The Diffiety Institute Preprint Series

Preprint DIPS-4/96 October 1, 1996

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Differential operators over quantum spaces

by

Alexander Verbovetsky

DIFFERENTIAL OPERATORS OVER

QUANTUM SPACES

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

Abstract. The topic of this paper is the development of a differential calculus on a quantum space covariant with respect to the action of a quantum group. Quantized differential operators, jets, the de Rham and Spencer complexes, etc. are constructed. Also the integration over a quantum space, adjoint operators, integral forms, Green’s formula are discussed. The Wess-Zumino de Rham complex and the algebra of differential operators are treated as a basic example.

Introduction

The aim of this paper is to develop a differential calculus over quantum spaces (by “differential calculus” we mean the theory of differential operators, jets, forms, the Spencer complexes, the integration, adjoint operators, etc., and not only the de Rham complex). The constructions has been principally motivated by the paper [23] by Wess and Zumino. In fact, the present work provides an algebraic framework that allows one to treat quantized differential calculus (à la Wess-Zumino) very much like the commutative or graded-commutative calculus (see

1991 Mathematics Subject Classification. 16S32, 16S80, 17B37, 16W30, 18D10, 35A99, 58A99, 58G05, 58G99, 81R50.

Key words and phrases. Quantum space, quantum group, differential operator, jets, differential forms, adjoint operator, Berezinian, integral forms, Hopf algebras, Wess-Zumino calculus.

2 Alexander Verbovetsky

[18, 19, 20, 21, 22, 16] as well as others articles in this issue of Acta Appl. Math.). The precise description of our setting is as follows.

Start with a Hopf algebra H over a commutative ring 𝕜. We shall imagine H as an algebra of functions on a quantum group acting on a “quantum space”. Let A denote an associative unital 𝕜-algebra that plays the rôle of an “algebra of functions” on this “quantum space”. Correspondingly, we endow A with a comodule-algebra structure δ: A → A ⊗ H. Now, for any H-module X there exists a natural transposition operator σ_{X}: X ⊗ A → A ⊗ X defined by

x ⊗ a ↦ a_{(0)} ⊗ a_{(1)}x.

(For notation and facts related to Hopf algebras see the Appendix). This suggests the possibility of exploring a differential calculus in a category of H-modules provided that the algebra A belong to this category. Hence, we assume that A is equipped with an H-module structure such that the multiplication m: A⊗A → A and the unit morphism u: 𝕜 → A are H-linear and A is a crossed H-bimodule. The last assumption implies that the transpositions σ_{X} are H-linear.

The following statement shows that σ_{X} satisfies all natural properties one could expect.

Proposition. (1) σ is an isomorphism of the functors · ⊗ A and A ⊗ ·

(2) (σ_{A} ⊗ 1) (1 ⊗ σ_{X}) (σ_{X} ⊗ 1) = (1 ⊗ σ_{X}) (σ_{X} ⊗ 1) (1 ⊗ σ_{A}) : X ⊗ A ⊗ A → A ⊗ A ⊗ X (the Yang-Baxter equation)

(3) σ_{X_{1}⊗X_{2}} = (σ_{X_{1}} ⊗ 1) (1 ⊗ σ_{X_{2}}) : X_{1} ⊗ X_{2} ⊗ A → A ⊗ X_{1} ⊗ X_{2}

(4) σ_{X} (x ⊗ 1) = 1 ⊗ x, x ∈ X σ_{X} (1 ⊗ m) = (m ⊗ 1) (1 ⊗ σ_{X}) (σ_{X} ⊗ 1) : X ⊗ A ⊗ A → A ⊗ X

Proof. (1) The inverse to σ_{X} is given by

σ_{X}^{−1} (a ⊗ x) = ã_{(1)}x ⊗ a_{(0)},

where the tilde stands for the antipode in H. The functoriality of σ_{X} is immediate.

(2) The evaluation of the Yang-Baxter equation on an element x ⊗ a ⊗ b ∈ X ⊗ A ⊗ A leads to

b_{(0)} ⊗ b_{(1)}a_{(0)} ⊗ b_{(2)}a_{(1)}x = b_{(0)} ⊗ (b_{(2)}a)_{(0)} ⊗ (b_{(2)}a)_{(1)}b_{(1)}x.

This is a consequence of the definition of crossed H-bimodule.

(3) σ_{X_{1}⊗X_{2}} (x_{1} ⊗ x_{2} ⊗ a) = a_{(0)} ⊗ a_{(1)} (x_{1} ⊗ x_{2}) = a_{(0)} ⊗ a_{(1)}x_{1} ⊗ a_{(2)}x_{2} = σ_{X_{1}} (x_{1} ⊗ a_{(0)}) ⊗ a_{(1)}x_{2} = (σ_{X_{1}} ⊗ 1) (1 ⊗ σ_{X_{2}}) (x_{1} ⊗ x_{2} ⊗ a) .

(4) We prove the second relation. (m ⊗ 1) (1 ⊗ σ_{X}) (σ_{X} ⊗ 1) (x ⊗ a ⊗ b) = (m ⊗ 1) (1 ⊗ σ_{X}) (a_{(0)} ⊗ a_{(1)}x ⊗ b) = (m ⊗ 1) (a_{(0)} ⊗ b_{(0)} ⊗ b_{(1)}a_{(1)}x) = a_{(0)}b_{(0)} ⊗ b_{(1)}a_{(1)}x = (ab)_{(0)} ⊗ (ab)_{(1)}x = σ_{X} (x ⊗ ab) . □

Remark. Indeed, in what follows one can use a family of operators σ_{X}, obeying the properties from this Proposition, rather than H-module and H-comodule structures.

Here are some examples of the above described setting.

Example 1. H = 𝕜, A is an arbitrary algebra, σ is the usual permutation operator.

Example 2. H = 𝕜 [G] is the group algebra of an abelian discrete group G. Consider the category of G-graded 𝕜-modules, with the H-module structure on the objects given by

g_{1} · x_{g} = ε (g_{1}, g) x_{g},  (1)

where g_{1}, g ∈ G, x_{g} is an element of an object X of grading g, ε: G × G → 𝕜^{∗} is a commutation factor (i.e., a mapping, satisfying two properties: ε (g_{1} + g_{2}, g_{3}) = ε (g_{1}, g_{3}) ε (g_{2}, g_{3}), ε (g_{1}, g_{2})^{−1} = ε (g_{2}, g_{1}); for details see [2]). Further, A = ⨁_{g∈G} A_{g} is an arbitrary graded algebra, with the comodule structure δ (a_{g}) = a_{g} ⊗ g, where a_{g} is of grading g. Thus the transposition mapping σ_{X}: X ⊗ A → A ⊗ X has the standard form:

σ_{X} (x_{g_{1}} ⊗ a_{g_{2}}) = ε (g_{2}, g_{1}) a_{g_{2}} ⊗ x_{g_{1}}  (2)

4 Alexander Verbovetsky

Example 3. The previous example can be generalized for the case of non-abelian group G (see [7, 10]). Namely, let us assume that the group G acts on the objects of the category under consideration in such a way that g_{1} (X_{g}) ⊂ X_{g_{1}^{−1}gg_{1}}. Then we can change (1) in order to get the same grading in the both sides of (2):

g_{1} · x_{g} = ε (g_{1}, g) g_{1} (x_{g}) ,

with ε(g_{1}g_{2}, g_{3}) = ε(g_{1}, g_{2}^{−1}g_{3}g_{2})ε(g_{2}, g_{3}) and ε(g_{1}, g_{2}g_{3}) = ε(g_{1}, g_{2})ε(g_{1}, g_{3}) Example 4. Let 𝕜 be a field. Consider an R-matrix bialgebra M (see [14, 6, 3, 4]) generated by the elements of a matrix T = ∥t_{i}^{j}∥, i, j = 1,…, n, obeying the relations

R̂_{ij}^{αβ}t_{α}^{k}t_{β}^{l} = t_{i}^{α}t_{j}^{β}R̂_{αβ}^{kl}  (3)

(throughout this paper we assume summation over repeated indices which occur in both upper and lower positions). Here R̂ = ∥R_{kl}^{ij}∥ is a non-singular solution of the Yang-Baxter equation:

R̂_{ij}^{αβ}R̂_{rα}^{kε}R̂_{εβ}^{lm} = R̂_{ri}^{αε}R̂_{εj}^{βm}R̂_{αβ}^{kl}  (4)

(we use the hat over R in order to conform with notation in [14, 23]). The simplest example is the R-matrix of GL_{q} (n) introduced in [5]:

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) ,  (5)

where q is an invertible element of 𝕜, θ (i) = \{ \begin{matrix}1, & for i > 0 \\ 0, & for i ⩽ 0. \end{matrix}

The comultiplication in M is given by δ(t_{j}^{i}) = t_{α}^{i} ⊗ t_{j}^{α}. the counit is ε(t_{i}^{j}) = δ_{i}^{j}. We obtain an R-matrix quantum group H by taking the Hopf envelope of M.

Now we can get examples of H-comodule-algebras by setting A to be an algebra generated by the set x = (x^{i}), i = 1,…, n, obeying the relations and H-coactions

f(R̂)_{αβ}^{ij}x^{α}x^{β} = 0, δ (x^{i}) = x^{α} ⊗ t̃_{α}^{i},

where f is a polynomial. However, it is clear that A is non-trivial if f is a factor of the minimal polynomial of R̂. In particular, let q be an eigenvalue of R̂ and f (ξ) = ξ − q. Then the algebra A obeys the relations

R̂_{αβ}^{ij}x^{α}x^{β} = qx^{i}x^{j}

DIFFERENTIAL OPERATORS OVER QUANTUM SPACES 5

and is called the algebra of functions on the quantum plane. We can obtain an H-multiplication on A from the matrix R̂ by putting

t_{j}^{i}x^{k} = c_{αj}^{ik}x^{α}

where the matrix C = ∥c_{ij}^{kl}∥ is defined by c_{jβ}^{iα}(R̂^{−1})_{lα}^{kβ} = γδ_{j}^{k}δ_{l}^{i}, γ is a constant. Relations (3) guarantee the compatibility of the multiplication and comultiplication (i.e., that A is a crossed H-bimodule). The multiplication is well-defined due to the Yang-Baxter condition (4).

One comment is in order now. In our setting the operators σ play the rôle similar to that of braiding in the theory of braded differential operators [8-11]. The difference is that σ is not defined for all pairs of modules. Although in commutative and graded-commutative cases (or, more generally, in a symmetric tensor category) both approaches yield the same calculus, in non-symmetric case they seem to be quite different.

Now we add the last component to our setting. One knows that in supergeometry a special rôle is played by the parity inversion functor. Its natural substitute in our setting is a commuting group-like invertible element φ ∈ H. In graded case (Examples 2 and 3) such elements form the centre of the group G. In particular, in supercase G = ℤ_{2} there is only one nontrivial element φ. In R-matrix case one can extend H by adjoining a single invertible and commuting generator φ, with δ (φ) = φ⊗φ, ε (φ) = 1, φ̃ = φ^{−1}, and action on A φx^{i} = λx^{i}, λ is a constant. We shall assume that we are given an element φ and enter it in the very definition of all fundamental objects. Indeed, this is a very important novelty as compared with commutative or graded-commutative case: in general, the transposition mapping can be incompatible with φ = 1. For instance, in the case of R-matrix quantum group with R̂ satisfying a quadratic equation (R̂ − μ_{1})(R̂ − μ_{2}) = 0, the constant λ in the definition of φ must be equal to the ratio of the eigenvalues λ = −μ_{1}/μ_{2}. The main place where the element φ should be entered is the formula for transposition on ℤ-graded object X = ∑_{i} X_{i}. Namely, we set

σ_{X} (x_{i} ⊗ a) = φ^{i}a_{(0)} ⊗ a_{(1)}x_{i}, x_{i} ∈ X_{i}.

6 Alexander Verbovetsky

One can easily prove the above Proposition in this generalized case. Note that in supercase our definition of σ_{X} corresponds to that used in the Bernshtein-Leites construction of exterior algebra (see [12]).

The paper is organized as follows. In Section 2 we define differential operators over an algebra A which is commutative with respect to σ_{A} and develop the theory of jets, forms, and so on. In Section 3 we consider adjoint operators and Berezinian, and in Section 4 we use them to describe the Spencer complex and obtain Green’s formula. In the Appendix we have collected all definitions related to Hopf algebras needed in the main text.

Acknowledgments. The author is grateful to Professor A. M. Vinogradov for his attention to this work and to D. M. Guessler for many helpful discussions. Most of results of this paper were obtained during author’s stay at the Scuola Internazionale Superiore di Studi Avanzati (SISSA) in Trieste. It is a great pleasure to thank SISSA for the hospitality and the Italian Ministero degli Afari Esteri for the financial support.

2. Differential Operators

2.1. Suppose we are given an algebra A. We shall assume that it is commutative with respect to σ_{A}, i.e., m = mσ_{A}. Consider the category Mod_{A} of modules over A. It is easy to prove that any left module P can be transformed canonically into a right module, pa = a_{(0)} (a_{(1)}p) for a ∈ A, p ∈ P, so that we may consider P as a bimodule. A map f: P → Q is said to be an A-homomorphism if f (ap) = a_{(0)} (a_{(1)}f) (p). The module of A-homomorphisms is denoted by Hom_{A} (P, Q).

Now we state the basic definition of differential operator. Let Δ ∈ Hom_{𝕜} (P, Q) be a 𝕜-homomorphism, P and Q being A-modules. For every element a ∈ A define a 𝕜-homomorphism [Δ, a]_{k} : P → Q, k = 0, 1, 2,…, by setting

[Δ, a]_{k} (p) = Δ (ap) − (φ^{k}a_{(0)}) (a_{(1)}Δ) (p) , p ∈ P.

Lemma. For Δ_{1} ∈ Hom_{𝕜} (Q, R) and Δ_{2} ∈ Hom_{𝕜} (P, Q) we have

[Δ_{1} ∘ Δ_{2}, a]_{k+l} = Δ_{1} ∘ [Δ_{2}, a]_{l} + [Δ_{1}, φ^{l}a_{(0)}]_{k} ∘ (a_{(1)}Δ_{2}) .

Proof is immediate. □

The set of differential operators (d.o.) of order ⩽ k is defined by the following inductive procedure. A 𝕜-homomorphism Δ ∈ Hom_{𝕜} (P, Q) is a d.o. of order zero if Δ ∈ Hom_{A} (P, Q). In general, Δ ∈ Hom_{𝕜} (P, Q) is a d.o. of order ⩽ k if [Δ, a]_{k} is a d.o. of order k − 1 for all a ∈ A. From the Lemma it follows in a straightforward way that the composition of two d.o.’s of order ⩽ k and of order ⩽ l is a d.o. of order ⩽ k +l. In particular, the set of all d.o.’s of order ⩽ k, from P to Q, is closed under the left and right multiplications by an element a ∈ A given by aΔ = a ∘ Δ and Δa = Δ ∘ a, where a is understood as the operator of multiplication by a. The modules that arise in this way are denoted by Diff_{k} (P, Q) and Diff_{k}^{+} (P, Q) respectively. Also, Diff_{k}^{(+)} (P, Q) is closed under the mapping Δ ↦ [Δ, a]_{m} ∀a ∀m, hence Diff_{k}^{(+)} (P, Q) ⊂ Diff_{l}^{(+)} (P, Q) for k ⩽ l, so that we may consider the union Diff^{(+)} (P, Q) = ⋃_{k⩾0} Diff_{k}^{(+)} (P, Q)

Proposition. (1) The identical map Diff_{k}^{+} → Diff_{k} is a d.o. of order ⩽ k. (2) There exists a canonical isomorphism

Diff_{k}^{+} (P, Q) → Hom_{A} (P, Diff_{k}^{+} (Q)) ,

where Diff_{k}^{+} (Q) = Diff_{k}^{+} (A, Q). To every d.o. Δ: P → Q there corresponds the homomorphism ψ_{Δ} ∈ Hom_{A} (P, Diff_{k}^{+} (Q)), ψ_{Δ} (p) (a) = Δ (pa) under this isomorphism. The inverse mapping takes a homomorphism ψ: P → Diff_{k}^{+} (Q) to an operator D_{k} ∘ψ, where D_{k}: Diff_{k}^{+} (Q) → Q is a d.o. of order ⩽ k defined by the formula D_{k} (∇) = ∇ (1), ∇ ∈ Diff_{k}^{+} (Q).

Proof consists of a series of automatic verifications. □

Remark. Unlike the commutative case, the map Diff_{k} (P, Q) → Diff_{k}^{+} (P, Q) is in general not a d.o.

This proposition has the following:

Corollary. The commutative diagram

Diff_{k,l}^{+} (P) ⟶^{D_{k}} Diff_{l}^{+} (P)
c_{k,l}↓ ↓^{D_{l}}
Diff_{k+l}^{+} (P) ⟶^{D_{k+l}}  P

where Diff_{k,l}^{+} (P) = Diff_{k}^{+} (Diff_{l}^{+} (P)), uniquely defines the A-homomorphism c_{k,l}, which is said to be glueing operator.

2.2. Denote by D_{1} (P) the kernel of the A-homomorphism Diff_{1} (A, P) → P, Δ ↦ Δ (1). Elements of D_{1} (P) are said to be P -valued derivations of A and can be equivalently defined by the Leibnitz rule:

∇ (ab) = ∇ (a) b + (φa_{(0)}) (a_{(1)}∇) (b) .

To construct multiderivations in more than one argument we need the following remark. Let F be a functor F: Mod_{A} → Mod_{A}. For any module P denote by Ḟ (P) the 𝕜-module F (P) supplied with the A-module structure induced by the Amodule structure on P. Then any natural 𝕜-homomorphism f: F_{1} (P) → F_{2} (P) gives rise to the A-homomorphism ḟ: Ḟ_{1} (P) → Ḟ_{2} (P).

Now we inductively define for k = 2, 3,… the modules of P -valued multiderivations D_{k} (P) together with an A-linear embeddings

i_{k}: Ḋ_{k} (P) → Ḋ_{k−1} (Diff_{1}^{+} (P))

(the dot in the right-hand side is applied to the whole functor D_{k−1} (Diff_{1}^{+})) as the kernel of the composition of A-homomorphisms Ḋ_{k−1} (Diff_{1}^{+} (P)) ⟶^{i_{k−1}}

Ḋ_{k−2} (Diff_{1,1}^{+} (P)) ⟶^{D_{k−2}(c_{1,1})} Ḋ_{k−2} (Diff_{2}^{+} (P)).

The Spencer Diff-operator S: Ḋ_{k} (Diff_{l}^{+} (P)) → Ḋ_{k−1} (Diff_{l+1}^{+} (P)) is defined as the composition

S: Ḋ_{k} (Diff_{l}^{+} (P)) → Ḋ_{k−1} (Diff_{1,l}^{+} (P)) → Ḋ_{k−1} (Diff_{l+1}^{+} (P))

Proposition. S^{2} = 0 □

This Proposition can be proved directly, but also it follows from Remark 2.3 below. The complex

0 ⟵ P ⟵ Diff (P) ⟵^{S} Ḋ_{1} (Diff^{+} (P)) ⟵^{S} Ḋ_{2} (Diff^{+} (P)) ⟵ …

is called the Spencer Diff-complex.

2.3. Let us give a more direct description of the modules D_{k} (P). Namely, by the very definition each element of the module D_{k} (P) can be identified with a k-multilinear over 𝕜 mapping ∇: A × A × … × A → P, ∇ (a_{1},…, a_{k}) = i_{k} (∇) (a_{1},…, a_{k−1}) (a_{k}). Then the elements of D_{k} (P)are described as follows.

Proposition. A k-multilinear over 𝕜 mapping ∇: A × A ×… × A → P belongs to D_{k} (P) if and only if the following conditions hold:

1. ∇ (a_{1},…, a_{i}, a_{i+1},…, a_{k}) = −∇ (a_{1},…, ã_{i}_{(1)}a_{i+1}, φa_{i}_{(0)},…, a_{k})
2. ∇ (a_{1},…, a_{i−1}, ab, a_{i+1},…, a_{k}) =
∇ (a_{1},…, a_{i−1}, a, b̃_{(k−i)}a_{i+1},…, b̃_{(1)}a_{k}) φ^{k−i}b_{(0)} +
∇ (a_{1},…, a_{i−1}, ã_{(k−i+1)}b, ã_{(k−i)}a_{i+1},…, ã_{(1)}a_{k}) φ^{k−i+1}a_{(0)}.

Proof. For k = 1 the statement is trivial. Suppose inductively that it holds true for all k < l. Take ∇_{1} ∈ D_{l−1} (Diff_{1}^{+} (P)) such that ∇_{1} (a_{1},…, a_{l−1}) (1) = 0 and set ∇ (a_{1},…, a_{l}) = ∇_{1} (a_{1},…, a_{l−1}) (a_{l}). Then

\begin{matrix}∇ (a_{1},…, a_{l−1}, a_{l}) = (∇_{1} (a_{1},…, a_{l−1}) a_{l}) (1) \\ = ∇_{1} (a_{1},…, a_{l−1}a_{l}) (1) − ∇_{1} (a_{1},…, a_{l−2}, ã_{l−1}_{(1)}a_{l}) (φa_{l−1}_{(0)}) \end{matrix}

= −∇ (a_{1},…, a_{l−2}, ã_{l−1}_{(1)}a_{l}, φa_{l−1}_{(0)}) ,

so that ∇ satisfies the first property. Further, observe that the second property

10 Alexander Verbovetsky

with ab in the i^{st} place is true if it is true with ab in the (i + 1)^{th} place: ∇ (a_{1},…, a_{i−1}, ab, a_{i+1},…, a_{k}) = − ∇ (a_{1},…, a_{i−1}, ã_{(1)}b̃_{(1)}a_{i+1}, (φa_{(0)}) (φb_{(0)}) , a_{i+2},…, a_{k}) = − ∇ (a_{1},…, a_{i−1}, ã_{(1)}b̃_{(k−i)}a_{i+1}, φa_{(0)}, b̃_{(k−i−1)}a_{i+2},…, b̃_{(1)}a_{k}) φ^{k−i}b_{(0)}− ∇ (a_{1},…, a_{i−1}, ã_{(k−i+1)}b̃_{(1)}a_{i+1}, φã_{(k−i)}b_{(0)}, ã_{(k−i−1)}a_{i+2},…, ã_{(1)}a_{k}) φ^{k−i+1}a_{(0)} = ∇ (a_{1},…, a_{i−1}, a, b̃_{(k−i)}a_{i+1}, b̃_{(k−i−1)}a_{i+2},…, b̃_{(1)}a_{k}) φ^{k−i}b_{(0)} + ∇ (a_{1},…, a_{i−1}, ã_{(k−i+1)}b, ã_{(k−i)}a_{i+1}, ã_{(k−i−1)}a_{i+2},…, ã_{(1)}a_{k}) φ^{k−i+1}a_{(0)}.

Thus it is sufficient to verify the second property for ∇ with ab in the last place. We get

\begin{matrix}∇ (a_{1},…, a_{l−1}, ab) = ∇_{1} (a_{1},…, a_{l−1}) (ab) = \\ ∇_{1} (a_{1},…, a_{l−1}) (a) b + ∇_{1} (a_{1},…, a_{l−1}) (ã_{(1)}b) (φa_{(0)}) = \end{matrix}

∇ (a_{1},…, a_{l−1}, a) b + ∇ (a_{1},…, a_{l−1}, ã_{(1)}b) φa_{(0)}.

So, if ∇ ∈ D_{l} (P) then it satisfies the properties. Now let us check the converse. We are to prove that if ∇ enjoys the properties then it belongs to D_{l−1} (Diff_{1}^{+} (P)). Clearly, ∇ is a d.o. of order 1 in the last argument and is skew symmetric. It remains to check the second property. We have

∇ (a_{1},…, a_{l−1}, ab, c) =

∇ (a_{1},…, a_{l−1}, a, b̃_{(1)}c) φb_{(0)} + ∇ (a_{1},…, a_{l−1}, ã_{(2)}b, ã_{(1)}c) (φ^{2}a_{(0)}) =
∇ (a_{1},…, a_{l−1}, a, b) c + ∇ (a_{1},…, a_{l−1}, a, b̃_{(1)}c) φb_{(0)}+
∇ (a_{1},…, a_{l−1}, ã_{(1)}b, φa_{(0)}) c + ∇ (a_{1},…, a_{l−1}, ã_{(2)}b, ãc) (φ^{2}a_{(0)}) =
∇ (a_{1},…, a_{l−1}, a, bc) + ∇ (a_{1},…, a_{l−1}, ã_{(1)}b, (φa_{(0)}) c)
□

Remark. It can be easily shown that the Spencer operator S: Ḋ_{k} (Diff_{l}^{+} (P)) → Ḋ_{k−1} (Diff_{l+1}^{+} (P)) acts by the rule:

S (∇) (a_{1},…, a_{k−1}) (a) = ∇ (a_{1},…, a_{k−1}, a) (1) .

2.4. In the sequel, the module D_{∗} (P) = ∑_{k=0}^{∞} D_{k} (P) will be considered as a graded module, with A-module structure being as follows:

a ∧ ∇ = a∇,
∇ ∧ a = ∇ (φ^{k}a) , ∇ ∈ D_{k} (P) .

Note that the Spencer operators are A-homomorphisms of degree −1. Proposition. There exists a natural embedding

D_{k+l} (P) → D_{k} (D_{l} (P)) .

Proof. Let ∇ be an element of D_{k+l} (P) and □ be the corresponding mapping \underbrace{A ×… × A}_{k times} → D_{l} (P). We have to show that □ is a derivation in the last argument.

We get □ (a_{1},…, a_{k−1}, ab) (c_{1},…, c_{l}) = ∇ (a_{1},…, a_{k−1}, ab, c_{1},…, c_{l}) = ∇ (a_{1},…, a_{k−1}, a, b̃_{(l)}c_{1},…, b̃_{(1)}c_{l}) φ^{l}b_{(0)}+ ∇ (a_{1},…, a_{k−1}, ã_{(l+1)}b, sa_{(l)}c_{1},…, ã_{(1)}c_{l}) φ^{l+1}a_{(0)} = (□ (a_{1},…, a_{k−1}, a) ∧ b) (c_{1},…, c_{l}) + (□ (a_{1},…, a_{k−1}, ã_{(1)}b) ∧ φa_{(0)}) (c_{1},…, c_{l}) .

□

2.5. Let us consider d.o.’s on the quantum plane, with γ = q^{−1}, λ = q^{2} (for details see [17]). Assume that the coordinates x^{1},…, x^{n} on the quantum plane are chosen in such a way that the monomial (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. Then one can prove the following: 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.

The operators ∂_{i} satisfy the Wess-Zumino Leibnitz rule:

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

12 Alexander Verbovetsky

For the matrix R̂ of the form (5) it can be shown that if q^{2} is not a root of unity or q^{2} = 1 and char 𝕜 = 0, then all d.o.’s are compositions of operators of order ⩽ 1. If q^{2} is a root of unity this is not the case. 2.6. Here we define the modules of jets J^{k} (P) as representing objects for the functors Diff_{k} (P, ·). Theorem. There exists an A-module J^{k} (P), called the module of jets, such that the functor Q ↦ Diff_{k} (P, Q) is isomorphic to the functor Q ↦ Hom_{A}(J^{k} (P) , Q). Proof. Consider the tensor product A ⊗_{𝕜} P and equip it with the A-module structure by a′ (a ⊗ p) = a′a ⊗ p. For any a′ ∈ A denote by δ_{k}^{a′}, k = 0, 1, 2,…, the endomorphism of the module A ⊗_{𝕜} P:

δ_{k}^{a′} (a ⊗ p) = a_{(0)} ⊗ (a_{(1)}a′) p − (φ^{k}a′) a ⊗ p.

Consider the subspaces μ_{k} of A ⊗_{𝕜} P generated by elements of the form

(δ_{k}^{a_{k}} ∘… ∘ δ_{0}^{a_{0}}) (a ⊗ p) .

From the equality a′′δ_{k}^{a′} (a ⊗ p) = δ_{k}^{ã′′_{(1)}a′} (a′′_{(0)}a ⊗ p) it follows that μ_{k} are submodules of A ⊗_{𝕜} P. Define the module of jets by J^{k} (P) = A ⊗_{𝕜} P/μ_{k} and consider the mappings j_{k}: P → J^{k} (P), j_{k} (p) = (1 ⊗ p) mod μ_{k}. Let us show that j_{k} is a d.o. of order ⩽ k. First note that the map δ_{k}^{a}: A ⊗_{𝕜} P → A ⊗_{𝕜} P induces an A-homomorphism δ_{k}^{a}: J^{k−1} (P) → J^{k} (P). Then we have [j_{k}, a]_{k} (p) = j_{k} (ap) − (φ^{k}a_{(0)}) (a_{(1)}j_{k}) (p) = (1 ⊗ ap − (φ^{k}a) ⊗ p) mod μ_{k} = δ_{k}^{a}j_{k−1} (p) and one obtains the statement by induction.

Clearly, the A-module J^{k} (P) is generated by the elements of the form j_{k} (p), p ∈ P. Now to any d.o. Δ ∈ Diff (P, Q) assign an A-homomorphism ψ^{Δ} : J^{k} (P) → Q such that Δ = ψ^{Δ} ∘ j_{k}. Obviously, if such a homomorphism exist, it is unique. To prove existence consider the isomorphism

η: Hom_{A} (A ⊗_{𝕜} P, Q) → Hom_{𝕜} (P, Q) ,

η (f) (p) = f (1 ⊗ p), f ∈ Hom_{A} (A ⊗_{𝕜} P, Q). We have η (f ∘ δ_{k}^{a}) = [η (f) , a]_{k}, so that η induce an isomorphism

\overline{η}: Hom_{A} (J^{k} (P) , Q) → Diff_{k} (P, Q) , with Δ = \overline{η} (ψ^{Δ}) = ψ^{Δ} ∘ j_{k}. □

Natural embeddings Diff_{k} (P, Q) → Diff_{l} (P, Q), k ⩽ l, give rise to projections ν_{l,k}: J^{l} (P) → J^{k} (P), which are uniquely determined by the following commutative diagram:

P ⟶^{j_{l}} J^{l} (P)
∥ ↓^{ν_{l,k}}
P ⟶_{j_{k}} J^{k} (P)

The homomorphism c^{k,l}: J^{k+l} (P) → J^{k} (J^{l} (P)) dual to c_{k,l} is defined by following commutative diagram:

P  ⟶^{j_{l}} J^{l} (P)
j_{k+l}↓  ↓^{j_{k}}
J^{k+l} (P) ⟶_{c^{k,l}} J^{k} (J^{l} (P))

2.7. Now we introduce the modules of differential forms.

Theorem. There exists an A-module Λ^{k}, called the module of k-forms, such that the functor P ↦ D_{k} (P) is isomorphic to the functor P ↦ Hom_{A} (Λ^{k}, P ).

Proof. Consider the homomorphism β_{1} : A → J^{1} (A) given by β_{1} (a) = aj_{1} (1) and define Λ^{1} = J^{1} (A) / im β_{1}. Clearly D_{1} (P) = Hom_{A} (Λ^{1}, P ) and a homomorphism h ∈ Hom_{A} (Λ^{1}, P ) corresponds to the derivation h ∘ d, where d = α_{1} ∘ j_{1} : A → Λ^{1}, α_{1} is the projection J^{1} (A) → Λ^{1}. Suppose inductively that we have defined the modules Λ^{2}, Λ^{3}, …, Λ^{k−1} together with A-linear surjections α_{l}: J^{1} (Λ^{l−1}) → Λ^{l}, l = 1, 2,…, k − 1, such that i_{l} = Hom (α_{l}, P) : D_{l} (P) → Ḋ_{l−1} (Diff_{1}^{+} (P)). Then we can define Λ^{k} as the cokernel of the composition of homomorphisms

J^{2} (Λ^{k−2}) ⟶^{c^{1,1}} J^{1} (J^{1} (Λ^{k−2})) ⟶^{J^{1}(α_{k−1})} J^{1} (Λ^{k−1}) .

It is obvious that D_{k} (P) = Hom_{A} (Λ^{k}, P ). □

2.8. The direct sum Λ^{∗} = ∑_{k=0}^{∞} Λ^{k} can be considered as a graded module, with Λ^{k} being of grading −k and A-module structure being as follows:

a ∧ ω = (φ^{k}a) ω,
ω ∧ a = ωa, ω ∈ Λ^{k}.

Clearly, D_{∗} (P) = Hom_{A} (Λ^{∗}, P) in the graded sense. Moreover, the natural inclusion D_{k+l} (P) → D_{k} (D_{l} (P)) from Proposition 2.4 by duality gives rise to a mapping Λ^{k} ⊗ Λ^{l} → Λ^{k+l}. This mapping, multiplied by (−1)^{kl}, is called the wedge product and is denoted by ∧: Λ^{k} ⊗ Λ^{l} → Λ^{k+l}, ω_{1} ⊗ ω_{2} ↦ ω_{1} ∧ ω_{2}. Obviously, the module Λ^{k} is generated by the elements of the form a_{0} ∧ da_{1} ∧… ∧ da_{k}, a_{i} ∈ A. From Proposition 2.3 it follows that for all a, b ∈ A

da ∧ db + ã_{(1)}db ∧ φda_{(0)} = 0.

By definition, the operator d: A → Λ^{1} is a derivation:

d (ab) = (da) b + (φa) db = da ∧ b + a ∧ db.

More general, the exterior derivation d: Λ^{k−1} → Λ^{k} is defined as the composition

Λ^{k−1} ⟶^{j_{1}} J^{1} (Λ^{k−1}) ⟶^{α_{k}} Λ^{k}.

Proposition. d^{2} = 0.

Proof. This follows from the following commutative diagram:

Λ^{k−1}  ⟶^{j_{1}} J^{1} (Λ^{k−1}) ⟶^{α_{k}} Λ^{k}
j_{2}↓ ^{j_{1}}↓ ^{j_{1}}↓
J^{2} (Λ^{k−1}) ⟶_{c^{1,1}} J^{1} (J^{1} (Λ^{k−1})) ⟶_{J^{1}(α_{k})} J^{1} (Λ^{k}) ⟶_{α_{k+1}} Λ^{k+1}
□

2.9. Proposition. For any ω_{1} ∈ Λ^{k} and ω_{2} ∈ Λ^{l} we have

d (ω_{1} ∧ ω_{2}) = dω_{1} ∧ ω_{2} + (−1)^{k} ω_{1} ∧ dω_{2}.

Proof. Clearly it is sufficient to consider the case l = 0, i.e., ω_{2} = a ∈ A. Note that the inclusion D_{k+1} (P) → D_{k} (D_{1} (P)) is the composition

D_{k+1} (P) ⟶^{i_{k+1}} Ḋ_{k} (Diff_{1}^{+} (P)) ⟶^{λ} D_{k} (D_{1} (P)) ,

with the map λ given by λ (∇) (a_{1},…, a_{k}) = ∇ (a_{1},…, a_{k}) − ∇ (a_{1},…, a_{k}) (1). Therefore the wedge product can be written as the composition Λ^{k} ⊗ Λ^{1} ⟶^{λ′}

J^{1} (Λ^{k}) ⟶^{α_{k+1}} Λ^{k+1}, with λ′ given by λ′ (ω ⊗ da) = (−1)^{k} (j_{1} (ωa) − j_{1} (ω) a).

Thus for any ω ∈ Λ^{k} we obtain (−1)^{k} ω ∧ da = (−1)^{k} α_{k+1} (λ′ (ω ⊗ da)) = α_{k+1} (j_{1} (ωa) − j_{1} (ω) a) = d (ωa) − dω ∧ a. □

2.10. For the case of quantum plane from the preceding example we have

x^{i} ∧ dx^{j} = qR̂_{αβ}^{ij}dx^{α} ∧ x^{β},
dx^{i} ∧ dx^{j} = −qR̂_{αβ}^{ij}dx^{α} ∧ dx^{β},

This together with Propositions 2.8 and 2.9 proves that our de Rham complex is the same as that of Wess-Zumino.

3. Adjoint Operators and Berezinian

3.1. It turns out that the operator adjoint to a d.o. is not a usual d.o., but a right differential operator. Let us introduce this concept. We shall write right operator on the right of the argument and mark with the arrow: p ↦ (p) \overleftarrow{Δ}. Define the space \overleftarrow{Hom}_{𝕜} (P, Q) of right 𝕜-homomorphisms as the 𝕜-module Hom_{𝕜} (P, Q) endowed with an H-module structure given by (p) (h\overleftarrow{Δ}) = h_{(2)} ((S^{−1} (h_{(1)}) p) \overleftarrow{Δ}), which is equivalent to h ((p) \overleftarrow{Δ}) = (h_{(1)}p) (h_{(2)}\overleftarrow{Δ}). Here S is the antipode in H. The two A-module structures on \overleftarrow{Hom}_{𝕜} (P, Q) are defined in a natural way:

(p) (a\overleftarrow{Δ}) = (pa) \overleftarrow{Δ} and (p) (\overleftarrow{Δ}a) = ((p) \overleftarrow{Δ}) a. The definition of right d.o.’s is as follows. Let \overleftarrow{Δ} ∈ \overleftarrow{Hom}_{𝕜} (P, Q) be a 𝕜-homomorphism. For every element a ∈ A define 𝕜-homomorphisms [a, \overleftarrow{Δ}]_{k} : P → Q, k = 0, 1, 2,… by putting

16 Alexander Verbovetsky

(p) [a, \overleftarrow{Δ}]_{k} = (pa) \overleftarrow{Δ} − (p) (ã_{(1)}\overleftarrow{Δ}) (φ^{−k}a_{(0)}). Then \overleftarrow{Δ} ∈ \overleftarrow{Hom}_{𝕜} (P, Q) is a right d.o. of order ⩽ k if for any a_{0},…, a_{k} ∈ A we have

[a_{0},…, [a_{k−1}, [a_{k}, \overleftarrow{Δ}]_{k}]_{k−1} ,…, ]_{0} = 0.

The bimodule of right d.o.’s of order ⩽ k from P to Q will be denoted by \overleftarrow{Diff}_{k}^{(+)} (P, Q). Below the usual d.o.’s will be sometimes called left d.o.’s. Starting with this definition one can easily get “right” counterparts for all objects introduced above such as multiderivations, jets, forms and so on. In particular, consider the right de Rham complex. Similarly to the left case we have

a ∧ ω = ã_{(1)}ω ∧ φ^{k}a_{(0)}, a ∈ A, ω ∈ Λ_{right}^{k}
a\overleftarrow{d} ∧ b\overleftarrow{d} + φ^{−1}b_{(0)}\overleftarrow{d} ∧ b_{(1)}a\overleftarrow{d} = 0, a, b ∈ A
(ω_{1} ∧ ω_{2}) \overleftarrow{d} = ω_{1} ∧ ω_{2}\overleftarrow{d} + (−1)^{l} ω_{1} ∧ ω_{2}\overleftarrow{d}, ω_{1} ∈ Λ_{right}^{k}, ω_{2} ∈ Λ_{right}^{l}

From this it easily follows an important fact that the right de Rham complex is isomorphic to the left one in such a way that dω = (−1)^{k} ω\overleftarrow{d}, ω ∈ Λ^{k}.

Below we shall formulate all result for the left case. 3.2. Given an A-module P, consider the complex of homomorphisms

0 ⟶ Diff^{+} (P, A) ⟶^{w} Diff^{+} (P, Λ^{1}) ⟶^{w} Diff^{+} (P, Λ^{2}) ⟶^{w} …,

where w (∇) = d ∘ ∇ ∈ Diff^{+} (P, Λ^{k}), ∇ ∈ Diff^{+} (P, Λ^{k−1}). Let us denote the cohomology module at the term Diff^{+} (P, Λ^{n}) by P̂_{n}, n ⩾ 0. Every d.o. Δ: P → Q generates the natural map of the complexes:

… ⟶ Diff^{+} (Q, Λ^{k−1}) ⟶^{w} Diff^{+} (Q, Λ^{k}) ⟶ …
\overleftarrow{Δ}′↓  \overleftarrow{Δ}′↓
… ⟶ Diff^{+} (P, Λ^{k−1}) ⟶^{w} Diff^{+} (P, Λ^{k}) ⟶ …,

where (∇) \overleftarrow{Δ}′ = ∇ ∘ Δ ∈ Diff^{+} (P, Λ^{k}), ∇ ∈ Diff^{+} (Q, Λ^{k}). Definition. The operator \overleftarrow{Δ}_{n}^{∗}: Q̂_{n} → P̂_{n} induced by \overleftarrow{Δ}′ is called the (n-th) adjoint operator.

Below we assume that an integer n is fixed and omit the corresponding index to simplify notation.

Proposition. (1) If Δ ∈ Diff_{k} (P, Q) then \overleftarrow{Δ}^{∗} ∈ \overleftarrow{Diff}_{k} (Q̂, P̂).

(2) For all Δ_{1} ∈ Diff (P, Q) and Δ_{2} ∈ Diff (Q, R) we have

\overleftarrow{(Δ_{2} ∘ Δ_{1})}^{∗} = \overleftarrow{Δ}_{2}^{∗} ∘ \overleftarrow{Δ}_{1}^{∗},

where the composition of right operators is understood in the following way:

\binom{}{r} (\overleftarrow{Δ}_{2} ∘ \overleftarrow{Δ}_{1}) = (\binom{}{r} \overleftarrow{Δ}_{2}) \overleftarrow{Δ}_{1}, r̂ ∈ R̂.

Proof. (1) Denote by [∇] the cohomology class of an operator ∇ ∈ Diff^{+} (Q, Λ^{n}), w (∇) = 0. Then ([∇]) \overleftarrow{Δ}^{∗} = [∇ ∘ Δ] and we have

([∇]) [a, \overleftarrow{Δ}^{∗}]_{k} = ([∇] a) \overleftarrow{Δ}^{∗} − [∇] (ã_{(1)}\overleftarrow{Δ}^{∗}) (φ^{−k}a_{(0)}) =

[∇ ∘ a ∘ Δ] − [∇ ∘ (ã_{(1)}Δ) ∘ (φ^{−k}a_{(0)})] = − [∇ ∘ [ã_{(1)}Δ, φ^{−k}a_{(0)}]_{k}] = − ([∇]) [ã_{(1)}Δ, φ^{−k}a_{(0)}]_{k}^{∗} .

This implies the statement.

(2) ([∇]) \overleftarrow{(Δ_{2} ∘ Δ_{1})}^{∗} = [∇ ∘ Δ_{2} ∘ Δ_{1}] = ([∇ ∘ Δ_{2}]) \overleftarrow{Δ}_{1}^{∗} = (([∇]) \overleftarrow{Δ}_{2}^{∗}) \overleftarrow{Δ}_{1}^{∗}. □

3.3. Let us consider some examples of adjoint operators.

Example 1. Let a: P → P be the operator of left multiplication by a ∈ A. Then ([∇]) \overleftarrow{a}^{∗} = [∇ ∘ a] = [∇] a, i.e., \overleftarrow{a}^{∗} is the right multiplication by a.

Example 2. Let p: A → P be the operator p (a) = pa, a ∈ A, p ∈ P. Then ([∇]) \overleftarrow{p}^{∗} = [∇ ∘ p], \overleftarrow{p}^{∗} ∈ Hom (P̂, Â). Thus we have a natural pairing ⟨· , ·⟩ : P̂⊗

P → Â, ⟨p̂, p⟩ = \binom{}{p} \overleftarrow{p}^{∗}, p̂ ∈ P̂.

Example 3. Berezinian and Integral Forms. Let … ⟶ P_{k−1} ⟶^{Δ_{k}} P_{k} ⟶ … be a complex of d.o.’s. Since \overleftarrow{Δ}_{k+1}^{∗} ∘ \overleftarrow{Δ}_{k}^{∗} = \overleftarrow{(Δ_{k+1} ∘ Δ_{k})}^{∗} = 0 we get a complex … ⟵ P̂_{k−1} ⟵^{\overleftarrow{Δ}_{k}^{∗}} P̂_{k} ⟵ …, which is called dual to the given one. The complex dual to the de Rham complex is said to be the complex of integral forms and is denoted by

0 ⟵ Σ_{0} ⟵^{\overleftarrow{δ}} Σ_{1} ⟵ …,

18 Alexander Verbovetsky

where Σ_{i} = Λ̂^{i}, \overleftarrow{δ} = \overleftarrow{d}^{∗}. The module Σ_{0} = Â is called the Berezinian (or the module of volume forms ) and will be denoted by B.

The d.o. D: Diff^{+} (Λ^{n}) → Λ^{n} induces the cohomology map: ∫ : B → H^{n} (Λ^{∗}), so that to any element ω ∈ B (volume form) corresponds the n-th de Rham cohomology class ∫ ω. This is an algebraic version of the integration operation. Clearly, ∫ [∇] = ∇ (1) mod dΛ^{n−1}, ∇ ∈ Diff (A, Λ^{n}).

Proposition. ∫ ω\overleftarrow{δ} = 0.

Proof. Suppose ω = [∇], then ω\overleftarrow{δ} = [∇ ∘ d]. Hence ∫ ω\overleftarrow{δ} = [∇ ∘ d (1)] = 0. □

3.4. Proposition (“Integration by parts”). For any d.o. Δ: P → Q and p ∈ P, q ∈ Q, we have

∫ ⟨q̂, Δ (p)⟩ = ∫ ⟨\binom{}{q} \overleftarrow{Δ}^{∗}, p⟩ .

Proof. Suppose q̂ = [∇], ∇: Q → Λ^{n}. Then ∫ ⟨q̂, Δ (p)⟩ = ∫ [∇ ∘ Δ (p)] =

∫ [∇ ∘ Δ ∘ p] = ∫ ⟨[∇ ∘ Δ] , p⟩ = ∫ ⟨\binom{}{q} \overleftarrow{Δ}^{∗}, p⟩. □

4. Spencer Complex and Green’s Formula

From now onwards we assume that

1. all A-modules under consideration are projective and the Spencer Diffcomplex (for such a module) is exact;

2. the right Berezinian is isomorphic to the left one and the correspondent

integral mappings coincide. In this case it is expedient to consider another variant of construction of adjoint

operator.

Note first that for any module P we have P̂ = Hom_{A} (P, B). In particular,

Σ_{i} = Λ̂^{i} = D_{i} (B), \begin{matrix} \\ J & k & (A) \end{matrix} = Diff_{k} (A, B).

Definition. For an operator Δ ∈ Diff_{k} (A, B) we define the operator \overleftarrow{Δ^{∘}}: A → B,

(a) \overleftarrow{Δ^{∘}} = (aΔ) \overleftarrow{j_{k}^{∗}}, a ∈ A, which for economy of language, will be called adjoint.

DIFFERENTIAL OPERATORS OVER QUANTUM SPACES 19

Proposition. (1) If Δ ∈ Diff_{k} (A, B) then \overleftarrow{Δ^{∘}} ∈ \overleftarrow{Diff}_{k} (A, B).

(2) For ω ∈ B = Diff_{0} (A, B), \overleftarrow{ω^{∘}} is the operator of right multiplication by ω.

(3) \overleftarrow{(aΔ)^{∘}} = a ∘ \overleftarrow{Δ^{∘}}.

(4) \overleftarrow{(Δ ∘ a)^{∘}} = \overleftarrow{Δ^{∘}}a.

(5) (Δ^{∘})^{∘} = Δ.

Proof. (1) The operator \overleftarrow{Δ^{∘}} is the composition of the right homomorphism from A to Diff_{k} (A, B), a ↦ aΔ, with the right d.o. \overleftarrow{j}_{k}^{∗}: Diff (A, B) → B, therefore it is a right d.o. of order ⩽ k.

(2) Obvious.

(3) (b) \overleftarrow{(aΔ)^{∘}} = (baΔ) \overleftarrow{j}_{k}^{∗} = (ba) \overleftarrow{Δ^{∘}} = (b) (a ∘ \overleftarrow{Δ^{∘}}).

(4) If Δ (b) = [∇_{b}], then (Δ ∘ a) \overleftarrow{j}^{∗} = [□], where □ (b) = ∇_{ab} (1). Hence (Δ ∘ a) \overleftarrow{j}^{∗} = (Δ) \overleftarrow{j}^{∗}a. This implies the statement.

(5) Consider Δ ∈ Diff (A, B). It is easy to see that if Δ (a) = [∇_{a}], ∇_{a} ∈ Diff^{+} (A, Λ^{n}), then (Δ) \overleftarrow{j}^{∗} = [□], where □ (a) = ∇_{a} (1). Note that the map a ↦ ∇_{a} needs not be a d.o. One can represent elements of B by right operators. Then □ (a) = (1) \overleftarrow{∇}_{a} if Δ (a) = [\overleftarrow{∇}_{a}]. This implies that (a) \overleftarrow{Δ^{∘}} = [□_{a}], where □_{a} (b) = (a) \overleftarrow{∇}_{b}. A similar consideration shows that (Δ^{∘})^{∘} (a) = [\overleftarrow{γ}_{a}], where (b) \overleftarrow{γ}_{a} = □_{b} (a). Thus \overleftarrow{γ}_{a} = \overleftarrow{∇}_{a}. □ 4.2. Now let us define the adjoint operator Δ^{∘} in the general case when Δ ∈ Diff (P, Q̌), where Q̌ = \overleftarrow{Hom}_{A} (Q, B), P, Q are A-modules. To do this consider the family of operators _{q}Δ_{p} ∈ Diff (A, B), p ∈ P, q ∈ Q, (_{q}Δ_{p}) (a) = ⟨q, Δ (pa)⟩. Obviously the operator Δ is determined by the family _{q}Δ_{p}. It is also obvious, that for any family of operators _{q}Δ_{p} ∈ Diff (A, B), such that the map q ↦ _{q}Δ_{p} is a right A-homomorphism from Q to Diff (A, B) and the map p ↦ _{q}Δ_{p} is a left Ahomomorphism from P to Diff^{+} (A, B), one can find the corresponding operator Δ ∈ Diff (P, Q̌). For a right operator \overleftarrow{Δ} ∈ \overleftarrow{Diff} (Q, P̂) there is also similar description by means of the family _{q}\overleftarrow{Δ}_{p} ∈ \overleftarrow{Diff} (A, B), (a)_{q}\overleftarrow{Δ}_{p} = ⟨(aq) \overleftarrow{Δ}, p⟩. In this case the map q ↦ _{q}Δ_{p} is a right A-homomorphism from Q to \overleftarrow{Diff}^{+} (A, B) and the map p ↦ _{q}Δ_{p} is a left A-homomorphism from P to \overleftarrow{Diff} (A, B). In view

20 Alexander Verbovetsky

of this the family of operators _{q}(\overleftarrow{Δ^{∘}})_{p} = \overleftarrow{(_{q}Δ_{p})^{∘}} uniquely defines the operator \overleftarrow{Δ^{∘}} ∈ \overleftarrow{Diff} (Q, P̂). Proposition. (1) For every Δ ∈ Diff (P, Q̌) we have (Δ^{∘})^{∘} = Δ.

(2) \overleftarrow{(aΔ)^{∘}} = a ∘ \overleftarrow{Δ^{∘}}, \overleftarrow{(Δ ∘ a)^{∘}} = \overleftarrow{Δ^{∘}}a.

(3) For Δ ∈ Diff (P, Q) the adjoint operator \overleftarrow{Δ^{∘}}: Q → P̂ coincides with the composition Q ⟶ ˆ Q̌ ⟶^{\overleftarrow{Δ}^{∗}} P̂, where Q → ˆ Q̌ is the natural right Ahomomorphism.

Proof. Statements (1) and (2) are straightforward.

(3) Take Δ: P → Q̌, p ∈ P, q ∈ Q. We have Δ = ψ_{Δ}j_{k}, so ⟨(q) \overleftarrow{Δ}^{∗}, p⟩ = ⟨(q) \overleftarrow{ψ_{Δ}}^{∗}\overleftarrow{j}_{k}^{∗}, p⟩ = (((q) \overleftarrow{ψ_{Δ}}^{∗}) (p)) \overleftarrow{j}_{k}^{∗} = (_{q}Δ_{p})^{∘} (1). □ 4.3. The following Theorem gives a fundamental description of the Spencer Diffcomplex. Theorem. There is an isomorphism of complexes:

0 ⟵ P̌ ⟵ Diff (A, P̌) ⟵^{S} Ḋ_{1} (Diff^{+} (A, P̌)) ⟵^{S} Ḋ_{2} (Diff^{+} (A, P̌)) ⟵ …
∥  ^{ψ}↓  ^{ψ}↓  ^{ψ}↓
0 ⟵ P̌ ⟵^{μ} \overleftarrow{Diff}^{+} (P, B) ⟵^{ω}  \overleftarrow{Diff}^{+} (P, Σ_{1}) ⟵^{ω}  \overleftarrow{Diff}^{+} (P, Σ_{2}) ⟵ …,

where (\overleftarrow{∇}) \overleftarrow{ω} = \overleftarrow{∇} ∘ \overleftarrow{δ}, ∇ ∈ Diff^{+} (P, Σ_{k}), μ (\overleftarrow{∇}) = ∇^{∘} (1), the maps ψ are the composition of isomorphisms:

Ḋ_{k} (Diff^{+} \binom{}{P̌}) ⟶^{D(∘)} D_{k} (\overleftarrow{Diff} (P, B)) ⟶ \overleftarrow{Diff}^{+} (P, D_{k} (B)) ⟶ \overleftarrow{Diff}^{+} (P, Σ_{k})

Proof. Clearly the Spencer Diff-complex is dual to the complex

0 ⟶ A ⟶^{s} J (A) ⟶^{s} J (Λ^{1}) ⟶^{s} J (Λ^{2}) ⟶^{s} …,

where s (aj (ω)) = aj (dω). On the other hand we get Hom_{A} (J (Λ^{k}) , P̌) = Diff (Λ^{k}, P̌) = \overleftarrow{Diff}^{+} (P, Σ_{k}) and the operator \overleftarrow{s}^{∗}: \overleftarrow{Diff}^{+} (P, Σ_{k}) → \overleftarrow{Diff}^{+} (P, Σ_{k−1}) has the form (\overleftarrow{∇}) \overleftarrow{s}^{∗} = \overleftarrow{∇} ∘ \overleftarrow{δ}, \overleftarrow{∇} ∈ Diff (P, Σ_{k}); (∇) \overleftarrow{j}^{∗} = ∇^{∘} (1). This completes the proof. □

This Theorem has the following:

DIFFERENTIAL OPERATORS OVER QUANTUM SPACES 21

Corollary (Green’s Formula). For any Δ ∈ Diff (P, Q̌), p ∈ P, q ∈ Q, we have

⟨q, Δ (p)⟩ − ⟨(q) \overleftarrow{Δ^{∘}}, p⟩ = G\overleftarrow{δ},

where G ∈ Σ_{1} is an integral 1-form. Proof. Consider ∇ ∈ Diff^{+} (B); then (∇ − ∇ (1)) ∈ ker D, hence there exists □ ∈ D_{1} (Diff^{+} (B)) such that S (□) = ∇ − ∇ (1). It follows from the Theorem that (1) \overleftarrow{∇^{∘}} − \overleftarrow{∇ (1)} = ψ ∘ S (□) = ω (ψ (□)). Therefore (1) \overleftarrow{∇^{∘}} − \overleftarrow{∇ (1)} = G\overleftarrow{δ}, where G = ψ (□) (1). Letting ∇ = _{q}Δ_{p}, we get the Green formula. □ Remark. Consider an A-homomorphism κ: ker D → D_{1} (Diff^{+} (B)), such that S ∘ κ = id, and we can take □ = κ (Δ − Δ (1)) and, therefore, G = G_{κ} = ψ (κ (_{q}Δ_{p} − ⟨q, Δ (p)⟩)) (1). For any two homomorphisms κ and κ′ one can find a homomorphism f: ker D → D_{2} (Diff^{+} (B)) such that κ − κ′ = S ∘ f. Hence G_{κ} − G_{κ′} = F \overleftarrow{δ}, where F = ψ (f (_{q}Δ_{p} − ⟨q, Δ (p)⟩)) (1). 4.4. In this subsection we describe a spectral sequence, which establishes a relationship between the de Rham cohomology and the homology of the complex of integral forms. Proposition (Poincaré duality). There is a spectral sequence, (E_{∗,∗}^{r}, d_{∗}^{r}), with

E_{p,q}^{2} = H_{p} ((Σ_{∗})_{−q}) ,

the homology of complexes of integral forms, and converging to the de Rham cohomology H (Λ^{∗}). Proof. Let K_{p,q} = D_{p} (Diff^{+} (Λ^{−q})), d′ be the Spencer operator d′: K_{p,q} → K_{p−1,q}, and d′′ = (−1)^{p} D_{p} (Diff^{+} (d)), d′′: K_{p,q} → K_{p,q−1}. Then {K_{∗,∗}, d′, d′′} is a double complex. Obviously, H^{I} (K) = H (K_{∗,∗}, d′) = Λ^{−q} for p = 0 and 0 for p ≠ 0. Therefore in the second spectral sequence ^{II}E_{p,q}^{2} = ^{II}H_{p,q} (^{I}H (K)) = H_{p,q} (^{I}H_{∗,∗} (K) , d′′) = H^{−q} (Λ^{∗}) for p = 0 and ^{II}E_{p,q}^{2} = 0 for p ≠ 0. Thus the second spectral sequence converges to the de Rham cohomology. From the first spectral sequence we get ^{I}E_{p,q}^{2} = ^{I}H_{p,q} (^{II}H (K)) = ^{I}H_{p,q} (H (K_{∗,∗}, d′′)) = H_{p,q} (D_{∗} (B) , d′) = H_{p} ((Σ_{∗})_{−q}). □

Corollary. If Â_{i} = 0 for all i ≠ n, then H_{i} (Σ_{∗}) = H^{n−i} (Λ^{∗}).

Appendix: Hopf Algebras

This appendix provides definitions and basic facts related to Hopf albegras, which are extensively used in the text. For more details see, e.g., [15, 1, 6, 3, 4].

1. Definition. An asociative unital algebra H over a commutative unital ring 𝕜 is called bialgebra if algebra homomorphisms δ: H → H ⊗ H, called the comultiplication, and ε: H → 𝕜, called the counit, are given, such that the following diagrams are commutative:

\begin{matrix}H & ⟶^{δ} & H ⊗ H \\ ^{δ}↓ & & ↓^{1⊗δ} \\ H ⊗ H & ⟶^{δ⊗1} & H ⊗ H ⊗ H \end{matrix}  \begin{matrix}𝕜 ⊗ H & ⟵^{ε⊗1} & H ⊗ H & ⟶^{1⊗ε} & H ⊗ 𝕜 \\ ↓ & & ↑^{δ} & & ↓ \\ H & & H & & H \end{matrix}

A bialgebra H is called a Hopf algebra if it is endowed with an antihomomorphism of bialgebras S: H → H, called the antipode, such that

S ∗ 1_{H} = 1_{H} ∗ S = η ∘ ε: H → H,

where ∗ stands for the convolution product on the space Hom (H, H) defined as the composition f ∗g = m∘(f ⊗ g)∘δ, f, g ∈ Hom (H, H). Here m is the multiplication and η is the unit in the algebra H.

If an antipode for a bialgebra exists, it is unique. We usually consider Hopf algebras with (composition) invertible antipode.

2. Heyneman-Sweedler Notation. Let H be any bialgebra with comultiplication δ: H → H ⊗ H. For any h ∈ H, we write

δ (h) = ∑ h_{(1)} ⊗ h_{(2)},

and usually even omit the summation sign. Thus, element of H, subscripted with a positive number in parentheses, is not a particular element of H, but is understood to be obtained from a coproduct, with the appropriate summation implied.

DIFFERENTIAL OPERATORS OVER QUANTUM SPACES 23

More generally, let δ_{n−1} (h) be the (necessarly unique) element obtained by applying comultiplication (n − 1) times. Then we adopt the notation

δ_{n−1} (h) = h_{(1)} ⊗… ⊗ h_{(n)}.

To illustrate the use of the Heyneman-Sweedler notation, the counit axiom takes the form

ε (h_{(1)}) h_{(2)} = ε (h_{(2)}) h_{(1)} = h

and the convolution product is given by

(f ∗ g) (h) = f (h_{(1)}) g (h_{(2)}) .

3. Category of modules over a Hopf algebra. For any Hopf algebra H and two (left) H-modules M and N the 𝕜-modules M ⊗_{𝕜} N and Hom_{𝕜} (M, N) can also be endowed with structures of H-modules:

h (m ⊗ n) = h_{(1)}m ⊗ h_{(2)}n, (hf) (m) = h_{(1)} (f (S (h_{(2)}) m))) ,

where h ∈ H, m ∈ M, n ∈ N, f ∈ Hom_{𝕜} (M, N). Thus the category of modules over a Hopf algebra is a closed tensor category [3, 6, 8, 11, 13].

4. Comodules. Definition. A 𝕜-module M is called a (right) comodule over the bialgebra H if a linear map δ: M → M ⊗ H is given, such that the following diagrams commute:

\begin{matrix}M & ⟶^{δ} & M ⊗ H \\ ^{δ}↓ & & ↓^{δ⊗1} \\ M ⊗ H & ⟶^{1⊗δ} & M ⊗ H ⊗ H \end{matrix}  \begin{matrix}M ⊗ H & M ⊗ H \\ ^{δ}↑ & ↓^{1⊗ε} \\ M & M \end{matrix}

The Heyneman-Sweedler notation for comodule is given as follows: for any m ∈ M, we write

δ (m) = m_{(0)} ⊗ m_{(1)} ∈ M ⊗ H,

preserving the conventions that m_{(i)} ∈ H for i > 0.

Choose a basis m^{1},…, m^{n} in M and an n × n matrix Y = ∥y_{j}^{i}∥, y_{j}^{i} ∈ H, such that δ (m^{i}) = m^{α} ⊗ y_{α}^{i}. It follows from the definition of a comodule that δ (Y) = Y ⊗ Y, δ (y_{j}^{i}) = y_{j}^{α} ⊗ y_{α}^{i}, and ε (Y) = I, ε (y_{j}^{i}) = δ_{j}^{i}. Such matrices Y are called multiplicative.

24 Alexander Verbovetsky

5. Modules-algebras, comodule-algebras, and crossed bimodules. Definition. For any bialgebra H, an algebra A is a (left) H-module-algebra if

(1) A is a (left) H-module

(2) h (ab) = (h_{(1)}a) (h_{(2)}b)

(3) h1 = ε (h) 1 for all h ∈ H, a, b ∈ A.

Definition. An algebra A is a (right) H-comodule-algebra if (1) A is a (right) H-comodule

(2) δ (ab) = a_{(0)}b_{(0)} ⊗ b_{(1)}a_{(1)}

(3) δ (1) = 1 ⊗ 1

Remark. Here our terminology is slightly non-standard. Condition (2) is often written as δ (ab) = a_{(0)}b_{(0)} ⊗ a_{(1)}b_{(1)}, so that our definition corresponds to H^{op}- comodule-algebra. Definition. A crossed bimodule over H is a 𝕜-module M that is both a left Hmodule and a right H-comodule and satisfies the compatibility condition

h_{(1)}m_{(0)} ⊗ h_{(2)}m_{(1)} = (h_{(2)}m)_{(0)} ⊗ (h_{(2)}m)_{(1)}h_{(1)}.  (6)

(Crossed bimodules are also called Yetter-Drinfeld modules).

From (6) one easily gets

h_{(1)}S (m_{(1)}) ⊗ h_{(2)}m_{(0)} = S ((h_{(1)}m)_{(1)}) h_{(2)} ⊗ (h_{(1)}m)_{(0)}.

6. Coquasitriangular Structures. Definition. A bialgebra H is called coquasitriangular if there exists a non-degene-rate bilinear form ⟨· , ·⟩ : H ⊗ H → 𝕜 such that for all h, g, f ∈ H,

(1) ⟨h_{(1)}, g_{(1)}⟩ h_{(2)}g_{(2)} = g_{(1)}h_{(1)} ⟨h_{(2)}, g_{(2)}⟩

(2) ⟨h, gf⟩ = ⟨h_{(1)}, f⟩ ⟨h_{(2)}, g⟩

(3) ⟨hg, f⟩ = ⟨h, f_{(1)}⟩ ⟨g, f_{(2)}⟩.

Remark. Coquasitriangular Hopf algebras are also called (co)braided or dual quasitriangular.

DIFFERENTIAL OPERATORS OVER QUANTUM SPACES 25

7. Hopf Envelope. Here we outline a general construction for making a Hopf algebra from a bialgebra, suggested by Manin [13].

Let E be a bialgebra generated by entries of a multiplicative n × n matrix Z = ∥z_{j}^{i}∥. Consider the category of bialgebra morphisms E → G, where G is a Hopf algebra. This category has an initial object E → H, called the Hopf envelope of E. The explicit construction of H is as follows.

Suppose that E=𝕜 ⟨Z⟩ /I_{E}, where 𝕜 ⟨Z⟩ is the free associative algebra generated by z_{j}^{i} and I_{E} is the ideal of relations on z_{j}^{i}. Introduce an infinite sequence of n × n matrices Z_{0}, Z_{1}, Z_{2},… with independent entries. Set H = 𝕜 ⟨Z_{0}, Z_{1},…⟩ /I_{H}, where the ideal I_{H} is generated by following relations:

the relations for Z, written for Z_{2l},

the relations for Z with reversed order of multiplication, written for Z_{2l+1}, Z_{k}Z_{k+1} − I, Z_{k+1}Z_{k} − I for even k, Z_{k}^{t}Z_{k+1}^{t} − I, Z_{k+1}^{t}Z_{k}^{t} − I for odd k.

The Hopf algebra structure on H is the following: δ (Z_{k}) = Z_{k} ⊗ Z_{k} for even k, δ (Z_{k}) = (Z_{k}^{t} ⊗ Z_{k}^{t})^{t} for odd k, ε (Z_{k}) = I, S (Z_{k}) = Z_{k+1}.

Theorem. A coquasitriangular structure on E can be uniquely extended to a coquasitriangular structure on H.

If ⟨· , ·⟩ is a coquasitriangular structure on E with ⟨z_{j}^{i}, z_{l}^{k}⟩ = r_{jl}^{ki}, then the coquasitriangular structure on H is given by

⟨z_{j}^{i}, z_{l}^{k}⟩ = r_{jl}^{ki}  for s ≡ t mod 2,
⟨z_{j}^{i}, z_{l}^{k}⟩ = (r^{−1})_{lj}^{ik}  for s odd and t even,
⟨z_{j}^{i}, z_{l}^{k}⟩ = c_{lj}^{ik}  for s even and t odd,

where z_{j}^{i} ∈ Z_{s}, z_{l}^{k} ∈ Z_{t}, (r^{−1})_{αβ}^{ij} r_{kl}^{αβ} = δ_{k}^{i}δ_{l}^{j}, c_{jβ}^{iα}r_{lα}^{kβ} = δ_{l}^{i}δ_{j}^{k}.

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