The Diffiety Institute Preprint Series

Preprint DIPS-5/96 October 1, 1996

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On quantized algebra of Wess-Zumino differential operators at roots of unity

by

Alexander Verbovetsky

ON QUANTIZED ALGEBRA OF WESS-ZUMINO

DIFFERENTIAL OPERATORS AT ROOTS OF UNITY

Alexander Verbovetsky Moscow State Technical University and The Diffiety Institute Moscow, Russia

1. Motivation. In [13] J. Wess and B. Zumino constructed a broad family of quantum deformations of the algebra of scalar differential operators on the affine space. If the deformation parameter q is generic, this algebra has the same dimension as in non-deformed case; if q is a root of unity, then the dimension of deformed algebra is less than that of non-deformed algebra. However, this does not quite agree with the intuition gained in the classical theory of differential operators. In fact, due to the analogy between quantum case at roots of unity and the classical case in positive characteristic, one should expect the appearance of differential operators that are not compositions of operators of order ⩽ 1, whereas the dimension of the whole algebra must be the same as in non-deformed case. In this short paper it is shown that this intuition is a true one: we quantize the standard algebraic definition of differential operator [… [[D, f_{0}] , f_{1}] …, f_{k}] = 0 by replacing the usual commutators by twisted ones and obtain the algebra of differential operators that has the classical dimension for any q and coincides with the Wess-Zumino algebra at generic parameter values. A detailed description of this

1991 Mathematics Subject Classification. Primary 16S32, 16S80; Secondary 17B37, 16W30, 81R50.

Key words and phrases. Quantum space, quantum group, differential operator, Wess-Zumino calculus.

2 Alexander Verbovetsky

algebra is presented. Note also that having such quantum differential operators one can construct quantum jets, de Rham and Spencer complexes, integral forms, Euler operator, and so on, following the same logic as in non-deformed case (see [9, 10, 12, 11, 7] as well as others articles in this issue of Acta Appl. Math.). A realization of this program can be found in [8].

2. Here we recall some standard facts about quantum affine spaces. We refer to [6, 2, 1, 4] for further details.

Let 𝕜 be a ground field. Consider a free 𝕜-module V and a non-degenerate linear operator R̂: V ⊗ V → V ⊗ V. Throughout this paper we assume that the operator R̂ satisfies the Yang-Baxter equation

R̂_{12}R̂_{23}R̂_{12} = R̂_{23}R̂_{12}R̂_{23}

(here R̂_{12} = R̂ ⊗ 1, R̂_{23} = 1 ⊗ R̂) and the Hecke condition

R̂^{2} = (q − q^{−1}) R̂ + 1

for some invertible element q of 𝕜.

Basic example ([3]). Let {ξ_{1},…, ξ_{n}} be a basis of V and {x^{1},…, x^{n}} be the dual basis of V^{∗}. Let ∥R̂_{kl}^{ij}∥ be the matrix of R̂ in this basis. Choose an invertible q ∈ 𝕜 and put

R̂_{kl}^{ij} = δ_{l}^{i}δ_{k}^{j} (1 + (q − 1) δ^{ij}) + (q − q^{−1}) δ_{k}^{i}δ_{l}^{j}θ (j − i) ,  (1)

where θ (i) = \{ \begin{matrix}1, & for i > 0 \\ 0, & for i ⩽ 0. \end{matrix}

The algebra A of a quantum affine space is defined as the quotient algebra of the tensor algebra T (V^{∗}) by the ideal generated by the image of q − R̂^{∗}: V^{∗} ⊗ V^{∗} → V^{∗} ⊗ V^{∗}. In coordinates, A is generated by x^{1},…, x^{n} subject to the relations

R̂_{αβ}^{ij}x^{α}x^{β} = qx^{i}x^{j}

(we assume summation over repeated indices which occur in both upper and lower positions).

Example. Assume that R̂ is given by (1). Then A is generated by x^{1},…, x^{n} modulo the following relations:

x^{i}x^{j} = qx^{j}x^{i}, i < j.

Consider the matrix algebra M defined as the quotient of the tensor algebra T (End (V ⊗ V)) by the ideal generated by the elements R̂F − F R̂, where F ∈ End (V ⊗ V). The algebra M is generated by t_{j}^{i}, 1 ⩽ i, j ⩽ n, obeying the relations

R̂_{αβ}^{ij}t_{k}^{α}t_{l}^{β} = t_{α}^{i}t_{β}^{j}R̂_{kl}^{αβ}.

There is a natural bialgebra structure on M, with the comultiplication given by Δt_{j}^{i} = t_{α}^{i} ⊗ t_{j}^{α} and the counit εt_{j}^{i} = δ_{j}^{i}. One obtains a matrix quantum group H by taking the Hopf envelope of M, that is an initial object in a category of bialgebra morphisms M → C, with C being a Hopf algebra (see [5]).

The algebra A is a (right) comodule-algebra over H, with the coaction given by

Δ (x^{i}) = x^{α} ⊗ t̃_{α}^{i},

where the tilde stands for the antipode.

Remark. The antipode was inserted here to make this transformation right coaction.

We shall assume that the Hopf envelope H of the matrix bialgebra is endowed with a cobraided structure, i.e., that there exists a non-degenerate bilinear form ⟨· , ·⟩ on H satisfying the conditions:

\begin{aligned}⟨h, h_{1}h_{2}⟩ & = ⟨h_{i}, h_{2}⟩ ⟨h^{i}, h_{1}⟩ , \\ ⟨h_{1}h_{2}, h⟩ & = ⟨h_{1}, h_{i}⟩ ⟨h_{2}, h^{i}⟩ , \end{aligned}

g_{j}h_{i} ⟨h^{i}, g^{j}⟩ = ⟨h_{i}, g_{j}⟩ h^{i}g^{j}

for all h, h_{1}, h_{2}, g ∈ H and Δh = h_{i} ⊗ h^{i}, Δg = g_{j} ⊗ g^{j}. On the generators this form will be given by

⟨t_{j}^{i}, t_{l}^{k}⟩ = q(R̂^{−1})_{jl}^{ki}.

4 Alexander Verbovetsky

It is well-known that such a structure does exist in all examples one is likely to encounter, provided H was constructed in an appropriate category.

Having this form we can introduce a (left) H-module structure on the algebra A:

h · f = f_{i} ⟨h, h^{i}⟩ ,

where h ∈ H, f ∈ A, Δf = f_{i} ⊗ h^{i}. In coordinates we have

t_{j}^{i}x^{k} = C_{αj}^{ik}x^{α}, t_{j}^{i} · 1 = δ_{j}^{i},
t̃_{j}^{i}x^{k} = q(R̂^{−1})_{jα}^{ki}x^{α},
˜t̃_{j}^{i}x^{k} = q^{−1}R̂_{αj}^{ik}x^{α},

where the matrix C is defined by C_{jβ}^{iα}(R̂^{−1})_{lα}^{kβ} = q^{−1}δ_{l}^{i}δ_{j}^{k}. Note that this makes A a crossed H-bimodule. Lemma. For any f ∈ A we have

x^{i}f = (˜t̃_{α}^{i}f) x^{α},
fx^{i} = x^{α} (t̃_{α}^{i}f) .

Proof. We prove the first relation, the second can be proved in the same way. For f = 1 the relation is obvious. Suppose inductively that it is true for all f such that deg f < m. Take f = x^{j}g with deg g < m. Then we get

(˜t̃_{α}^{i}f) x^{α} = (˜t̃_{β}^{i}x^{j}) (˜t̃_{α}^{β}g) x^{α} = (˜t̃_{β}^{i}x^{j}) x^{β}g = q^{−1}R̂_{αβ}^{ij}x^{α}x^{β}g = x^{i}x^{j}g = x^{i}f.

Therefore the relation holds true for anyf of degree m. □

Having an H-module structure on A we can define an H-module structure on Hom_{𝕜} (A, A) by the formula:

(t_{j}^{i}D) (f) = t_{α}^{i} (D (t̃_{j}^{α}f)) , D ∈ Hom_{𝕜} (A, A) .

Notice that Hom_{𝕜} (A, A) is a module-algebra over H:

(t_{j}^{i} (D_{1} ∘ D_{2})) = t_{α}^{i}D_{1}D_{2}t̃_{j}^{α} = t_{β}^{i}D_{1}t̃_{ε}^{β}t_{α}^{ε}D_{2}t_{j}^{α} = (t_{ε}^{i}D_{1}) ∘ (t_{j}^{ε}D_{2}) .

QUANTIZED DIFFERENTIAL OPERATORS 5

3. Now we define quantum algebra of (scalar) differential operators. For any 𝕜linear mapping D: A → A and for all k ⩾ 0, 1 ⩽ i ⩽ n, define the mappings [D, x^{i}]_{k} : A → A by the formula:

[D, x^{i}]_{k} (f) = D (x^{i}f) − q^{2k}x^{α} (t̃_{α}^{i}D) (f) .

The set of differential operators Diff_{k} (A) of order ⩽ k can be defined by the following inductive procedure: Diff_{0} (A) = {D ∈ Hom_{𝕜} (A, A) | [D, x^{i}]_{0} = 0 ∀i}. In general, Diff_{k+1} (A) = {D ∈ Hom_{𝕜} (A, A) | [D, x^{i}]_{k+1} ∈ Diff_{k} (A) ∀i}. Proposition. (1) If D ∈ Diff_{k} (A) then h · D ∈ Diff_{k} (A) ∀h ∈ H.

(2) If D_{1} ∈ Diff_{k} (A) and D_{2} ∈ Diff_{l} (A), then D_{1} ∘ D_{2} ∈ Diff_{k+l} (A).

Proof. (1) It is sufficient to prove that

t_{l}^{j} [D, x^{i}]_{k} = C_{βl}^{αi} [t_{α}^{j}D, x^{β}]_{k} .

We have

t_{l}^{j} [D, x^{i}]_{k} = (t_{α}^{j}D) (t_{l}^{α}x^{i}) − q^{2k} (t_{β}^{j}x^{α}) (t_{l}^{β}t̃_{α}^{i}D)
= C_{βl}^{αi} (t_{α}^{j}D) x^{β} − q^{2k}C_{εβ}^{jα}x^{ε} (t_{l}^{β}t̃_{α}^{i}D)
= C_{βl}^{αi} (t_{α}^{j}D) x^{β} − q^{2k}C_{βl}^{αi}x^{ε} (t̃_{ε}^{β}t_{α}^{j}D)
= C_{βl}^{αi} [t_{α}^{j}D, x^{β}]_{k} .
(2) [D_{1} ∘ D_{2}, x^{i}]_{k+l}

= D_{1}D_{2}x^{i} − q^{2l}D_{1}x^{α} (t̃_{α}^{i}D_{2}) + q^{2l}D_{1}x^{α} (t̃_{α}^{i}D_{2}) − q^{2k+2l}x^{β} (t̃_{β}^{α}D_{1}) (t̃_{α}^{i}D_{2})

= D_{1} ∘ [D_{2}, x^{i}]_{l} + q^{2l} [D_{1}, x^{α}]_{k} ∘ t̃_{α}^{i}D_{2}

This implies the statement in a standard way. □

This Proposition has the following: Corollary. (1) There exists an A-bimodule structure on Diff_{k} (A) :

a · D = a ∘ D, D · a = D ∘ a.

(2) For any k < l we have Diff_{k} (A) ⊂ Diff_{l} (A).

We denote Diff (A) = ⋃_{k⩾0} Diff_{k} (A).

6 Alexander Verbovetsky

4. Now we give a detailed description of the modules Diff_{k} (A). We shall assume that the space V^{∗} has an ordered basis x^{1},…, x^{n} such that the monomials (x^{1})^{i_{1}} … (x^{n})^{i_{n}} form a basis of A and R̂_{kl}^{ij} = 0 if i > j and either k > l or k ⩾ i.

Clearly, any differential operator D ∈ Diff_{k} (A) is completely determined by its values on the basic monomials of degree ⩽ k. The following theorem states that these values can be taken in an arbitrary way. Theorem. The A-module Diff_{k} (A) has a basis 1, ∂_{σ}, |σ| ⩽ k, where σ = j_{1} … j_{r}, 1 ⩽ j_{l} ⩽ j_{l+1} ⩽ n, is a multi-index, |σ| = r, and the operator ∂_{σ} is equal to 1 at x^{j_{1}} … x^{j_{r}} and vanishes on the other monomials of degree ⩽ k. Proof. We have to show that the defining relations [⋯ [D, x^{i_{k}}]_{k} ⋯ x^{i_{0}}]_{0} = 0 with i_{l+1} ⩽ i_{l} will be sufficient for D ∈ Diff_{k} (A). For this we verify that

[[D, x^{i}]_{l+1} , x^{j}]_{l} = q^{−1}R̂_{αβ}^{ij} [[D, x^{α}]_{l+1} , x^{β}]_{l} .

We have

[[D, x^{i}]_{l+1} , x^{j}]_{l}

= Dx^{i}x^{j} − q^{2l+2}x^{α} (t̃_{α}^{i}D) x^{j} − q^{2l}x^{α} (t̃_{α}^{γ}D) (t̃_{γ}^{j}x^{i}) + q^{4l+2}x^{α} (t̃_{α}^{γ}x^{ε}) (t̃_{γ}^{j} t̃_{ε}^{i}D) = Dx^{i}x^{j} − q^{2l+1} (qδ_{γ}^{i}δ_{ε}^{j} + (R̂^{−1})_{γε}^{ij}) (x^{α} (t̃_{α}^{γ}D) x^{ε}) + q^{4l+2}x^{ε}x^{γ} (t̃_{γ}^{j} t̃_{ε}^{i}D) = q^{−1}R̂_{αβ}^{ij}Dx^{α}x^{β} − q^{2l}R̂_{αβ}^{ij} (qδ_{γ}^{α}δ_{ε}^{β} + (R̂^{−1})_{γε}^{αβ}) (x^{κ} (t̃_{κ}^{γ}D) x^{ε})

+ q^{4l+1}R̂_{αβ}^{ij}x^{ε}x^{γ} (t̃_{γ}^{β}t̃_{ε}^{α}D) = q^{−1}R̂_{αβ}^{ij} [[D, x^{α}]_{l+1} , x^{β}]_{l} . □

5. We can define a (right) H-comodule structure on Diff_{k} (A) by the following property:

h · D = D_{i} ⟨h, h^{i}⟩ ,

where h ∈ H, D ∈ Diff_{k} (A), ΔD = D_{i} ⊗ h^{i}. In particular, for the operators ∂_{i} this yields:

Δ∂_{i} = ∂_{α} ⊗ t_{i}^{α}.

This structure makes Diff (A) an H-comodule-algebra:

h · (D_{1} ∘ D_{2}) = (h_{i}D_{1}) (h^{i}D_{2}) = (D_{1})_{k} (D_{2})_{l} ⟨h_{i}, g_{1}^{k}⟩ ⟨h^{i}, g_{2}^{l}⟩ = (D_{1})_{k} (D_{2})_{l} ⟨h, g_{2}^{l}g_{1}^{k}⟩ , so that

Δ (D_{1} ∘ D_{2}) = (D_{1})_{k} ∘ (D_{2})_{l} ⊗ g_{2}^{l}g_{1}^{k},

where Δh = h_{i} ⊗ h^{i}, ΔD_{r} = (D_{r})_{k} ⊗ g_{r}^{k}, r = 1, 2.

Proposition (Covariance of differential operators). For any D ∈ Diff_{k} (A) and f ∈ A

Δ (D (f)) = Δ (D) (Δ (f)) ,

where the action in right-hand side is given by (∇ ⊗ h_{1}) (a ⊗ h_{2}) = ∇ (a) ⊗ h_{2}h_{1}, ∇ ∈ Diff_{k} (A), a ∈ A, h_{1}, h_{2} ∈ H.

Proof. For f of degree zero the statement is trivial. Suppose inductively that it is true for deg f ⩽ m and take an element f of degree m. We get

Δ (D (x^{i}f)) = Δ ((Dx^{i}) (f)) = Δ (Dx^{i}) (Δ (f)) = Δ (D) Δ (x^{i}) Δ (f) = Δ (D) (Δ (x^{i}f)) . □

6. Consider the first order operators ∂_{i}. It follows easily from the very definition that they satisfy the Wess-Zumino Leibnitz rule:

∂_{i}x^{j} = δ_{i}^{j} + qR̂_{iα}^{jβ}x^{α}∂_{β}.

Further, one has the following commutation relations between ∂_{i}:

R̂_{ji}^{βα}∂_{α}∂_{β} = q∂_{i}∂_{j}.

Indeed, it is obvious that the second order operator R̂_{ji}^{βα}∂_{α}∂_{β} − q∂_{i}∂_{j} vanishes on the monomials of degree 0 and 1. For the degree 2 we have:

(R̂_{ji}^{βα}∂_{α}∂_{β} − q∂_{i}∂_{j}) (x^{k}x^{l}) = R̂_{ji}^{kl} + qR̂_{ji}^{βα}R̂_{βα}^{kl} − qδ_{i}^{l}δ_{j}^{k} − q^{2}R̂_{ji}^{kl} = q(R̂^{2} − (q − q^{−1}) R̂ − 1)_{ji}^{kl} = 0.

Definition ([2]). The quantum Weyl algebra D is the algebra with 2n generators x^{1},…, x^{n}, ∂_{1},…, ∂_{n} satisfying the following commutation relations:

R̂_{αβ}^{ij}x^{α}x^{β} = qx^{i}x^{j},
R̂_{ji}^{βα}∂_{α}∂_{β} = q∂_{i}∂_{j},
∂_{i}x^{j} = δ_{i}^{j} + qR̂_{iα}^{jβ}x^{α}∂_{β}.

From the previous discussion it follows that there exists an algebra morphism w: D → Diff (A) such that w (x^{i}) = x^{i}, w (∂_{j}) = ∂_{j}. The image of w consists of differential operators that are compositions of operators of order ⩽ 1. One can also show that coker w is exactly the Spencer zero cohomology of the algebra A (see [12]). For the matrix R̂ of the form (1) from the above-stated Theorem and a result of E. Demidov [2, Corollary 12.6] it follows the following:

Proposition. If q^{2} is not a root of unity, or q^{2} = 1 and char 𝕜 = 0, then w: D → Diff (A) is an isomorphism, and if q^{2} is a primitive root of unity of degree ℓ, then the kernel of w is generated by ∂_{1}^{ℓ},…, ∂_{n}^{ℓ}.

Thus, in the root of unity case there exist operators that can not be presented as a composition of operators of order ⩽ 1.

Example. Consider the operators D_{j} = ∂\underbrace{j… j}_{ℓ times}. It is easy to see that ∂_{i} ((x^{j})^{ℓ})

= 0 ∀i. Hence D_{j} is not a composition of operators of order ⩽ 1. In fact, any operator is a composition of D_{j} and operators of order ⩽ 1.

Acknowledgments. The author is grateful to the Italian Ministero degli Afari Esteri for the financial support and to the Scuola Internazionale Superiore di Studi Avanzati (SISSA) in Trieste, where this work was done, for the excellent working conditions.

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Address for correspondence: Profsoyuznaya 98-9-132, 117485 Moscow, Russia E-mail address: verbovet@ecfor.msk.su