Seminar Sophus Lie 1 (1991) 23–30

An example of a differential calculus on the quantum complex n-space*

Rainer Matthes

Abstract. Modified notions of derivation and differential form on the noncommutative space ℂ_{q}^{n} are introduced. It is shown that in this way a first order differential calculus in the sense of Woronowicz is obtained.

Introduction

Recently there is a growing number of articles concerning the theory of so-called “quantum groups”. For a survey and general references see the article [9] in this volume. As mentioned there one considers not only quantum groups but also other noncommutative spaces which are comodules of a quantum group. ℂ_{q}^{n} is the most simple example of such an object.

It is an important task to introduce analogues of notions of differential geometry on such noncommutative spaces ([2]). The most fundamental notions of this kind are those of tangent vectors and differential forms, whose definition will be our concern in this note. Since all the information about a noncommutative space is encoded in an algebra one has to define these notions in purely algebraic terms. By an algebra we always mean an associative algebra over the complex numbers ℂ with unity I. The symbols ℤ and ℕ denote the integers and the nonnegative integers respectively. Throughout the paper we use the so-called Einstein convention: We always form sums over pairs of equal upper and lower indices.

On a classsical space (differentiable manifold) one introduces tangent vectors as equivalence classes of curves starting from a point and defines differential forms as duals to vector fields. From this geometric point of view, everything is based on the notion of a vector. For commutative algebras A (with the example A = C^{∞}(M), M a manifold, in mind) it is quite clear how to translate this approach into an algebraic language: One starts with Der(A), the Lie algebra of derivations of the algebra, and defines ⋀^{1}(A), the space of 1-forms on A, as the space of A-linear maps Der(A) ⟶ A. This is possible because Der(A) is an A-(bi)module for commutative A. For noncommutative A, Der(A) isn’t a left or right A-module in a natural way, and one could define ⋀^{1}(A) only as the space of ℂ-linear maps Der(A) ⟶ A which is much to big in the classical case (contains even nonlocal mappings). One can try to replace the A-linearity condition by another one using e.g. certain maximal ideals of A as analogues of points (see [5]). However, this defect may be taken equally well as a hint to replace the notion of derivation by something else.

There is another approach to differential forms on a general associative algebra which, at first sight, A does not use derivations ([12]):

Definition 1. A pair (Γ, d) is called first order differential calculus on A if 1. Γ is an A-bimodule.

2. d : A ⟶ Γ is ℂ-linear and satisfies (i) d(ab) = d(a)b + ad(b), (ii) Any α ∈ Γ can be written α = ∑_{k=1}^{N} a_{k}db_{k} with a_{k}, b_{k} ∈ A, k =

1, …, N.

Obviously, the classical example A = C^{∞}(M)(M a manifold), Γ = sec(T^{∗}M), d the standard differential, is of this type. (Condition 2.(ii) is a mild restriction. It is satisfied at least for manifolds which can be covered by a finite number of charts.)

Now, let us assume that (Γ, d) is a given first order differential calculus on A, and let Γ be in particular a free A-bimodule with a finite basis (ω^{i})_{i=1,...,n}. This means that any α ∈ Γ can be written α = a_{i}ω^{i} = ω^{i}b_{i} (summation over pairs of equal upper and lower indices, see remark at the end of the second paragraph) with uniquely determined a_{i} and b_{i} ∈ A. Then we must have

(1)  aω^{i} = ω^{j}C_{j}^{i}(a)

with C_{j}^{i} ∈ L(A) =^{def} End_{ℂ}(A),

(2)  da = ω^{i}X_{i}^{r}(a)

with X_{i}^{r} ∈ L(A). It follows immediately that

(3)  C_{j}^{i}(ab) = C_{j}^{k}(a)C_{k}^{i}(b)

and

(4)  X_{i}^{r}(ab) = X_{i}^{r}(a)b + C_{i}^{j}(a)X_{j}^{r}(b)

(Notice that we could obtain similar relations starting from ω^{i}a = B_{j}^{i}(a)ω^{j} and da = X_{i}^{l}(a)ω^{i}.)

The examples of noncommutative algebras appearing in the theory of quantum groups are typically generated by elements x^{i} (subject to quadratic relations) which may be interpreted as analogues of the classical coordinate functions. In a differential calculus (Γ, d) one would like to have the dx^{i} as a basis of Γ. Then the above X_{i}^{r} should be interpreted as analogues of the partial derivatives ∂_{i}, and formula (4) says that the “partial derivatives” X_{i}^{r} do not satisfy the usual Leibniz rule, i.e. they aren’t derivations. This is a typical phenomenon for quantum groups (cf. [12]).

We will now present a special example of a differential calculus on ℂ_{q}^{n} starting from a generalized notion of derivation. These q-derivations have properties analogous to all properties of usual derivations. In particular, they form a left ℂ_{q}^{n} -module which makes it possible to define differential forms in the classical spirit. In this way we obtain a differential calculus in the sense of Definition

The algebra ℂ_{q}^{n}

We define ℂ_{q}^{n}, q ∈ ℂ \ {0} as the quotient algebra

ℂ⟨x^{1}, …, x^{n}⟩/I_{R}

where ℂ⟨x^{1}, …, x^{n}⟩ is the free associative algebra with unity generated by the elements x^{1}, …, x^{n}, and I_{R} is the twosided ideal generated by the relations

(5)  x^{i}x^{j} = qx^{j}x^{i} , i < j

(see [8]). The elements (x^{1^{i_{1}}} ⋯ x^{n^{in}})_{(i_{1},...,i_{n})∈ℕ^{n}} form a basis (as a vector space) of ℂ_{q}^{n}. We will consider ℂ_{q}^{n} as an ℕ^{n} -graded algebra with homogeneous components ℂ_{q}^{n(i_{1},...,i_{n})} = {λx^{1^{i_{1}}} ⋯ x^{n^{in}} | λ ∈ ℂ}. Further, ℂ_{q}^{nh} denotes the set of homogeneous elements of ℂ_{q}^{n}. For homogeneous (basis) elements x = x^{1^{i_{1}}} ⋯ x^{n^{in}} , y = x^{1^{k_{1}}} ⋯ x^{n^{kn}} with degrees g(x) = (i_{1}, …, i_{n}), g(y) = (k_{1}, …, k_{n}) one immediately obtains the following commutation rule

(6)  xy = q^{m(g(x),g(y))}yx

(7) m(g(x), g(y)) = (i_{1}+…+i_{n−1})k_{n}+(i_{1}+…+i_{n−2}−i_{n})k_{n−1}+…+(−i_{2}−…−i_{n})k_{1}

The map m : ℕ^{n} × ℕ^{n} ⟶ ℤ^{n} has two important properties:

1. m is additive in both arguments.

2. m is antisymmetric.

We will even consider ℂ_{q}^{n} as a ℤ^{n} -graded algebra setting ℂ_{q}^{n(i_{1},...,i_{n})} = 0 if i_{k} < 0 for some k ∈ {1, …, n}. Obviously, m can be extended to a mapping ℤ^{n} × ℤ^{n} ⟶ ℤ defined by (7).

q-derivations

We will follow the idea that any object related to the algebra ℂ_{q}^{n} should also be ℤ^{n} -graded and that a commutation of any two homogeneous objects a, b in algebraic manipulations should yield a factor according to the rule ab = q^{m(g(a),g(b))}ba as in the algebra ℂ_{q}^{n}. This is a generalization of the well known case of ℤ_{2} -graded algebras ([1, 3, 4]), which corresponds to q = −1 in our case. (The relations x^{i^{2}} = 0 are missing here.)

Definition 2. An element ∂ ∈ L(ℂ_{q}^{n}) is called homogeneous q-derivation of degree g(∂) ∈ ℤ^{n} if

1. g(∂(x)) = g(∂) + g(x),

2. ∂(xy) = ∂(x)y + q^{m(g(∂),g(x))}x∂(y)

for x ∈ ℂ_{q}^{nh}. A q-derivation is a finite sum of homogeneous q-derivations. Also, Der_{q}^{h}(ℂ_{q}^{n}) denotes the set of homogeneous q-derivations, Der_{q}(C_{q}^{n}) the vector space of all q-derivations.

Notice that the definition is made in such a way that every q-derivation is a finite sum of homogeneous components. The same remark applies to the definition of q-differential forms to be given later. Therefore, all propositions in this paper need a proof only for homogeneous elements.

Proposition 1. Let X_{1} , X_{2} ∈ Der_{q}^{h}(ℂ_{q}^{n}). Then we have

[X_{1}, X_{2}]_{q} =^{def} X_{1}X_{2} − q^{m(g(X_{1}),g(X_{2}))}X_{2}X_{1} ∈ Der_{q}^{h}(ℂ_{q}^{n})

with g([X_{1}, X_{2}]_{q}) = g(X_{1}) + g(X_{2}). Proof. For x ∈ ℂ_{q}^{nh}, y ∈ ℂ_{q}^{n} we have

(X_{1}X_{2} − q^{m(g(X_{1}),g(X_{2}))}X_{2}X_{1})(xy) = X_{1}(X_{2}(x)y + q^{m(g(X_{2}),g(x))}xX_{2}(y))

− q^{m(g(X_{1}),g(X_{2}))}X_{2}(X_{1}(x)y + q^{m(g(X_{1}),g(x))}xX_{1}(y))

= X_{1}X_{2}(x)y + q^{m(g(X_{1}),g(X_{2})+g(x))}X_{2}(x)X_{1}(y) + q^{m(g(X_{2}),g(x))}(X_{1}(x)X_{2}(y)

+ q^{m(g(X_{1}),g(x))}xX_{1}X_{2}(y)) − q^{m(g(X_{1}),g(X_{2}))}(X_{2}X_{1}(x)y

+ q^{m(g(X_{2}),g(X_{1})+g(x))}X_{1}(x)X_{2}(y) + q^{m(g(X_{1}),g(x))}(X_{2}(x)X_{1}(y)

+ q^{m(g(X_{2}),g(x))}xX_{2}X_{1}(y))).

The coefficient of the term X_{2}(x)X_{1}(y) is

q^{m(g(X_{1}),g(X_{2})+g(x))} − q^{m(g(X_{1}),g(X_{2}))+m(g(X_{1}),g(x))} = 0,

because m is additive. In the same way the coefficient of X_{1}(x)X_{2}(y) is zero. The remaining terms are

\begin{matrix}(X_{1}X_{2} − q^{m(g(X_{1}),g(X_{2}))}X_{2}X_{1})(x)y \\ + q^{m(g(X_{1})+g(X_{2}),g(x))}x(X_{1}X_{2}(y) − q^{m(g(X_{1}),g(X_{2}))}X_{2}X_{1}(y)), \end{matrix}

which proves the proposition.

The binary operation [·, ·]_{q} is called q-commutator. In this terminology, ℂ_{q}^{n} itself is q-commutative. The proofs of the following propositions are as simple and direct as the preceding one and make use of the two properties of m only. We therefore leave them to the reader.

Proposition 2. For X, X_{1}, X_{2}, X_{3} ∈ Der_{q}^{h}(ℂ_{q}^{n}) we have 1. [X, X]_{q} = 0.

2. [X_{2}, X_{1}]_{q} = −q^{m(g(X_{2}),g(X_{1}))}[X_{1}, X_{2}]_{q}.

3. q^{m(g(X_{3}),g(X_{1}))}[[X_{1}, X_{2}]_{q}, X_{3}]_{q} + q^{m(g(X_{1}),g(X_{2}))}[[X_{2}, X_{3}]_{q}, X_{1}]_{q} + q^{m(g(X_{2}),g(X_{3}))}[[X_{3}, X_{1}]_{q}, X_{2}]_{q} = 0.

These are modifications of the usual properties of a Lie algebra. Thus, Der_{q}(ℂ_{q}^{n}) could be called a q-Lie algebra.

Proposition 3. For a ∈ ℂ_{q}^{nh}, X, X_{1}, X_{2} ∈ Der_{q}^{h}(ℂ_{q}^{n}) we have

1. aX ∈ Der_{q}^{h}(ℂ_{q}^{n}) with g(aX) = g(a) + g(X).

2. [aX_{1}, X_{2}]_{q} = a[X_{1}, X_{2}]_{q} − q^{m(g(X_{1})+g(a),g(X_{2}))}X_{2}(a)X_{1} .

Condition 1. says that Der_{q}(ℂ_{q}^{n}) is a left ℂ_{q}^{n} -module. Condition 2. is a modification of well known properties of vector fields.

We will now define analogues of partial derivatives and prove that they form a basis of the left ℂ_{q}^{n} -module Der_{q}(ℂ_{q}^{n}). First let us notice that derivations (in the usual sense) ∂_{i} of ℂ_{q}^{n} with the property ∂_{i}(x^{j}) = δ_{i}^{j}I do not exist: For example for i < j would follow ∂_{i}(x^{i}x^{j}) = x^{j} = ∂_{i}(qx^{j}x^{i}) = qx^{j} (no summation over i!). We will define ∂_{i} first as operators on ℂ⟨x^{1}, …, x^{n}⟩ by

\begin{aligned}∂_{i}(x^{j}) & = δ_{i}^{j}I, ∂_{i}(I) = 0, \\ ∂_{i}(x^{j}x) & = δ_{i}^{j}x + q^{m(g(∂_{i}),g(x^{j}))}x^{j}∂_{i}(x) \end{aligned}

with g(∂_{i}) = −g(x^{i}) = (0, …, 0, −1, 0, …, 0) (-1 at i-th position). Here, a homogeneous element of ℂ⟨x^{1}, …, x^{n}⟩ is any product of the x^{1}, …, x^{n} times a scalar, and the j -th component of the degree of such an element is the number of factors x^{j} appearing in this element.

Proposition 4. The following statements hold:

1. ∂_{i}(xy) = ∂_{i}(x)y + q^{m(g(∂_{i}),g(x))}x∂_{i}(y) for homogeneous x, i .e., ∂_{i} is a q-derivation on ℂ⟨x^{1}, …, x^{n}⟩).

2. ∂_{i}(I_{R}) ⊂ I_{R}, where I_{R} is the ideal defined by the relations (5).

This means that ∂_{i} project to q-derivations of ℂ_{q}^{n}.

Theorem 1. Let X ∈ Der_{q}(ℂ_{q}^{n}) (not necessarily homogeneous). Then we have

X = X(x^{1})∂_{1} + … + X(x^{n})∂_{n}.

The proof can be performed by a direct computation. It is easy to show that the ∂_{i} are linearly independent as elements of the left ℂ_{q}^{n} -module Der_{q}(ℂ_{q}^{n}):

a^{i}∂_{i} = 0 ⇒ a^{i}∂_{i}(x^{j}) = a^{j}I = 0 ⇒ a^{j} = 0.

Thus, Der_{q}(ℂ_{q}^{n}) is a free left ℂ_{q}^{n} -module with basis (∂_{i})_{i=1,...,n}.

q-differential forms

Definition 3. We denote by ⋀_{0q}^{1h}(ℂ_{q}^{n}) the set of ℂ- linear maps

α: Der_{q}(ℂ_{q}^{n}) → ℂ_{q}^{n}

with

1. X ∈ Der_{q}^{h}(ℂ_{q}^{n}) ⇒ α(X) ∈ ℂ_{q}^{nh}, g(α(X)) = g(α) + g(X). (This defines

g(α) ∈ ℤ^{n}.)
2. α(aX) = q^{m(g(α),g(a))}aα(X) for a ∈ ℂ_{q}^{nh}.

Further, ⋀_{0q}^{1}(ℂ_{q}^{n}) is the vector space of finite sums of elements of ⋀_{0q}^{1h}(ℂ_{q}^{n}). Elements of ⋀_{0q}^{1}(ℂ_{q}^{n}) and ⋀_{0q}^{1h}(ℂ_{q}^{n}) are called q -differential 1-forms and homogeneous q -differential 1-forms, respectively.

Condition 2. replaces A-linearity which for commutative algebras A gives a locality condition. Obviously, ⋀_{0q}^{1}(ℂ_{q}^{n}) becomes a ℂ_{q}^{n} -bimodule with

aα = q^{m(g(a),g(α))}αa, αa = q^{m(g(α),g(a))}aα

for a ∈ ℂ_{q}^{nh}, α ∈ ⋀_{0q}^{1h}(ℂ_{q}^{n}).

Definition 4. We define d: ℂ_{q}^{n} → ⋀_{0q}^{1}(ℂ_{q}^{n}) by

da(X) = q^{m(g(a),g(X))}X(a)

for a ∈ ℂ_{q}^{n}, X ∈ Der_{q}^{h}(ℂ_{q}^{n}).

Obviously, we have g(da) = g(a) and

da(bX) = q^{m(g(a),g(b)+g(X))}bX(a) = q^{m(g(a),g(b))}bda(X),

i. e., d indeed has its values in ⋀_{0q}^{1}(ℂ_{q}^{n}).

Theorem 2. (⋀_{0q}^{1}(ℂ_{q}^{n}), d) is a first order differential calculus on ℂ_{q}^{n} in the sense of Definition 1. Proof. Firstly, we know that ⋀_{0q}^{1}(ℂ_{q}^{n}) is a ℂ_{q}^{n} -bimodule. Secondly, we compute

d(ab)(X) = q^{m(g(a)+g(b),g(X))}X(ab)
= q^{m(g(a)+g(b),g(X))}(X(a)b + q^{m(g(X),g(a))}aX(b))
= q^{m(g(a)+g(b),g(X))}X(a)b + q^{m(g(b),g(X))}aX(b).
(d(a)b)(X) = q^{m(g(a),g(b))}bda(X) = q^{m(g(a),g(b))+m(g(a),g(X))}bX(a)
= q^{m(g(a),g(b)+g(X))+m(g(b),g(a)+g(X))}X(a)b
= q^{m(g(a)+g(b),g(X))}X(a)b.
adb(X) = q^{m(g(b),g(X))}aX(b).

Also we have dx^{i}(∂_{j}) = q^{m(g(x^{i}),g(∂_{j}))}∂_{j}(x^{i}) = q^{m(g(x^{j}),g(x^{i}))}δ_{i}^{j}I = δ_{i}^{j}I. By Theorem 1, X = X^{i}∂_{i} for any X ∈ Der_{q}(ℂ_{q}^{n}). We have

α(X^{i}∂_{i}) = q^{m(g(α),g(X^{i}))}X^{i}α(∂_{i}).

With α_{i} = α(∂_{i}) and g(α) = g(α_{i}) − g(∂_{i}) we obtain

α(X^{i}∂_{i}) = q^{m(g(α),g(X^{i}))+m(g(X^{i}),g(x)+g(∂_{i}))}α_{i}X^{i} = q^{m(g(X^{i}),g(∂_{i}))}α_{i}X^{i}.

On the other hand,

α_{j}dx^{j}(X^{i}∂_{i}) = α_{j}q^{m(g(x^{j}),g(X^{i}))}X^{i}dx^{j}(∂_{i})
= q^{m(g(x^{i}),g(X^{i}))}α_{i}X^{i} = q^{m(g(X^{i}),g(∂_{i}))}α_{i}X^{i}.

Therefore, α = α(∂_{i})dx^{i} for homogeneous α, thus also for nonhomogeneous α.

Thus, the dx^{i} form a basis of the ℂ_{q}^{n} -bimodule ⋀_{0q}^{1}(ℂ_{q}^{n}) as linear independence follows immediately from dx^{i}(∂_{j}) = δ_{j}^{i}I. The calculus presented here corresponds to C_{j}^{i}(a) = δ_{j}^{i}q^{m(g(a),g(x^{i}))}a for a ∈ ℂ_{q}^{nh} in formula (1) of the introduction.

Remarks

1. Following the same ideas as above one can introduce q-differential forms of higher than first degree on ℂ_{q}^{n}, and define analogues of the usual operations (exterior derivative, Lie derivative, inner derivative). In particular, the q-differential forms on ℂ_{q}^{n} form a q-Graßmann algebra. This will be part of [6].

2. The calculus presented here can be extended to noncommutative tori. The algebra 𝕋_{q}^{n} corresponding to such a torus is obtained from ℂ_{q}^{n} by first adding the inverses x^{i^{−1}} and imposing further relations according to the rule (6). The resulting algebra consists of Laurent polynomials in x^{1}, …, x^{n}. For | q |= 1 this algebra can be completed in a certain topology. This completion, being the algebra 𝕋_{q}^{n}, is the algebra of Laurent series in x^{1}, …, x^{n} with fastly decreasing coefficients ([2, 8]). It is rather evident that the above notions are meaningful at least for the algebraic extension (see [6]).

3. From the viewpoint of quantum groups, ℂ_{q}^{n} appears as a left comodule, and one would be interested in differential calculi adapted to this structure, i.e. left covariant differential calculi (see [12] for the definition of left and right covariance for calculi on a quantum group, the definition for comodules is analogous). Unfortunately, though our calculus looks very natural it is not left or right covariant with respect to the coactions of M_{q}(n) or SU_{q}(n) on ℂ_{q}^{n}. In [7] all SU_{q}(n)-bicovariant calculi on ℂ_{q}^{n} are classified (see also [11]). For n ≥ 3 there are exactly two calculi, for n = 2 two one-parameter families of calculi. The two calculi given in [11] are the only M_{q}(2)-bicovariant calculi on ℂ_{q}^{2} ([10]).

Acknowledgements

I would like to thank K. Schmüdgen and G. Rudolph for their interest and several discussions and K. Voigt for his help in preparing the text on the computer.

References

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[2] Connes, A., Noncommutative differential geometry, Publ. Math. IHES62 (1986), 41–144.

[3] Kostant, B., “Graded manifolds, graded Lie theory, and prequantisation”, Lecture Notes in Math. 570, 1977.

[4] Lejtes, D. A., Vvedenie v teoriju supermnogoobrazij, Usp. Mat. Nauk 35,1 (1980),3–57.

[5] Matthes, R., “A general approach to connections: algebra and geometry”, Preprint KMU-NTZ 89-03, to appear in Rep. Math. Phys.

[6]  —, work in preparation.

[7] Pusz, W. and S. L. Woronowicz, “Twisted second quantization”, Preprint Warszawa, 1988.

[8] Reshetikhin, N. Yu., L. A. Takhtajan, and L. D. Faddeev, Kvantovanie grupp Li i algebr Li, Algebra i analiz, 1(1989),178–206.

[9] Schmüdgen, K., Eine Einführung in die Theorie der Quantenmatrixgruppen, in this volume.

[10]  —, Vortrag Leipzig 1991.

[11] Wess, J. and B. Zumino, “Covariant differential calculus on the quantum hyperplane”, Preprint 1990.

[12] Woronowicz, S. L., Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989),125–170.

Institut für Mechanik der AdW Chemnitz Current address: Sektion Physik Universität Leipzig Augustusplatz 10/11 O-7010 Leipzig

Received February 18, 1991