Non-Associative Geometry of Quantum Tori

We describe how to obtain the imprimitivity bimodules of the noncommutative torus from a"principal bundle"construction, where the total space is a quasi-associative deformation of a 3-dimensional Heisenberg manifold.


Introduction
In differential geometry a standard way to construct vector bundles is from a principal bundle and a representation of the structure group. This construction works in noncommutative geometry too [6,13,15], with spaces replaced by algebras, vector bundles replaced by finitely generated projective modules, structure groups replaced by Hopf algebras or compact quantum groups, and principal bundles replaced by algebra extensions with suitable additional properties (see, e.g., [3,14]). When the structure group is U (1), it is possible to reconstruct the total space of the "bundle" (more precisely, a strongly graded C * -algebra) from the base space and a noncommutative "line bundle" (a self-Morita equivalence bimodule), cf. [2] (see also [1,7]).
A case study is provided by the C * -algebra of the noncommutative torus A θ , θ ∈ R\Q [5,20]. The group SL 2 (Z) acts on R\Q by fractional linear transformations,  twisted module algebras, namely Proposition 3.2. In Section 4 we will give a concrete realization of h 3 (R) in terms of differential operators on the total space of a principal U (1)-bundle M 3 → T 2 on the 2-torus, and apply to it the deformation recipe of Section 2 with the twist introduced in Section 3: the total space becomes a quasi-associative Z-graded algebra, which in degree 0 is the algebra of smooth functions on the noncommutative torus; in degree n = 0 we get bimodules that we compare with Connes-Rieffel imprimitivity bimodules. In Section 5 we give a slightly different version of the construction, more natural if one is interested in complex structures: we show that, in this case, the twisted algebra of functions on the Heisenberg manifold has an associative commutative subalgebra given by ordinary theta functions. In Section 6 we study vector bundles of any rank.
Notations. By a algebra we shall always mean a unital algebra (not necessarily associative nor commutative) over a commutative unital ring R; the algebraic tensor product over R will be denoted by ⊗ R , or simply ⊗ if there is no risk of confusion; by a Hopf algebra over C[[ν]] we shall always mean a topological Hopf algebra, completed in the h-adic topology, and by ⊗ C[[ν]] the completed tensor product [4, Section 4.1A].

Mathematical preliminaries
In this section, we recall some definitions and properties of quasi-Hopf algebras and Drinfeld twists, from [4,10,11,16].

Monoidal categories
A monoidal category is a category C equipped with a functor ⊗ : C × C → C which, modulo natural isomorphisms, is associative and unital [4, Section 5.1]. More precisely, there is an object I and three natural isomorphisms -Φ between the functors ( ⊗ ) ⊗ and ⊗ ( ⊗ ), λ between I ⊗ and the identity, ρ between ⊗ I and the identity -such that the diagrams 1a and 1b commute for all objects A, B, C, D. Examples are the category of modules over a field (vector spaces), over a group or a Hopf algebra (representations), over a topological Hopf algebra (with completed tensor product).
A monoid [16, Section VII.3] in a monoidal category is an object A together with two arrows m : A ⊗ A → A and η : I → A, the "multiplication" and "unit", satisfying the usual axioms of a unital associative algebra modulo natural transformations, namely the diagrams 2a and 2b must commute. Monoids in the category of vector spaces are associative algebras, in the category of representations of a Hopf algebra H are H-module algebras.

Quasi-Hopf algebras
Let H be an algebra and ∆ : H → H ⊗H a linear map. For all n ≥ m and all 1 ≤ i 1 < i 2 < · · · < i m ≤ n, we will denote by h → h i 1 ...im the linear map H ⊗m → H ⊗n defined on homogeneous tensors h = a 1 ⊗ a 2 ⊗ · · · ⊗ a m as follows: we put a k in the leg i k of the tensor product for all k = 1, . . . , m, and fill the additional n − m legs with 1. The subscript (i k i k+1 ) in parenthesis means that we apply ∆ to a k and put the first leg of ∆(a k ) in position i k and the second in A quasi-bialgebra (over a commutative ring R) is an associative algebra H together with two homomorphisms ∆ : H → H ⊗ H and : H → R (the "coproduct" and "counit") and an invertible element Φ ∈ H ⊗ H ⊗ H (the "coassociator") satisfying the following conditions (here we follow the notations of [10,11]): for all h ∈ H. A quasi-Hopf algebra is a quasi-bialgebra satisfying an additional condition which for Φ = 1 reduces to the existence and bijectivity of an antipode, see, e.g., [4,Definition 16.1.1]. Any Hopf algebra is a quasi-Hopf algebra with trivial coassociator, Φ = 1. In fact, in the Hopf case any Φ which is invariant (i.e., commutes with the image of the iterated coproduct) does the job. If Φ is invariant, (2.3) reduces to the coassociativity condition (id ⊗ ∆)∆ = (∆ ⊗ id)∆ and we get the usual definition of bialgebra/Hopf algebra.
If H is a commutative Hopf algebra, (2.1) can be interpreted as a 3-cocycle condition in the Hopf algebra cohomology of H [17, Section 2.3]. In general, if A, B, C are three H-modules, we may think of Φ as a module map (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C), with the natural isomorphism of vector spaces understood. Condition (2.1) becomes the associativity condition 1a for the category of H-modules, which is then monoidal with unit object I = R given by the ground ring (with module structure being given by the counit, so that (2.2) implies the commutativity of the diagram 1b, and module structure on a tensor product defined by the coproduct).
A monoid A in this category is a H-module algebra, that is an algebra A with multiplication satisfying the quasi-associativity condition (cf. diagram 2a): for all h ∈ H. Equation (2.4) becomes the usual associativity condition if the action of Φ on A ⊗ A ⊗ A is trivial (a sufficient condition of course is that Φ = 1). The above definitions remain valid if the ground ring is R = C[[ν]], with ⊗ R the completed tensor product.
The advantage of quasi-bialgebras (resp. quasi-Hopf algebras) over ordinary bialgebras (resp. Hopf algebras) is that there is a "gauge action" of the group of invertible elements in H ⊗ H that doesn't change the category of modules.
Let H be a quasi-bialgebra (resp. quasi-Hopf algebra) and F ∈ H ⊗ H an invertible element satisfying (id ⊗ )(F ) = ( ⊗ id)(F ) = 1. One can define a new quasi-bialgebra (resp. quasi-Hopf algebra) H F given by H as an algebra and with the same counit, but with a new coproduct ∆ F and coassociator Φ F defined by It turns out that the categories of H-modules and H F -modules are equivalent as monoidal categories. In particular, if (A, m) is a H-module algebra, there is a H F -module algebra (A, m F ) given by A as a vector space, with the same unit element, and with multiplication: We will refer to H F as a twist deformation of H, and to F as a twisting element based on H.

Quantum universal enveloping algebras
By a deformation of a quasi-bialgebra over a field k we mean a topological quasi-bialgebra H ν over k If H = U(g) is the universal enveloping algebra of a Lie algebra g, with standard Hopf algebra structure and trivial coassociator, a deformation H ν with coassociator Φ ν ≡ 1 mod ν 2 will be called a quantum universal enveloping algebra (or QUEA). Any twist of U(g) by a twisting element F ν based on U(g)[[ν]] (and satisfying F ν ≡ 1 mod ν) is a QUEA, and roughly speaking every QUEA arises in this way (see Theorem 16.1.11 of [4] for the precise statement).

A twist based on the Heisenberg Lie algebra
The example which is of interest to us is based on the Lie algebra h 3 (R) of the 3-dimensional Heisenberg group, or more precisely on the Hopf algebra U(h 3 (R)).
The Lie algebra h 3 (R) has a basis of three elements p, q, t, with t central and ] the ideal: Note that for all {a n } n≥0 belonging to a complex vector space A and for all θ ∈ Θ, the expression n≥0 a n θ n is a well defined element of A[[ν]] (for each N ≥ 0, the coefficient of ν N is a finite linear combination of elements of A).
There is an action α of Z by automorphisms on the vector space Θ given by Clearly α maps Θ into itself, and Let A be any U(h 3 (R))-module algebra (associative, since the coassociator is trivial), and for all n ∈ Z define is a yet smaller subalgebra.
Proof . From (3.4), recalling that p, q are primitive elements Next, we use Baker-Campbell-Hausdorff formula, which for two elements X, Y of an associative algebra with central commutator [X, Y ] reduces to e νX e νY = e ν 2 [X,Y ] e νY e νX . We get Using this in (3.6) we get Now all the exponents mutually commute, and after some simplification we arrive at (3.5).
In particular, we can observe that the coassociator: of the QUE algebra is not trivial nor invariant, so that we are dealing with a genuine quasi-Hopf deformation of U(h 3 (R)). Many properties of A θ can be deduced from Lemma 3.1.
Proposition 3.2. If θ = α n (θ), with α as in (3.2), one has the generalized associativity law: Proof . If θ = α n (θ), from the definition of the star product and the observation that t.b = nb, we deduce that Φ θ,θ is the identity on a ⊗ b ⊗ c and then (3.8) holds.
where the left and right module structure are given by the * θ multiplication).

4)
For all m, n, p ∈ Z and θ = α n (θ), the following diagram commutes: With a slight abuse of notations, we identify elements of U(h 3 (R)) with left invariant vector fields on H 3 (R), that are generated by the differential operators The choice of normalization will be clear later on (one can check that (3.1) is satisfied). Let H 3 (Z) := {(x, y, t) ∈ H 3 (R) : x, y, t ∈ Z}. By left invariance, the above vector fields descend to the 3-dimensional Heisenberg manifold M 3 := H 3 (Z)\H 3 (R). Thinking of functions on K\G as left K-invariant functions on G, we get where T 2 = R 2 /Z 2 (so f ∈ C ∞ (M 3 ) is periodic with period 1 in y and t). The action of central elements (0, 0, t) ∈ H 3 (R) descends to a principal action of U (1) on M 3 , and U (1)\M 3 T 2 . We identify C ∞ (T 2 ) with the subset of f ∈ C ∞ (M 3 ) that do not depend on t (and so are periodic in both x and y): Let A := C ∞ (M 3 ). In the notation of previous section, A 0 = C ∞ (T 2 ) and, for n = 0, every element f ∈ A n can be written in the form f (x, y, t) = k∈Z f x + k n ; k e 2πi(ky+nt) (4.1) for a unique Schwartz function f : R × Z/nZ → C. The bijection A n → S(R × Z/nZ), f → f , is known as Weil-Brezin-Zak transform [12, Section 1.10]. Functions (4.1) can be interpreted as smooth sections of a non-trivial smooth line bundle on T 2 [8]. The algebra A • = n∈Z A n is dense in A (in the uniform topology); indeed, by periodicity in t, the only weight spaces of t appearing in the decomposition of A are those with integer weight. It is strongly Z-graded, which is the algebraic counterpart of the principality of the bundle M 3 → T 2 (see, e.g., [1] or [7]). By point (1) of Corollary 3.3, A θ 0 is an associative subalgebra of A θ . By standard Fourier analysis, it is not difficult to verify that A θ 0 is generated by two unitary elements, the functions with relation It is then the formal analogue of the smooth algebra of the noncommutative torus.
We can extend C[[ν]]-linearly the map f → f in (4.1) to a bijective map For θ, θ ∈ Θ, we can define a left action of A θ 0 on E n and a right action of A θ 0 on E n by a. f := a * θ f , f .a := f * θ a, for all f ∈ E n and a ∈ A 0 . A computation using (4.1) gives the explicit formulas (for n = 0): where for a smooth function ψ, by ψ(y + θ ) we mean the formal power series ψ(y + θ ) := e θ ∂y ψ(y) = k≥0 θ k k! ∂ k y ψ(y).
(If we replace θ by a real number, although the above series is convergent only for ψ analytic, the formulas (4.4) are well defined for any Schwartz function f .) By comparing these formulas with equations (2.1)-(2.5) of [18] we recognize the formal analogue of Connes-Rieffel imprimitivity bimodule E g (θ), in the special case Finally, for all f 1 ∈ E n 1 and f 2 ∈ E n 2 , one can compute f 1 * θ f 2 ∈ E n 1 +n 2 using (4.1) and find the explicit formula x + (1 + n 1 θ) k 1 n 2 −k 2 n 1 n 1 (n 1 +n 2 ) ; k 1 f 2 x − k 1 n 2 −k 2 n 1 n 2 (n 1 +n 2 ) ; k 2 , (4.6) which is valid for n 1 , n 2 , n 1 + n 2 = 0. If, on the other hand, f 1 ∈ E n and f 2 ∈ E −n (n = 0), using the identity valid for ψ ∈ S(R), we get where Equations We can now give an interpretation to Corollary 3.3: point (2) is the analogue of, e.g., equation (2.6) of [18], stating that the algebra of endomorphisms of a finitely generated projective module over a noncommutative torus is another noncommutative torus with a different deformation parameter; point (4)

Complex structures and theta functions
In order to include a complex structure in the construction, it is convenient to start from a different realization of the Heisenberg group and of the principal bundle of previous section.
Let τ ∈ C be a complex number with imaginary part (τ ) > 0 and Λ := Z + τ Z ⊂ C a lattice. We now construct a principal U (1) bundle over the elliptic curve E τ := C/Λ, isomorphic to T 2 if one forgets about the complex structure.
We parametrize the Heisenberg group as follows: we set H 3 (R) := C × R with multiplication (z 1 , t 1 ) · (z 2 , t 2 ) = z 1 + z 2 , t 1 + t 2 + (z 1 z 2 ) , where the bar denotes complex conjugation and the imaginary part. Right invariant vector fields are spanned by where we identify elements of U(h 3 (R)) with their representation as differential operators. Let H 3 (Z) ⊂ H 3 (R) be the subgroup generated by the elements (1, 0) and (τ, 0).
We define M 3 := H 3 (R)/H 3 (Z). As in previous section, the action of central elements (0, t) ∈ H 3 (R) descends to a principal action of U (1) on M 3 , and M 3 /U (1) E τ . In the notations of Section 2.3, we set A := C ∞ (M 3 ) and A n as in (3.3), so that A 0 C ∞ (E τ ). The right invariance of f ∈ A under the action of (0, 2 (τ )) ∈ H 3 (Z) proves that f is periodic in t with period 2 (τ ), hence the differential operator t in (5.2) has integer spectrum (this explains the choice of normalization) and the subalgebra A • = n∈Z A n is dense in A.
The map (4.1) is replaced by the bijection (5.3) below.
Proposition 5.2. For all n = 0 there is a bijection A n f → f ∈ S(R × Z/nZ) given by where the real coordinates (x, y) ∈ R 2 are defined by z := x + τ y.
Proof . This is essentially (4.1) modulo a reparametrization. f ∈ A iff it is right invariant under the action of the two generators (1, 0) and (τ, 0) of H 3 (Z). In real coordinates, we get the conditions f (x + 1, y, t − y (τ )) = f (x, y, t) and f (x, y + 1, t + x (τ )) = f (x, y, t). As mentioned above, f is also 2 (τ )-periodic in t, and the condition tf = nf says that every f ∈ A n is given by e  F (x, y). The two invariance conditions above become F (x + 1, y) = F (x, y) and F (x, y + 1) = F (x, y)e −2πinx . From the former, F (x, y) = k∈Z e 2πikx F k (y) for some functions F k .
The latter condition gives F k (y + 1) = F k+n (y); if we define f (y; k) := F k (y − k n ), the condition becomes f (y; k + n) = f (y; k). Thus, f (y; k) is periodic in k with period n. Finally, from [8,Lemma 3.2] it follows that f is C ∞ iff f is Schwartz.
We can apply the same recipe of previous section, and deform the algebra A with the twist F θ = exp θp ⊗ q , θ ∈ Θ. The advantage is that now A θ has, besides A θ 0 , two additional associative subalgebras. The coassociator Φ θ,θ = e θ 2 p⊗t⊗q , cf. (3.7), is 1 on the kernels of p, t and q. The second kernel is A θ 0 , the first and third are related by a conjugation z →z. We will focus on the latter.
As in previous section, A θ 0 is generated by two unitary functions U (x, y, t) := e 2πix , V (x, y, t) := e 2πiy .
(τ ) V U , we get the usual noncommutative torus commutation relation 1 hol and A θ hol is a commutative associative subalgebra of A θ . With an explicit computation we now check that elements of A hol are (essentially) classical theta functions on the torus. Clearly A 0 ∩ A hol = C is the set of constant functions. For n = 0, the set A n ∩ A hol is described by the following lemma. where q := e πiτ .
One has ∇f = 0 iff e −πinτ (y+ k n ) 2 f y + k n ; k =: c f (k) does not depend on y + k n , and in this case (5.7) reduces to (5.6). Since (τ ) > 0, for n > 0 the function f (y; k) = c f (k)e πinτ y 2 is of Schwartz class for any c f ∈ C Z/nZ , while for n < 0 it is of Schwartz class only if it is zero.
The series in (5.6) are the usual theta functions on E τ . If n = 1, for example, the series in (5.6) is proportional to the Jacobi's theta function ϑ(z; q) = k∈Z q k 2 /n e 2πikz .
The algebraic structure of theta functions is encoded in the formula (5.8) below.
where f → c f is the map in (5.6).
Remark 5.5. In the C * -algebraic setting, for any fixed modular parameter τ , any g as in (1.1) and θ ∈ R solution of gθ = θ, a ring of "quantum theta functions" B g (θ, τ ) can be defined as a suitable "holomorphic" subalgebra of the tensor algebra k≥0 E g (θ) ⊗k (with tensor product over the algebra A θ of the noncommutative torus and E g (θ) ⊗0 := A θ ). A product formula for quantum theta functions appeared first in [9] in terms of generators and relations (see also [19]), while an alternative formula which is closer to our notations is [22, equation (7.4)].
For g as in (4.5) and n = 1, as one can easily check, the product (c f 1 , c f 2 ) → c f 1 f 2 defined by (5.8) coincides with [22, equation (7.4)] and B g (θ, τ ) A hol (while B g (θ, τ ) is a subalgebra of A hol if n > 1 in (4.5)). However, this is not surprising since, for g as in (4.5), the only solution to gθ = θ is θ = 0. The construction in this section, on the other hand, works in a formal setting and there is no constrain on θ. For any θ ∈ Θ, A θ is well-defined, although not associative, and has an associative (and commutative) subalgebra given by classical theta functions.

On higher rank vector bundles
In this section, we describe how to derive (formal) imprimitivity bimodules associated to an arbitrary element of SL(2, Z).
Let us fix a g as in (1.1), with d = 0. We denote by M c,d ⊂ C ∞ (R × T) the set of functions satisfying f (x + d, y) = e −2πicy f (x, y), (6.1) This is a C ∞ (T 2 )-right module with product given by pointwise multiplication. One can verify that M c,d is isomorphic to the module of (smooth) sections of a rank |d| vector bundle on T 2 as follows. Note that M c,d M −c,−d , so from now on we can assume that d ≥ 1.
That M 0,d is a free module of rank d comes from the following observation.
Remark 6.1. By standard Fourier analysis, every smooth function ϕ with period d can be written (in a unique way) as with ϕ k of period 1. This gives a C ∞ (R/Z)-module isomorphism: (Functions of period d form a free module of rank d over functions of period 1.) For the reason above, M 0,d M 0,1 ⊗ C d as modules over M 0,1 = C ∞ (T 2 ). From now on we forget about free modules and assume that c = 0 (and d ≥ 1 as above). Note that the condition det(g) = 1 in (1.1) guarantees that c and d are coprime. Vice versa, by Bézout's lemma such a g exists for every coprime c, d (although it is not unique). Lemma 6.2. Every n ∈ Z can be written, in a unique way, as n = kc + md for some 1 ≤ k ≤ d and m ∈ Z.
Proof . The map Z 2 (k, m) → kc+md ∈ Z is surjective, due to the identity n = (−nb)c+(na)d following from the determinant condition in (1.1); since (k, m) and (k − d, m + c) have the same image, one can always choose 1 ≤ k ≤ d; restricted to [1, . . . , d] × Z the map is also injective, since kc + md = k + md -i.e., (k − k )c + (m − m )d = 0 -implies that d must divide k − k . But |k − k | ≤ d − 1, so it must be k − k = 0, which also implies m − m = 0. Remark 6.1 can then be rephrase as follows.