Heisenberg modules over real multiplication noncommutative tori and related algebraic structures
aa r X i v : . [ m a t h . QA ] D ec HEISENBERG MODULES OVER REAL MULTIPLICATIONNONCOMMUTATIVE TORI AND RELATED ALGEBRAICSTRUCTURES
JORGE PLAZAS
Abstract.
We review some aspects of the theory of noncommutative two-tori with real multiplication focusing on the role played by Heisenberg groupsin the definition of algebraic structures associated to these noncommutativespaces. Introduction
Noncommutative tori have played a central role in noncommutative geometrysince the early stages of the theory. They arise naturally in various context andhave provided a good testing ground for many of the techniques from which non-commutative geometry has developed ([1, 14]). Noncommutative tori are definedin terms of their algebras of functions. The study of projective modules over thesealgebras and the corresponding theory of Morita equivalences leads to the exis-tence of a class of noncommutative tori related to real quadratic extensions of Q .These real multiplication noncommutative tori are conjectured to provide the cor-rect geometric setting under which to attack the explicit class field theory problemfor real quadratic fields ([7]). The right understanding of the algebraic structuresunderlying these spaces is important for these applications.The study of connections on vector bundles over noncommutative tori givesrise to a rich theory which has been recast recently in the context of complexalgebraic geometry ([1, 3, 16, 5, 12]). The study of categories of holomorphic bundleshas thrown light about some algebraic structures related to real multiplicationnoncommutative tori ([10, 18, 9]). Some of these results arise in a natural way fromthe interplay between Heisenberg groups and noncommutative tori.I want to thank the organizers of the 2007 summer school “Geometric and topo-logical methods for quantum field theory” and I.H.E.S. for their support and hos-pitality. This work was supported in part by ANR-Galois grant NT05-2 44266.1. Noncommutative tori and their morphisms
In many situations arising in various geometric settings it is possible to charac-terize spaces and some of their structural properties in terms of appropriate ringsof functions. One instance of this duality is provided by Gelfand’s theorem whichidentifies the category of locally compact Hausdorff topological spaces with thecategory commutative C ∗ -algebras. This correspondence assigns to a space X thecommutative algebra C ( X ) consisting of complex valued continuous functions on Mathematics Subject Classification.
Primary 58B34, 46L87; Secondary 16S38.
Key words and phrases. noncommutative tori, real multiplication, Heisenberg groups. X vanishing at infinity. Topological invariants of the space X can be obtained bythe corresponding invariants of C ( X ) defined in the context of C ∗ -algebras. If X isa smooth manifold the smooth structure on X singles out the ∗ -subalgebra C ∞ ( X )consisting of smooth elements of C ( X ). Considering the space X in the frameworkof differential topology leads to structures defined in terms of the algebra C ∞ ( X ).Various geometric notions which can be defined in terms of rings of functions ona space do not depend on the fact that the rings under consideration are commuta-tive and can therefore be extended in order to consider noncommutative rings andalgebras. In noncommutative geometry spaces are defined in terms of their ringsof functions which are noncommutative analogs of commutative rings of functions.This passage is far from being just a translation of classical ideas to a noncommu-tative setting. Many extremely rich new phenomena arises in this context (see [2]).The noncommutative setting also enriches the classical picture since noncommuta-tive rings may arise in a natural way from classical geometric considerations.We will be considering noncommutative analogs of the two-torus T = S × S .The reader may consult [6, 1, 2, 4, 15, 14, 7] for the proofs of the results on non-commutative tori mentioned in this section.1.1. The C ∗ -algebra A θ . Under the Gelfand correspondence compact spaces cor-respond to commutative unital C ∗ -algebras and T is dual to C ( T ). At a topologi-cal level a noncommutative two-torus is defined in terms of a unital noncommutative C ∗ -algebra which plays the role of its algebra of continuous functions.Given θ ∈ R let A θ be the universal C ∗ -algebra generated by two unitaries U and V subject to the relation: U V = e πıθ V U.
Then(1) If θ ∈ Z the algebra A θ is isomorphic to C ( T ).(2) If θ ∈ Q the algebra A θ is isomorphic to the algebra of global sections ofthe endomorphism bundle of a complex vector bundle over T .(3) If θ ∈ R \ Q the algebra A θ is a simple C ∗ -algebra.For irrational values of θ we will refer to A θ as the algebra of continuous functionson the noncommutative torus T θ . Thus, as a topological space the noncommutativetorus T θ is defined as the dual object of C ( T θ ) := A θ .There is a natural continuous action of the compact group T on the algebra A θ .This action can be given in terms of the generators U and V by: α ϕ ( U ) = e πıϕ Uα ϕ ( V ) = e πıϕ V where ϕ = ( ϕ , ϕ ) ∈ T .One of the main structural properties of the algebra A θ is the existence of acanonical trace whose values at each element is given by the average over T of theabove action. Theorem 1.1.
Let θ be an irrational number. Then there exist a unique normalizedtrace: χ : A θ → C invariant under the action of T . ONCOMMUTATIVE TORI 3
For the remaining part of the article θ will denote an irrational number. Also,for any complex number z ∈ C we will use the notation:e( z ) = exp(2 πız ) , ¯e( z ) = exp( − πız ) . Smooth elements.
The above action of T on A θ induces a smooth structureon the noncommutative torus T θ . An element a ∈ A θ is called smooth if the map T −→ A θ ϕ α ϕ ( a )is smooth. The set of smooth elements of A θ is a dense ∗ -subalgebra which wedenote by A θ . Elements in this subalgebra should be thought as smooth functionson the noncommutative torus T θ thus we take C ∞ ( T θ ) := A θ . The algebra A θ canbe characterized in the following way: A θ = (cid:26) X n,m ∈ Z a n,m U n V m ∈ A θ | { a n,m } ∈ S ( Z ) (cid:27) where S ( Z ) denotes the space of sequences of rapid decay in Z .In the algebra A θ the trace χ is given by χ ( X a n,m U n V m ) = a , The Lie algebra L = R of T acts on A θ by derivations. A basis for this actionis given by the derivations: δ ( U ) = 2 πıU ; δ ( V ) = 0 δ ( U ) = 0; δ ( V ) = 2 πıV. A complex parameter τ ∈ C \ R induces a complex structure on L = R givenby the isomorphism: R −→ C x = ( x , x ) ˜ x = τ x + x . The corresponding complex structure on A θ is given by the derivation: δ τ = τ δ + δ . Vector bundles and K -theory. If X is a smooth compact manifold thespace of smooth sections of a vector bundle over X is a finite-type projective mod-ule over C ∞ ( X ) and any such a module arises in this way. In our setting finite-typeprojective right A θ -modules will play the role of vector bundles over the noncom-mutative torus T θ .We denote as above by L = R the Lie algebra of T acting as an algebra ofderivations on A θ . If P is a finite-type projective right A θ -module a connection on P is given by an operator: ∇ : P → P ⊗ L ∗ such that ∇ X ( ξa ) = ∇ X ( ξ ) a + ξδ X a for all X ∈ L, ξ ∈ P and a ∈ A θ .The connection ∇ is determined the operators ∇ i : P → P, i = 1 , JORGE PLAZAS giving its values on the basis elements δ , δ of L .The K group of A θ is by definition the enveloping group of the abelian semigroupgiven by isomorphism classes of finite-type projective right A θ -modules togetherwith direct sum. The trace χ extends to an injective morphismrk : K ( A θ ) → R whose image is Γ θ = Z ⊕ θ Z . Morphisms of noncommutative tori.
Since noncommutative tori are de-fined in terms of their function algebras one should expect a morphism T θ → T θ ′ between two noncommutative tori T θ and T θ ′ to be given by a morphism A θ ′ → A θ of the corresponding algebras of functions. It turns out that algebra morphisms arein general insufficient to describe the type of situations arising in noncommutativegeometry. The right notion of morphisms in our setting is given by Morita equiv-alences. A Morita equivalence between A θ ′ and A θ is given by the isomorphismclass of a A θ ′ - A θ -bimodule E which is projective and of finite-type both as a left A θ ′ -module and as a right A θ -module. If such bimodule exists we say that A θ ′ and A θ are Morita equivalent. We can consider a Morita equivalence between A θ ′ and A θ as a morphism between A θ ′ and A θ inducing a morphism between T θ and T θ ′ .Composition of morphisms is provided by tensor product of modules.Let SL ( Z ) act on R \ Q by fractional linear transformations, i.e. given g = (cid:18) a bc d (cid:19) ∈ SL ( Z ) , θ ∈ R \ Q we take gθ = aθ + bcθ + d . Morita equivalences between noncommutative tori are characterized by the follow-ing result:
Theorem 1.2. (Rieffel [13])
Let θ ′ , θ ∈ R \ Q . Then the algebras A θ ′ and A θ areMorita equivalent if and only if there exist a matrix g ∈ SL ( Z ) such that θ ′ = gθ . In Section 3 we will construct explicit bimodules realizing this equivalences. Inwhat follows whenever we refer to a right A θ -module (resp. left A θ ′ -module, resp. A θ ′ - A θ -bimodule) we mean a projective and finite-type right A θ -module (resp. left A θ ′ -module, resp. A θ ′ - A θ -bimodule)Given a irrational number θ a A θ - A θ -bimodule E induces a Morita self equiv-alences of A θ . We denote by End
Morita ( A θ ) the group of Morita self equivalenceof A θ . For example, given any positive integer n the free bimodule A nθ induces a aMorita self equivalences of A θ . A Morita self equivalence defined via a free moduleis called a trivial Morita self equivalence.A Morita self equivalence of A θ given by a A θ - A θ -bimodule E defines an endo-morphism of K ( A θ ) via φ E : [ P ] [ P ⊗ A θ E ]for P a right projective finite rank A θ module and [ P ] ∈ K ( A θ ) is its K -theoryclass. ONCOMMUTATIVE TORI 5
Via the map rk the endomorphism φ M becomes multiplication by a real number.Thus we get a map: φ : End
Morita ( A θ ) → { α ∈ R | α Γ θ ⊂ Γ θ } this map turns out to be surjective.We can summarize the situation as follows (see [7]): Theorem 1.3.
Let θ ∈ R be irrational. The following conditions are equivalent: • A θ has nontrivial Morita autoequivalences. • φ ( End
Morita ( A θ )) = Z . • There exist a matrix g ∈ SL ( Z ) such that θ = gθ. • θ is a real quadratic irrationality: [ Q ( θ ) : Q ] = 2 . If any of these equivalent conditions holds we say that the noncommutative torus T θ with algebra of smooth functions A θ is a real multiplication noncommutativetorus . If T θ is a real multiplication noncommutative torus then φ ( End
Mor ( A θ )) = { α ∈ R | α Γ θ ⊂ Γ θ } = Z + f O k where f ≥ O k is the ring of integers of the real quadratic field k = Q ( θ ).These results should be compared with the analogous results for elliptic curvesleading to the theory of complex multiplication. The strong analogy suggests thatnoncommutative tori may play a role in number theory similar to the role playedby elliptic curves. In particular noncommutative tori with real multiplication couldgive the right geometric framework to attack the explicit class field theory problemfor real quadratic fields (see [7]).2. Heisenberg groups and their representations
Various aspects of the theory of representations of Heisenberg groups arisenaturally when considering geometric constructions associated to noncommutativetori. This fact gives relations between noncommutative tori and elliptic curvesthrough theta functions and plays a relevant role in the study of the arithmeticnature of related algebraic structures. In this section we sketch the parts of thetheory of Heisenberg groups that are relevant in order to describe these results.We follow Mumford’s Tata lectures [8] which also recommend as a reference for thematerial in this section.Let G be a locally compact group lying in a central extension:1 → C ∗ → G → K → C ∗ is the group of complex numbers of modulus 1 and K is a locally compactabelian group. Assume that the exact sequence splits, so as a set G = C ∗ × K andthe group structure is given by:( λ, x )( µ, y ) = ( λµψ ( x, y ) , x + y ) JORGE PLAZAS where ψ : K × K → C ∗ is a two-cocycle in K with values in C ∗ . The cocycle ψ induces a skew multiplicative pairing e : K × K −→ C ∗ ( x, y ) ψ ( x, y ) ψ ( y, x ) . This pairing defines a group morphism ϕ : K → b K from K to its Pontrjagin dualgiven by φ ( x )( y ) = e ( x, y ). Definition 2.1. If ϕ is a isomorphism we say that G is a Heisenberg group .For a Heisenberg group G lying in a central extension as above we use the no-tation G = Heis( K ). The main theorem about the representation of Heisenberggroups states that these kind of groups admit a unique normalized irreducible rep-resentations which can be realized in terms of a maximal isotropic subgroup of K . A subgroup H of K is called isotropic if e | H × H ≡
1, this is equivalent to theexistence of a section of G over H : σ : K −→ Gx ( α ( x ) , x ) . We say that a subgroup H of K is maximal isotropic if it is maximal with thisproperty. A subgroup H of K is maximal isotropic if and only if H = H ⊥ wherefor S ⊂ H we have S ⊥ = { x ∈ K | e ( x, y ) = 1 for all y ∈ S } . Theorem 2.2. (Stone, Von Neumann, Makey)
Let G be a Heisenberg group. Then • G has a unique irreducible unitary representation in which C ∗ acts by mul-tiples of the identity. • Given a maximal isotropic subgroup H ⊂ K and a splitting σ as above let H = H H be the space of measurable functions f : K → C satisfying (1) f ( x + h ) = α ( h ) ψ ( h, x ) − f ( x ) for all h ∈ H . (2) R K/H | f ( x ) | dx < ∞ .Then G acts on H by U ( λ,y ) f ( x ) = λψ ( x, y ) f ( x + y ) . and H is an irreducible unitary representation of G . We call such representation a
Heisenberg representation of G . The followingtheorem will be useful later: Theorem 2.3. (Stone, Von Neumann, Makey)
Given two Heisenberg groups → C ∗ → G i → K i → i = 1 , with Heisenberg representations H and H then → C ∗ → G × G / { ( λ, λ − ) | λ ∈ C ∗ } → K × K → is a Heisenberg group and its Heisenberg representation is H ˆ ⊗H . ONCOMMUTATIVE TORI 7
Real Heisenberg groups.
Let K = R and let ε be a positive real number. Weendow G = K × C ∗ with the structure of a Heisenberg group defined by the cocycle ψ and the pairing e given by ψ ( x, y ) = e (cid:18) ε ( x y − y x )2 (cid:19) e ( x, y ) = e (cid:18) ε ( x y − y x ) (cid:19) . where x = ( x , x ) , y = ( y , y ) ∈ K .If we choose as maximal isotropic subgroup H = { x = ( x , x ) ∈ K | x = 0 } then the values of the functions in the corresponding Heisenberg representation(Theorem 2.2) are determined by their values on { x = ( x , x ) ∈ K | x = 0 } andwe may identify the space H H with L ( R ). The action of the G is given by U ( λ,y ) f ( x ) = λ e (cid:18) ε (cid:16) xy + y y (cid:17)(cid:19) f ( x + y ) . for ( λ, y ) = ( λ, ( y , y )) ∈ G and f ∈ L ( R ). In particular we have U (1 , ( y , f ( x ) = f ( x + y ) .U (1 , (0 ,y )) f ( x ) = e (cid:18) ε xy (cid:19) f ( x ) . We will denote this Heisenber representation by H ε .For any X ∈ Lie ( G ) and any Heisenberg representation H there is a dense subsetof elements f ∈ H for which the limit δU X ( f ) = lim t → U exp( tX ) f − ft exists. The above formula for δU X defines an unbounded operator on this set. Anelement f ∈ H is a smooth element for the representation H of G if δU X δU X · · · δU X n ( f )is well defined for any n and any X , X , . . . X n ∈ Lie ( G ). The set of smoothelement of H is denoted by H ∞ . We may realize Lie ( G ) as an algebra of operatorson H ∞ . If choose a basis { A, B, C } for the Lie algebra Lie ( G ) such thatexp( tA ) = (1 , ( t, , exp( tB ) = (1 , (0 , t )) , exp( tC ) = (e( t ) , (0 , . then a complex number τ ∈ C with nonzero imaginary part gives a decompositionof Lie ( G ) ⊗ C into conjugate abelian complex subalgebras: W τ = h δU A − τ δU B i W ¯ τ = h δU A − ¯ τ δU B i . Theorem 2.4.
Fix τ ∈ C with ℑ ( τ ) > then in any Heisenberg representation of G there exists an element f τ , unique up to a scalar, such that δU X ( f τ ) is definedand equal to for all X ∈ W τ . JORGE PLAZAS
In the Heisenberg representation H ε we have: δU A f ( x ) = ddx f ( x ) δU B f ( x ) = 2 πıxε f ( x ) δU B f ( x ) = 2 πı f ( x )and H ε, ∞ is the Schwartz space S ( R ). The element f τ in Theorem 2.4 is given by f τ = e (cid:18) ε τ x (cid:19) . Z /c Z ) ) . Let c be a positive integer and let K = ( Z /c Z ) . We endow G = K × C ∗ with the structure of a Heisenberg group defined by the cocycle ψ andthe pairing e given by ψ (([ n ] , [ n ]) , ([ m ] , [ m ])) = e (cid:18) c ( n m − m n ) (cid:19) e (([ n ] , [ n ]) , ([ m ] , [ m ])) = e (cid:18) c ( n m − m n ) (cid:19) where ([ n ] , [ n ]) , ([ m ] , [ m ]) ∈ K .If we choose as maximal isotropic subgroup H = { ([ n ] , [ n ]) ∈ K | [ n ] = 0 } wemay realize the Heisenberg representation as the action of G on C ( Z /c Z ) given by U ( λ, ([ m ] , [ m ])) φ ([ n ]) = λ e (cid:18) c (cid:16) nm + m m (cid:17)(cid:19) φ ([ n + m ]) . for ( λ, ([ m ] , [ m ])) ∈ G and φ ∈ C ( Z /c Z ). In particular we have U (1 , ([ m ] , φ ([ n ]) = φ ([ n + m ]) .U (1 , (0 , [ m ])) φ ([ n ]) = e (cid:18) c nm (cid:19) f ([ n ]) . Remark . This type of Heisenberg groups is related to algebraic Heisenberggroups or, more generally,
Heisenberg group schemes . These are given by centralextensions of the form 1 → G m → G → K → K is a finite abelian group scheme aver a base field k . These groups arise ina natural way by considering ample line bundles on abelian varieties over the basefield k . The corresponding Heisenberg representations can be realized as canonicalactions of G on the spaces of sections of these bundles. The action of Gal (¯ k/k )on the geometric points of G implies important algebraicity results about theserepresentations. The abelian varieties that play a role in the constructions thatfollow are the elliptic curves whose period lattice is spanned by the parameter τ which defines the complex structure on the noncommutative torus.3. Heisenberg modules over noncommutative tori with realmultiplication
Let θ ∈ R be a quadratic irrationality and let g = (cid:18) a bc d (cid:19) ∈ SL ( Z ) ONCOMMUTATIVE TORI 9 be a matrix fixing θ . In this section we describe the construction of a A θ - A θ -bimodule E g whose isomorphism class gives a Morita self equivalence of A θ . Inwhat follows we assume that c and cθ + d are positive. Let ε = cθ + dc and consider the following operators on the Schwartz space S ( R ):( ˇ U f )( x ) = f ( x − ε )( ˇ V f )( x ) = e( x ) f ( x )( ˆ U f )( x ) = f (cid:18) x − c (cid:19) ( ˆ V f )( x ) = e (cid:16) xcε (cid:17) f ( x ) . Note that each pair of operators corresponds to the Heisenberg group action of twogenerators of R where as above we identify the Schwartz space S ( R ) with the setof smooth elements of the Heisenberg representation H ε of Heis( R ).We consider also the following operators on C ( Z /c Z )(ˇ uφ )([ n ]) = φ ([ n − vφ )([ n ]) = ¯e (cid:18) dnc (cid:19) φ ([ n ])(ˆ uφ )([ n ]) = φ ([ n − a ])(ˆ vφ )([ n ]) = ¯e (cid:16) nc (cid:17) φ ([ n ]) . Since both a and d are prime relative to c each pair of operators correspondsto the Heisenberg group action of two generators of ( Z /c Z ) on the Heisenbergrepresentation C ( Z /c Z ) of Heis(( Z /c Z ) ).Taking into account the commutation relations satisfied between each of theabove pairs of operators and the fact that gθ = θ we see that the space E g = S ( R ) ⊗ C ( Z /c Z )becomes a A θ - A θ -bimodule by defining:( f ⊗ φ ) U = ( ˇ U ⊗ ˇ u )( f ⊗ φ )( f ⊗ φ ) V = ( ˇ V ⊗ ˇ v )( f ⊗ φ ) U ( f ⊗ φ ) = ( ˆ U ⊗ ˆ u )( f ⊗ φ ) V ( f ⊗ φ ) = ( ˆ V ⊗ ˆ v )( f ⊗ φ )where f ∈ S ( R ) and φ ∈ C ( Z /c Z ). Theorem 3.1. (Connes [1])
With the above bimodule structure E g is finite-typeand projective both as a right A θ -module and as a left A θ -module. Considered as aright module we have rk( E g ) = cθ + d . The left action of A θ gives an identification: End A θ ( E g ) ≃ A θ . We refer to this kind of modules as
Heisenberg modules . Taking into account the A θ - A θ -bimodule structure of E g we may consider thetensor product E g ⊗ A θ E g . This is one of the main consequences of the real multi-plication condition. There is a natural identification (see [5, 12]): E g ⊗ A θ E g ≃ E g . To see this consider first the completed tensor product over C of the space E g with it self: E g ˆ ⊗ E g = [ S ( R ) ⊗ C ( Z /c Z )] ˆ ⊗ [ S ( R ) ⊗ C ( Z /c Z )]= (cid:2) S ( R ) ˆ ⊗S ( R ) (cid:3) ⊗ [ C ( Z /c Z ) ⊗ C ( Z /c Z )]= [ S ( R × R )] ⊗ [ C ( Z /c Z × Z /c Z )] . The space S ( R × R ) is the space of smooth elements of the Heisenberg repre-sentation of Heis( R ) obtained as a product of the Heisenberg representations ofHeis( R ). Likewise C ( Z /c Z × Z /c Z ) is the Heisenberg representation of Heis(( Z /c Z ) )obtained as a product of the Heisenberg representations of Heis(( Z /c Z ) ).To pass from E g ˆ ⊗ E g to E g ⊗ A θ E g we have to quotient E g ˆ ⊗ E g by the spacespanned by the relations:[( f ⊗ φ ) U ] ˆ ⊗ [ g ⊗ ω ] = [ f ⊗ φ ] ˆ ⊗ [ U ( g ⊗ ω )][( f ⊗ φ ) V ] ˆ ⊗ [ g ⊗ ω ] = [ f ⊗ φ ] ˆ ⊗ [ V ( g ⊗ ω )]where f, g ∈ S ( R ) and φ, ω ∈ C ( Z /c Z ).At the level of the Heisenberg representations involved this amounts to restrictto the subspaces of S ( R × R ) and C ( Z /c Z × Z /c Z ) which are invariant under theaction of the subgroups of Heis( R ) and Heis(( Z /c Z ) ) generated by the elementsgiving these relations.The corresponding space of invariant elements in S ( R × R ) is canonically iso-morphic to the space S ( R ) of smooth elements of the Heisenberg representationof Heis( R ) with cε / ( a + d ) playng the role of ε . In C (( Z /c Z ) × ( Z /c Z )) thecorresponding invariant subspace is canonically isomorphic to the Heisenber repre-sentation C ( Z /c ( a + d ) Z ) of Heis(( Z /c ( a + d ) Z ) ). Thus we get: E g ⊗ A θ E g ≃ S ( R ) ⊗ C ( Z /c ( a + d ) Z )= E g The compatibility of the module structures in this isomorphism is implied by thethe compatibility of the Heisenberg representations involved.In a similar manner one may obtain isomorphisms: E g ⊗ A θ · · · ⊗ A θ E g | {z } n ≃ E g n . Some rings associated to noncommutative tori with realmultiplication
Noncommutative tori may be considered as noncommutative projective varieties.In noncommutative algebraic geometry varieties are defined in terms of categorieswhich play the role of appropriate categories of sheaves on them (see [17]). In[10] Polishchuk analyzed real multiplication noncommutative tori from this pointof view. Given a real quadratic irrationality θ and a complex structure δ τ on A θ Polishchuk constructs a homogeneous coordinate ring associated T θ and δ τ . ONCOMMUTATIVE TORI 11
During the rest of the section g will denote a matrix in SL ( Z ) fixing a quadraticirrationality θ . We will denote the elements of the powers of this matrix by g n = (cid:18) a n b n c n d n (cid:19) , n > . Given a Heisenberg A θ - A θ -bimodule E g we use the complex structure δ τ on A θ tosingle out a finite dimensional subspace in each of the graded pieces of E g = M n ≥ E g ⊗ A θ · · · ⊗ A θ E g | {z } n = M n ≥ E g n . This should be done in a way compatible with the product structure of E g .Given a Heisenberg A θ - A θ -bimodule E g we may define a connection on E g by:( ∇ f ⊗ φ )( x, [ n ]) = 2 πı (cid:16) xε (cid:17) ( f ⊗ φ )( x, [ n ])( ∇ f ⊗ φ )( x, [ n ]) = ddx ( f ⊗ φ )( x, [ n ]) . Connections of this kind were studied in [3] in the context of Yang Mils theory fornoncommutative tori. Note that this connection corresponds to the action of theLie algebra of Heis( R ) on the left factor of S ( R ) ⊗ C ( Z /c Z ) given by the derivations δU A and δU B . Once we choose a complex parameter τ ∈ C \ R giving a complexstructure on A θ the corresponding decomposition of the complexified Lie algebrasingles out the element f τ (Theorem 2.4) thus it is natural to consider the space: R g = { f τ ⊗ φ ∈ E g | φ ∈ C ( Z /c Z ) } = n e (cid:18) ε τ x (cid:19) ⊗ φ ∈ E g | φ ∈ C ( Z /c Z ) o . This are the the spaces of the holomorphic vectors considered in [16, 5, 12, 10].We denote by f τ,n the corresponding element on the left factor of E g n and let R g n = n f τ,n ⊗ φ ∈ E g n | φ ∈ C ( Z /c n Z ) o . Following [10] we define the homogeneous coordinate ring for the noncommutativetorus T θ with complex structure δ τ by: B g ( θ, τ ) = M n ≥ R g n . The following result characterizes some structural properties of B g ( θ, τ ) in termsof the matrix elements of g : Theorem 4.1. ([10] Theorem 3.5)
Assume g ∈ SL ( Z ) has positive real eigenval-ues: (1) If c ≥ a + d then B g ( θ, τ ) is generated over C by R g . (2) If c ≥ a + d + 1 then B g ( θ, τ ) is a quadratic algebra. (3) If c ≥ a + d + 2 then B g ( θ, τ ) is a Koszul algebra. Let X τ be the elliptic curve with complex points C / ( Z ⊕ τ Z ). Taking into accountthe remarks at the end of section Section 2 it is possible to realize each space R g n as the space of sections of a line bundle over X τ . For this we consider the matrix coefficients obtained by pairing f τ,n with functionals in the distribution completionof the Heisenberg representation which are invariant under the action of elements inHeis( R ) corresponding to a lattice in R associated to g n . This matrix coefficientscorrespond to theta functions with rational characteristics which form a basis forthe space of sections of the corresponding line bundle over X τ . In these bases thestructure constants for the product of B g ( θ, τ ) have the form ϑ r ( lτ )where l ∈ Z and ϑ r ( lτ ) is the theta constant with rational characteristic r ∈ Q defined by the series ϑ r ( lτ ) = X n ∈ Z exp[ πı ( n + r ) lτ ] . This fact has the following consequence:
Theorem 4.2. ([9])
Let θ ∈ R be a quadratic irrationality fixed by a matrix g ∈ SL ( Z ) and assume c ≥ a + d + 2 . Let k be the minimal field of definition of theelliptic curve X τ . Then the algebra B g ( θ, τ ) admits a rational presentation over afinite algebraic extension of k .Remark . Analogous results hold for the rings of quantum theta functions consid-ered in [18]. These rings correspond to Segre squares of the homogeneous coordinaterings B g ( θ, τ ) and can be analyzed in terms of the Heisenberg modules involved intheir construction. References [1] A. Connes, C ∗ -alg`ebres et g´eom´etrie diff´erentielle. C. R. Ac. Sci. Paris, t. (1980) 599-604.[2] A. Connes,
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