicovariant fodc is equivalent ad(I) ⊂ I ⊗ A which in turn is equivalent to g −1 Sg = S for all g ∈ G.
For our case of a Coxeter group G we choose Since σ −1 α = σ α , we have S −1 = S. Also g −1 σ α g = σ αg for all g ∈ G, which implies g −1 Sg = S.

The quantum connection
We continue considering the QPB introduced above. Let ω f be the canonical flat connection defined for all θ ∈ Γ inv by ω f (θ) := 1 ⊗ θ.
Then the covariant derivative D ω associated with the connection ω := ω f +λ (called a Dunkl connection) for all φ ∈ hor(P ) is where D : hor(P ) → hor(P ) is the standard de Rham derivative of classical differential geometry.

Concluding Remarks -and Another Theorem
MORAL: Without changing any part of the theories of Dunkl operators nor of QPB's we have found that Dunkl operators are a special example in the theory of QPB's. It is important to note that Dunkl operators can not be viewed as covariant derivatives in classical differential geometry, since the latter operators are local. Seen this way, Dunkl operators are a quantum phenomenon.
We have found a formula for D ω φ(x) for all elements φ ∈ hor(P ) of any degree. In analysis Dunkl operators are usually (maybe always?) considered as acting only on elements φ ∈ hor (0) (P ), that is on smooth functions φ : P → R.
Theorem 0.3 The condition r ω ≡ 0 implies that the family of operators {T j,κ |j=1, . . . , N } is commutative when applied to smooth functions φ : P → R.
This last result has been known since Dunkl's very first paper on these operators. However, we now have given a geometrical explanation (zero curvature) for why this turns out to be true.