Algebraic deformations of toric varieties I. General constructions
aa r X i v : . [ m a t h . QA ] J un ALGEBRAIC DEFORMATIONS OF TORIC VARIETIES I.GENERAL CONSTRUCTIONS
LUCIO CIRIO, GIOVANNI LANDI, AND RICHARD J. SZABO
Abstract.
We construct and study noncommutative deformations of toric varieties bycombining techniques from toric geometry, isospectral deformations, and noncommuta-tive geometry in braided monoidal categories. Our approach utilizes the same fan struc-ture of the variety but deforms the underlying embedded algebraic torus. We developa sheaf theory using techniques from noncommutative algebraic geometry. The casesof projective varieties are studied in detail, and several explicit examples are workedout, including new noncommutative deformations of Grassmann and flag varieties. Ourconstructions set up the basic ingredients for thorough study of instantons on noncom-mutative toric varieties, which will be the subject of the sequel to this paper.
Contents
Introduction 2Acknowledgments 41. Algebraic preliminaries 41.1. Twist deformations of symmetries 41.2. Braided monoidal categories of Hopf-module algebras 61.3. Ore localization 82. Algebraic torus deformations 82.1. The noncommutative algebraic torus 92.2. Twisted toric actions 112.3. The noncommutative variety GL θ ( n ) 122.4. Quantum determinants 143. Noncommutative toric varieties 173.1. Noncommutative deformations of toric varieties 173.2. Algebraic Moyal plane and D -modules 223.3. Noncommutative projective plane 223.4. Noncommutative orbifold 233.5. Noncommutative conifold 234. Sheaves on noncommutative toric varieties 244.1. Quasi-coherent sheaves 244.2. Equivariant sheaves 254.3. Invariant subschemes and ideal sheaves 264.4. K¨ahler differential forms 285. Noncommutative projective varieties 325.1. Noncommutative projective spaces CP nθ G r θ ( d ; n ) 345.4. Noncommutative flag varieties F l θ ( d , . . . , d r ; n ) 37 Date : January 2010 ; Modified February 2011 and June 2013 HWM–09–14 , EMPG–09–22 .
6. Geometry of noncommutative projective varieties 406.1. Cohomology of CP nθ CP nθ G r θ ( d ; n ) 456.4. Differential forms on G r θ ( d ; n ) 46References 52 Introduction
This paper is the first part of a series of articles in which we define and study a classof noncommutative toric varieties, and construct instantons thereon. Our approach is in-spired by the theory of isospectral deformations [13] and a construction due to Ingalls [25].We expand and elaborate on some of the constructions introduced in the latter paper us-ing techniques from noncommutative geometry in braided monoidal categories. We startwith a noncommutative deformation of an algebraic torus and use this to deform everytoric variety on which the torus acts. This is done in a fashion that does not alter thecombinatorial fan data describing the toric variety.Part of the motivation for our construction comes from enumerative geometry and at-tempts to provide physical interpretations of enumerative invariants of toric varieties.In [26, 10], it is argued that the computation of Donaldson–Thomas invariants of a toricthreefold X can be reduced to the problem of locally enumerating noncommutative in-stantons on each open patch of X , and then assembling the local contributions into aglobal quantity using the gluing rules of toric geometry. This heuristic construction worksbecause noncommutative deformations of C are simple enough to explicitly construct in-stantons thereon, but the construction utilizes commutative toric geometry techniques toglue together quantities which are locally constructed using methods of noncommutativegeometry. In the the present paper we define a precise notion of “noncommutative toricvariety” which leads to a more global picture of their noncommutative geometry and ofthe construction of instantons thereon. Although our main interest lies in the constructionof noncommutative instantons, the requisite building blocks turn out to be rather tech-nically involved and lengthy. Thus the present paper is a (partly expository) systematicdevelopment of the general machinery required. The treatment of instanton counting onthese varieties is defered to a sequel [12].Another motivation for our constructions comes from string geometry. Chiral fermionfields on a quantum curve can be embedded in string theory as an intersecting D-braneconfiguration together with a B -field [18]. Mathematically, this system is described by aholonomic D -module. In some instances, the category of D -modules is in correspondencewith the category of modules on a noncommutative variety, of which some of our con-structions furnish explicit examples and give precise realizations of the noncommutativegeometry alluded to in [18]. The simplest example of such a correspondence is betweenright ideals of the algebra of differential operators on the affine line and line bundles overa certain noncommutative deformation of the projective plane CP [6]. The classifica-tion of bundles on noncommutative CP is related to the construction of instantons on anoncommutative R [27].From a mathematical perspective, our general construction produces new examples ofnoncommutative varieties. In particular, by considering noncommutative deformations of LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 3 projective toric varieties, we give new examples of noncommutative grassmannians, andmore generally flag varieties. We use techniques of noncommutative algebraic geometryto develop a sheaf theory for our varieties. Our treatment of flag varieties includes anoncommutative twistor theory, while our development of sheaf theory also producessheaves of differential forms, all of which are instrumental in the analysis of instantons [12].An alternative approach to noncommutative toric varieties can be found in [8].The organisation of this paper is as follows. In § § C × ) n , which extends the standard (real) noncommutativetorus and is the basic building block for all constructions in this paper. We use this toconstruct a twist deformation of the algebraic group GL( n ), which requires a suitablenotion of quantum determinant. We give a new description of these noncommutativedeterminants. We also work out the related braided exterior algebras of noncommutativeminors. These ingredients are used in the description of the noncommutative geometryof Grassmann and flag varieties.In § § § CP n is equivalent to a“global” description which is a special instance of the noncommutative weighted projec-tive spaces considered in [5]. We use these spaces to define noncommutative Grassmannand flag varieties as noncommutative quadrics in projective space, through suitable defor-mations of Pl¨ucker embeddings. We study the embedding relations in detail and deriveconditions for the embeddings into noncommutative projective space to exist.Finally, in § LUCIO CIRIO, GIOVANNI LANDI, AND RICHARD J. SZABO
Acknowledgments.
We thank Simon Brain, Ugo Bruzzo, Brian Dolan, Michel Dubois-Violette and Chiara Pagani for helpful discussions. An anonymous referee made a numberof pertinent remarks that led to a much improved version of the paper, for which we aregrateful. The work of RJS was supported in part by grant ST/G000514/1 “String TheoryScotland” from the UK Science and Technology Facilities Council.1.
Algebraic preliminaries
This section summarizes the algebraic constructions which will be used throughout thispaper and its sequel [12]. We present a general framework for working with the symmetriesof the noncommutative varieties that we shall encounter later on. We also recall somenotions from the localization theory for noncommutative algebras.1.1.
Twist deformations of symmetries.
Let H be a Hopf algebra over C with co-product ∆ : H → H ⊗ H , counit ε : H → C , and antipode S : H → H . We will make useof the conventional Sweedler notation ∆( h ) = h (1) ⊗ h (2) (with implicit summation) and(id ⊗ ∆) ∆( h ) = (∆ ⊗ id) ∆( h ) = h (1) ⊗ h (2) ⊗ h (3) . Definition 1.1.
Let H ⊗ A → A , h ⊗ a h ⊲ a be a left action of the Hopf algebra H ona unital algebra A with product µ : A ⊗ A → A . The action is said to be covariant if thecompatibility conditions h ⊲ µ ( a ⊗ b ) = µ (cid:0) ∆( h ) ⊲ ( a ⊗ b ) (cid:1) := µ (cid:0) ( h (1) ⊲ a ) ⊗ ( h (2) ⊲ b ) (cid:1) , h ⊲ ε ( h ) 1(1.2) hold for all h ∈ H and a, b ∈ A . In this case A is called a left H -module algebra . Similarly, a left action ⊲ of the Hopf algebra H on a coalgebra ( C, δ, ǫ ) is said to becovariant, making the latter a left H -module coalgebra , if the compatibility conditions δ ( h ⊲ c ) = ∆( h ) ⊲ δ ( c ) := (cid:0) h (1) ⊲ c (1) (cid:1) ⊗ (cid:0) h (2) ⊲ c (2) (cid:1) , ǫ ( h ⊲ c ) = ε ( h ) ǫ ( c )hold for all h ∈ H and c ∈ C , with the notation δ ( c ) = c (1) ⊗ c (2) .The Hopf algebra H is itself an H -module algebra with respect to the left adjoint action h ⊲ ad g = ad h ( g ) := h (1) g S ( h (2) ) for h, g ∈ H . We recall next how to produce new Hopfalgebra structures on H by deforming the original one using two-cocycles of H . Definition 1.3.
An element F ∈ H ⊗ H is called a Drinfel’d twist element for H if ithas the following properties: (1) F is invertible; (2) F is counital: (id ⊗ ε )( F ) = ( ε ⊗ id)( F ) = 1 ; and (3) F obeys the cocycle condition: (1 ⊗ F ) (id ⊗ ∆)( F ) = ( F ⊗
1) (∆ ⊗ id)( F ) . In the category of left H -modules, a Drinfel’d twist in the Hopf algebra H generates adeformation of the product µ : A ⊗ A → A on every algebra object A . Similarly, the twistcan be used to deform the coproduct δ : C → C ⊗ C on every coalgebra object C . Theresults are H -module algebras or coalgebras respectively. In the present paper we shallconcentrate on the algebra cases. Theorem 1.4. (1)
A Drinfel’d twist element F = F (1) ⊗ F (2) ∈ H ⊗ H defines atwisted Hopf algebra structure H F with the same multiplication and counit as H ,but with new coproduct and antipode given for h ∈ H by (1.5) ∆ F ( h ) = F ∆( h ) F − , S F ( h ) = U F S ( h ) U − F LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 5 where U F = F (1) S (cid:0) F (2) (cid:1) . (2) If A is a left H -module algebra, the deformed product (1.6) a ⋆ F b := µ (cid:0) F − ⊲ ( a ⊗ b ) (cid:1) for a, b ∈ A makes A F = ( A, ⋆ F ) into a left H F -module algebra with respect to thesame action of H . There are analogous results for right actions. If A is an H -module algebra, then thecollection of left H -invariant elements H A forms an ideal of A in which the productassociated to a Drinfel’d twist for H by Theorem 1.4 coincides with the undeformedproduct [11].In general, the deformation of the H -module algebra structure of H itself providedby Theorem 1.4 need not be compatible with the Hopf algebra structure of H , becausegenerically one has ∆( h⋆ F g ) = ∆( h ) ⋆ F ∆( g ). In order to obtain a deformation of both theunderlying variety of H and the quantum group associated to H , we use a dual frameworkdealing with coactions. Definition 1.7.
Let
Φ : A → A ⊗ H , Φ( a ) = a (0) ⊗ a (1) be a right coaction of the Hopfalgebra H on a unital algebra A with product µ : A ⊗ A → A . The coaction is said to be covariant if the linear map Φ is a unital algebra morphism, Φ (cid:0) µ ( a ⊗ b ) (cid:1) = µ (cid:0) a (0) ⊗ b (0) (cid:1) ⊗ a (1) b (1) , Φ(1) = 1 ⊗ , (1.8) for all a, b ∈ A . In this case A is called a right H -comodule algebra . The initial coproduct ∆ of H defines a right coaction of the Hopf algebra H on itself,and it makes H into an H -comodule algebra. For dually paired Hopf algebras H and F , with nondegenerate pairing h− , −i : H × F → C , to a right coaction of F on (analgebra, a coalgebra, etc.) A there corresponds a left action of H on A . Thus, e.g., a right F -comodule algebra is a left H -module algebra. The left regular action of H on F : h ⊲ α = α (1) (cid:10) h , α (2) (cid:11) (1.9)for h ∈ H and α ∈ F , is a covariant action which makes F into a left H -module algebra. Definition 1.10.
A linear map F ∨ : H ⊗ H → C is called a dual Drinfel’d twist element for H if it has the following properties for all f, g, h ∈ H : (1) F ∨ is convolution-invertible: There exists a linear map F ∨ − : H ⊗ H → C suchthat F ∨ (cid:0) f (1) ⊗ g (1) (cid:1) F ∨ − (cid:0) f (2) ⊗ g (2) (cid:1) = F ∨ − (cid:0) f (1) ⊗ g (1) (cid:1) F ∨ (cid:0) f (2) ⊗ g (2) (cid:1) = ε ( f ) ε ( g ) ;(2) F ∨ is unital: F ∨ ( f ⊗
1) = F ∨ (1 ⊗ f ) = ε ( f ) ; and (3) F ∨ obeys the cycle condition: F ∨ (cid:0) f (1) ⊗ g (1) (cid:1) F ∨ (cid:0) f (2) g (2) ⊗ h (cid:1) = F ∨ (cid:0) g (1) ⊗ h (1) (cid:1) F ∨ (cid:0) f ⊗ g (2) h (2) (cid:1) . Theorem 1.11. (1)
A dual Drinfel’d twist element F ∨ for H defines a twisted Hopfalgebra structure H F ∨ with the same coproduct and counit as H , but with newalgebra structure and antipode given for g, h ∈ H by g × F ∨ h = F ∨ (cid:0) g (1) ⊗ h (1) (cid:1) (cid:0) g (2) · h (2) (cid:1) F ∨ − (cid:0) g (3) ⊗ h (3) (cid:1) ,S F ∨ ( g ) = U F ∨ (cid:0) g (1) (cid:1) S (cid:0) g (2) (cid:1) U F ∨ − (cid:0) g (3) (cid:1) (1.12) where U F ∨ ( g ) = F ∨ ( g (1) ⊗ S ( g (2) )) . LUCIO CIRIO, GIOVANNI LANDI, AND RICHARD J. SZABO (2) If A is a right H -comodule algebra, the deformed product (1.13) a ⋆ F ∨ b := µ (cid:0) a (0) ⊗ b (0) (cid:1) F ∨ − (cid:0) a (1) ⊗ b (1) (cid:1) for a, b ∈ A makes A F ∨ = ( A, ⋆ F ∨ ) into a right H F ∨ -comodule algebra. The proof of Theorem 1.11 can be found in [36]. Again, there is an analogous resultfor left coactions. If the two Hopf algebras H and F are dually paired, then to any twistelement F = F (1) ⊗ F (2) ∈ H ⊗ H there is a canonically associated dual twist element F ∨ : F ⊗ F → C defined by(1.14) F ∨ ( α ⊗ β ) = (cid:10) F , α ⊗ β (cid:11) := (cid:10) F (1) , α (cid:11) (cid:10) F (2) , β (cid:11) for α, β ∈ F . Every time an H -module algebra is also an F -comodule algebra (i.e. theaction determines a coaction of the dual Hopf algebra) any deformation obtained usingthe twist F of H can be equivalently described using the dual twist F ∨ of F definedby (1.14). However, the dual twist element depends only on the pairing, without anyreference to an action of F .In our main examples, we will use this Hopf algebraic approach as a means of deformingthe algebra of functions on a variety acted upon by a group. For the purpose of thepresent paper, we consider algebraic varieties and their polynomial coordinate algebras.However, with additional structure, the same constructions apply to algebras of functionson topological spaces, differentiable manifolds, and the like. Given a Lie group G , theenveloping algebra U ( g ) of the Lie algebra g of G is a Hopf algebra over C . This Hopfalgebra has coproduct given on primitive elements x ∈ g by ∆( x ) = 1 ⊗ x + x ⊗
1, counitby ε ( x ) = 0, and antipode by S ( x ) = − x . The adjoint action of H on itself extends theusual adjoint action of Lie algebra elements x ∈ g . When the group G acts on a variety X the algebra of functions on X is a U ( g )-module algebra.Let F = Fun( G ) be the algebra generated by commuting matrix elements g ij in finite-dimensional representations of G , with i, j = 1 , . . . , dim( G ) . Let g ij ( P ) ∈ C denote theirevaluations on group elements P ∈ G . The commutative algebra F is a Hopf algebra withcoproduct given by ∆ ∨ ( g ij ) = P k g ik ⊗ g kj , i.e. the transpose of the map given by matrixmultiplication, antipode S ∨ ( g ij )( P ) = g ij ( P − ) for P ∈ G , and counit ε ∨ ( g ij ) = δ ij . TheHopf algebra F is dual to the enveloping Hopf algebra H , with dual pairing h h, g i = h ( g )(1)the evaluation at the identity of the bi-invariant differential operator on G associated to h ∈ H acting on the function g ∈ F . When the group G acts on a space X , the algebraof functions on X is a Fun( G )-comodule algebra.As we will consider deformations depending on some (matrix of) complex parameters θ ,we will rather need to work in the quantum enveloping algebra H = U ( g )[[ θ ]], the algebraof formal power series in θ over U ( g ).1.2. Braided monoidal categories of Hopf-module algebras.
A useful unifyingframework in which to analyse our noncommutative deformations is provided by braidedmonoidal categories, wherein the noncommutativity is completely encoded in a braidingof a category whose objects are commutative varieties.
Definition 1.15. A braided monoidal (or quasitensor) category ( C , ⊗ , Ψ) is a monoidalcategory ( C , ⊗ ) with a natural equivalence between the two functors ⊗ , ⊗ op : C × C → C given by functorial isomorphisms (1.16) Ψ V,W : V ⊗ W −→ W ⊗ V LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 7 for all objects
V, W of C , obeying hexagon relations which express compatibility of Ψ withthe associativity structure of the tensor product ⊗ (see e.g. [36, Fig. 9.4] ). The operators(1.16) are called braiding morphisms . If in addition Ψ = id , the category ( C , ⊗ , Ψ) issaid to be a symmetric (or tensor) category . Our interest in braided monoidal categories stems from the category of Hopf-modulesintroduced in § H M the (sub)category of Hopf-module algebras.An algebra map A σ −→ B is a morphism of the category H M if and only if it fits into thecommutative diagram H ⊗ A id ⊗ σ / / (cid:15) (cid:15) H ⊗ B (cid:15) (cid:15) A σ / / B where the vertical arrows are the H -actions, i.e. σ is an H -equivariant map.On the tensor product of two Hopf-module algebras A ⊗ B we will consider the actionof the Hopf algebra H defined by(1.17) ∆( h ) ⊲ ( a ⊗ b ) = (cid:0) h (1) ⊲ a (cid:1) ⊗ (cid:0) h (2) ⊲ b (cid:1) for all a ∈ A , b ∈ B , and h ∈ H . Both the algebra structure of A ⊗ B and the braidingin the category are determined by a quasitriangular structure of H , i.e. an invertible R -matrix R = R (1) ⊗ R (2) in H ⊗ H obeying τ ◦ ∆( h ) = R ∆( h ) R − and (∆ ⊗ id) R = R (1) ⊗ R (1) ⊗ (cid:0) R (2) (cid:1) , (id ⊗ ∆) R = (cid:0) R (1) (cid:1) ⊗ R (2) ⊗ R (2) where τ : H ⊗ H → H ⊗ H is the flip map which interchanges the two factors of H .See [36] for proofs of the following results. Proposition 1.18. If ( H , R ) is a quasitriangular Hopf algebra, then the category of left H -module algebras H M is a braided monoidal category with braiding morphism (1.19) Ψ A,B ( a ⊗ b ) = (cid:0) R (2) ⊲ b (cid:1) ⊗ (cid:0) R (1) ⊲ a (cid:1) for all a ∈ A and b ∈ B . When the Hopf algebra is triangular, i.e. R − = R (2) ⊗ R (1) , or τ ◦ R − = R , thecategory H M is symmetric, i.e. the braiding in (1.19) squares to the identity: Ψ = id.If in addition H is cocommutative, like the classical enveloping algebras U ( g ), then the R -matrix can be taken to be R = 1 ⊗ τ , where τ A,B : A ⊗ B → B ⊗ A interchanges the factors as τ A,B ( a ⊗ b ) = b ⊗ a .In this case, the ordinary tensor algebra structure of A ⊗ B is compatible with the actionof H , i.e. ( a ⊗ b ) · ( a ⊗ b ) := ( a a ) ⊗ ( b b ). In the general case, the algebra structureon A ⊗ B which is acted upon covariantly by H depends on the quasitriangular structure. Proposition 1.20. If ( H , R ) is a quasitriangular Hopf algebra and A, B are H -modulealgebras, then the braided tensor product A b ⊗ B is the vector space A ⊗ B endowed withthe product (1.21) ( a ⊗ b ) · ( a ⊗ b ) := ( a ⊗
1) Ψ
B,A ( b ⊗ a ) (1 ⊗ b ) = a (cid:0) R (2) ⊲ a (cid:1) ⊗ (cid:0) R (1) ⊲ b (cid:1) b . With this product A b ⊗ B is an H -module algebra. LUCIO CIRIO, GIOVANNI LANDI, AND RICHARD J. SZABO
In a braided monoidal category of algebras it is natural to relate the notion of commu-tativity to the braiding morphism. The usual definition of commutativity of an algebra A may be expressed as the invariance of the multiplication µ : A ⊗ A → A under theflip morphism τ A,A : A ⊗ A → A ⊗ A , i.e. µ ◦ τ A,A = µ . In a braided monoidal category( C , ⊗ , Ψ) it is natural to replace τ , which is not necessarily a morphism in the category,by the braiding morphism Ψ. This motivates the following definition. Definition 1.22.
An algebra object A in the category H M is braided commutative if itsmultiplication map µ : A ⊗ A → A is invariant with respect to the braiding morphism Ψ A,A : A ⊗ A → A ⊗ A as (1.23) µ ◦ Ψ A,A = µ or a b = (cid:0) R (2) ⊲ b (cid:1) (cid:0) R (1) ⊲ a (cid:1) , for every a, b ∈ A . If A is an object in the category H M , and A F is the twisted Hopf-module algebradefined by a Drinfel’d twist element F = F (1) ⊗ F (2) ∈ H ⊗ H as in Theorem 1.4, thenthe braiding morphism Ψ F and tensor product b ⊗ F on the category H F M are defined as inPropositions 1.18 and 1.20 with respect to the twist deformed quasitriangular structure R F = (cid:0) F (2) ⊗ F (1) (cid:1) R F − . There is a natural equivalence between braided monoidal categories of left Hopf-modulealgebras defined by the functor F F : (cid:0) H M , b ⊗ , Ψ (cid:1) −→ (cid:0) H F M , b ⊗ F , Ψ F (cid:1) which acts as the identity on objects and morphisms of H M [28, Thm. XV.3.5], thenontriviality being contained in what happens to the braided monoidal structure. Thisfunctorial isomorphism implies that any H -covariant construction in the category H M of H -module algebras has a twisted analog in the category H F M of H F -module algebras.1.3. Ore localization.
Given a commutative unital algebra A over C which is a domain,one usually localizes with respect to a subset S ⊂ A which is closed under multiplication.For noncommutative algebras, the existence of the localization is guaranteed, for example,by an additional Ore condition on the subset S . Full details on the construction may befound in standard textbooks (see e.g. [30, § A [ S − ] = S − · A as a set of equivalence classes in S × A , regarded as“fractions” ( s, a ) = s − a , endowed with a suitable algebra structure. Geometrically, thelocalization A ֒ → A [ S − ] corresponds to deleting the locus specified by the vanishing ofelements of S in the variety dual to A .2. Algebraic torus deformations
This paper systematically combines constructions from toric geometry and the theoryof isospectral deformations. Isospectral deformations produce noncommutative geome-tries by using the isometric action of a real n -dimensional torus T n on a Riemannian(spin) manifold and its noncommutative deformation T nθ [13, 14]. We will extend theseconstructions to actions of the algebraic torus ( C × ) n , in order to obtain an analogousdeformation of toric algebraic varieties. In this section we spell out the various algebraicconstructions behind these deformations. Throughout this paper an implicit sum overrepeated upper and lower indices is always understood. LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 9
The noncommutative algebraic torus.
The definition of the noncommutativereal torus essentially relies on harmonic analysis and a choice of homomorphism of groupsbetween the space of characters and the torus itself. This procedure may be easily ex-tended to a generic locally compact abelian Lie group G . We are ultimately interestedin the case G = ( C × ) n . Let A ( G ) ⊂ C ∞ ( G ) be the commutative algebra of a class offunctions on G with a suitable growth condition “at infinity”. The Fourier transform on G provides a decomposition of every function f ∈ A ( G ) over a basis of functions { χ p } p ∈ b G labelled by the group of characters of G , i.e. its Pontrjagin dual b G = Hom C ( G, C × ). Forevery p ∈ b G , we set χ p to be the function on G defined by χ p ( g ) = h p, g i , for g ∈ G ,where h− , −i : b G × G → C × is the pairing between G and b G . This defines the Fouriercomponents b f : b G → C of f ∈ A ( G ) as b f ( p ) = Z G f ( g ) χ p ( g ) d g where p ∈ b G and d g denotes the bi-invariant Haar measure of G . Using L -orthonormalityof the characters, the inverse Fourier transformation is given by f ( g ) = Z b G b f ( p ) χ p ( g ) d p with d p the bi-invariant Haar measure of b G .In order to define a noncommutative associative product on A ( G ) it is enough to de-scribe it on the G -eigenbasis { χ p } p ∈ b G and then extend it to A ( G ) via the Fourier transform.Given a homomorphism of groups Θ : b G → G , we set χ p ⋆ Θ χ q := χ p · (cid:0) Θ( p ) ⊲ χ q (cid:1) = (cid:10) q , Θ( p ) (cid:11) χ p + q for p, q ∈ b G . Here the symbol ⊲ denotes the (left) action of the group G on A ( G ). Usingthe Fourier transformation this extends to a product on functions f, f ′ ∈ A ( G ):( f ⋆ Θ f ′ )( g ) = Z b G × b G b f ( p ) b f ′ ( q ) χ p + q ( g ) (cid:10) q , Θ( p ) (cid:11) d p d q . The vector space A ( G ) with this product defines a noncommutative associative algebradenoted A Θ ( G ). Example 2.1.
Let G = V be a locally compact abelian vector Lie group of (real) dimension n . Then b G ∼ = V ∗ = Hom R ( V, R ) . By choosing an R -basis of V , there are isomorphisms V ∼ = R n and V ∗ ∼ = R n . In this case the homomorphism Θ may be taken to be a linearendomorphism on V defined by a real skew-symmetric n × n matrix θ ∈ V V , and we getthe Moyal product on R n . Example 2.2.
Let G = V /L with V as in Example 2.1 and L ⊂ V a lattice of maximalrank n . Then b G ∼ = L ∗ = Hom Z ( L, Z ) . Upon choosing a Z -basis for L , there are isomor-phisms L ∼ = Z n , L ∗ ∼ = Z n and G ∼ = T n . In this case we put Θ( p ) = exp( i2 θ · p ) for p ∈ L ∗ with θ again a real skew-symmetric n × n matrix, and we obtain the noncommutativetorus T nθ . When G = T is an algebraic torus of (complex) dimension n over C , we proceed asfollows. Let L be a lattice of rank n . Let L ∗ = Hom Z ( L, Z ) be the dual lattice and denotethe canonical pairing between the lattices by h− , −i : L ∗ × L → Z . The dual lattice is thegroup of characters { χ p } p ∈ L ∗ which provide a basis of T -eigenfunctions on the algebraic torus T = L ⊗ Z C × , i.e. one has b G = L ∗ ∼ = Hom C ( T, C × ). Thus L ∼ = Hom C ( C × , T )is the lattice of one-parameter subgroups of T . Pick a Z -basis e , . . . , e n of L , withcorresponding dual basis e ∗ , . . . , e ∗ n for L ∗ . Then there is an isomorphism T ∼ = ( C × ) n . Set p = P i p i e ∗ i ∈ L ∗ and t = P i e i ⊗ t i ∈ T . Then the characters are given by χ p ( t ) = t p := t p · · · t p n n . (2.3)The Fourier components in this case are given by(2.4) b f ( p ) = Z T f ( t ) t p d × t with respect to the T -invariant measure d × t = (d t d t ) / | t | . Using the discrete measure onthe Pontrjagin dual b T = L ∗ , every function f : T → C with suitable growth “at infinity”can be written in terms of its Fourier components via the Laurent power series expansion f ( t ) = X p ∈ L ∗ b f ( p ) t p . The space C χ p is the eigenspace for the T -action corresponding to the character givenby h p, −i : T → C × in Hom C ( T, C × ) ∼ = L ∗ . Thus the L ∗ -grading gives precisely theeigenspace decompositions of algebraic objects, dual to T -invariant geometric objects.The homomorphism Θ : L ∗ → T is defined by a complex skew-symmetric n × n matrix θ via the usual relation Θ( p ) = exp( i2 θ · p ). The real part of θ again describes thedeformation of the compact real torus T n ⊂ ( C × ) n , while the imaginary part applies tothe “dilatation” part given by ( R + ) n , according to the polar decomposition( C × ) n = ( R + ) n × T n ∼ = R n × T n . In this way we may think of the deformation of ( C × ) n as a simultaneous and independentdeformation of R n and T n as given in Example 2.1 and Example 2.2. However, for concretecomputations this prescription is not very useful, because the Moyal deformation affectslog | t | for elements t ∈ ( C × ) n and thus leads to rather involved commutation relations.The transformation (2.4) with this decomposition of ( C × ) n is the Fourier transform withrespect to the real torus and the Mellin transform with respect to ( R + ) n .As an algebraic variety, the torus ( C × ) n is dual to the Laurent polynomial algebra in n variables C [ t ± , . . . , t ± n ]. The monomials in this coordinate algebra are the functionslabelled by the characters χ p ( t ) = t p that we introduced in (2.3). The deformation of theproduct between such functions may be written explicitly as(2.5) z p ⋆ θ w q = exp (cid:0) i2 X ij p i θ ij q j (cid:1) z p · w q where z = P i e i ⊗ z i , w = P i e i ⊗ w i ∈ T , and p, q ∈ L ∗ . The product (2.5) is extendedlinearly to all of C [ t ± , . . . , t ± n ]. Definition 2.6.
The vector space A ( T ) = C [ t ± , . . . , t ± n ] with the product ⋆ θ is called the quantum Laurent algebra A θ ( T ) = C θ [ t ± , . . . , t ± n ] and its elements are called quantumLaurent polynomials . It is dual to a noncommutative variety denoted ( C × θ ) n . Remember that θ is a complex matrix. As we show explicitly in § T on itself extends to an action on ( C × θ ) n . In particular, T acts by algebraautomorphisms with respect to the product ⋆ θ . LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 11
Twisted toric actions.
Using the Hopf algebraic approach described in § H of the algebraic torus group T . This is simply the polynomial algebra in n commuting elements H i , the infinitesimal generators of the group. In fact we rather needformal power series in some parameters θ , but we will abuse notation by simply writing H = H [[ θ ]].As twisting element we take the abelian Drinfel’d twist F = F θ := exp (cid:0) − i2 X ij θ ij H i ⊗ H j (cid:1) . (2.7)The infinitesimal action of T on characters is given by H i ⊲χ p = h p, e i i χ p for p ∈ L ∗ . Thenformula (1.6) for a = z p and b = w q monomials in the algebra A ( T ) = C [ t ± , . . . , t ± n ]coincides exactly with (2.5).On the other hand, in this case H = H θ := H F θ as Hopf algebras. Since the Lie algebraof T is abelian, the coproduct ∆ θ := ∆ F θ of H θ computed from (1.5) is unaffected by thedeformation and is given on generators by∆ θ ( H i ) = ∆( H i ) = H i ⊗ ⊗ H i . The antipode defined in (1.5) is also unaffected by the deformation, S F θ = S , as is alwaysthe case with Drinfel’d twist elements of the form (2.7) [11]. Indeed, one shows that theelement U F θ = F (1) θ S (cid:0) F (2) θ (cid:1) in this case is the identity by computing its n -th order termfor any n > θ . This term is proportional to X θ i j · · · θ i n j n H i · · · H i n S ( H j · · · H j n )= X ( − n θ i j · · · θ i n j n H i · · · H i n H j · · · H j n = 0 , and the vanishing follows from θ ij = − θ ji and H i H j = H j H i for each i, j = 1 , . . . , n .Thus H = H θ as a Hopf algebra, and the deformed algebra A θ ( T ) is also an H -modulealgebra with respect to the same (undeformed) toric action. In this case the deformationof the triangular structure R = 1 ⊗ H by the twist element (2.7) gives the twisted R -matrix R F θ = F − θ (1 ⊗ F − θ = F − θ , (2.8)so that the twisted enveloping algebra H θ is triangular, τ ◦ R − F θ = R F θ , but no longercocommutative, resulting in a nontrivial, albeit symmetric, braiding in the category H θ M .The coproduct on the algebra of functions A ( T ) on the torus T is given on characterelements χ p : T → C × , p ∈ L ∗ , by∆ ∨ ( χ p ) = χ p ⊗ χ p , (2.9)while the antipode is the inverse S ∨ ( χ p ) = χ − p in C × . For this undeformed case, the dualpairing between generators H i of T and the character algebra A ( T ) is provided by theevaluation of the Lie derivative L H i with respect to the invariant vector field associatedto H i ; in particular for the characters one finds: h H i , χ p i := L H i ( χ p )(1) = p i . Using the Drinfel’d twist (2.7) and its dual twist element F ∨ = F θ defined by (1.14),from Theorem 1.11 we obtain the twisted Hopf algebra Fun θ ( T ) with deformed product on characters given by χ p × θ χ q = F θ ( χ p ⊗ χ q ) ( χ p · χ q ) F θ − ( χ p ⊗ χ q )= (cid:10) F θ , χ p ⊗ χ q (cid:11) ( χ p · χ q ) (cid:10) F − θ , χ p ⊗ χ q (cid:11) = exp (cid:0) − i2 X ij p i θ ij q j (cid:1) ( χ p · χ q ) exp (cid:0) i2 X ij p i θ ij q j (cid:1) = χ p · χ q , which coincides with the undeformed product on the character algebra. The antipodeis also unaffected by the deformation, S F θ ∨ ( χ p ) = S ∨ ( χ p ), as can be checked directly byusing (1.12), or by using duality and the fact that the antipode in H θ is unchanged by thedeformation in this case. Thus the quantum group symmetry underlying the quantumLaurent algebra also coincides with the classical (undeformed) toric symmetry.2.3. The noncommutative variety GL θ ( n ) . Some of our constructions will rely ona noncommutative ( C × ) n deformation of the general linear group GL( n ) over C . Thedeformation is realized using the action of the algebraic torus by a (dual) Drinfel’d twiston the algebra of functions F n := Fun(GL( n )) on GL( n ), as described in § n × n skew-symmetric complex matrix θ . The Hopf algebra F n is dual tothe enveloping Hopf algebra H n = U ( gl ( n )). The left regular action of H n on F n , definedin general in (1.9), is a covariant action which makes F n into a left H n -module algebra.There is an analogous right regular covariant action of H n on F n which makes F n into aright H n -module algebra.The deformation of GL( n ) which we use in the following is the only one which deforms F n as a Hopf algebra, and also as an H n -bimodule algebra. Within the context of § § F n as a left H n -module algebra via eitherthe left regular action or the left adjoint action, or by their right acting versions. Forour purposes this is undesirable as it introduces an asymmetry between row and columnoperations on matrix elements considered in the following. The deformation we use iscompatible with the Hopf algebra structure, which is instrumental in some of our laterconstructions of differential forms, and moreover it is the one that is compatible with theembeddings we will consider into noncommutative projective spaces.We first twist the standard Hopf algebra structure of H n to obtain H nθ , using thetwist element (2.7), where the H i are the generators of the Lie algebra of the diagonallyembedded maximal torus ( C × ) n ⊂ GL( n ). Let { E ij } i,j =1 ,...,n be the standard basis of gl ( n ), with matrix elements ( E ij ) kl = δ ik δ jl and H i = E ii , and the commutation relations[ E ij , E kl ] = E il δ jk − E kj δ il , [ H k , E ij ] = E ij (cid:0) δ ki − δ kj (cid:1) . These are used to compute the twisted coproduct ∆ θ := ∆ F θ as in (1.5). A straightforwardcomputation, along the lines of [11], yields∆ θ ( E ij ) = E ij ⊗ λ − ij + λ ij ⊗ E ij with the group-like element λ ij defined by λ ij = exp (cid:0) i2 X kl θ kl ( δ ik − δ jk ) H l (cid:1) . As expected, the generators H i of the twist have undeformed coproduct.By the general discussion of § F n which preservesthe quantum group structure, we use the Drinfel’d twist F ∨ = F θ defined as in (1.14), LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 13 which is dual to the initial twist (2.7). As in § h H k , g ij i = H k ( g ij )(1) = L H k ( g ij )(1) = g ij ( H k ) = δ ik δ jk , with the generators g ij of the algebra F n . Using Theorem (1.11) we then obtain thetwisted Hopf algebra F θn still generated by elements g ij , but now with noncommutativerelations between them given by g ij × θ g kl = n X m,p,r,s =1 F θ ( g ir ⊗ g ks ) ( g rm · g sp ) F θ − ( g mj ⊗ g pl )= n X m,p,r,s =1 (cid:10) F θ , g ir ⊗ g ks (cid:11) ( g rm · g sp ) (cid:10) F − θ , g mj ⊗ g pl (cid:11) = n X m,p,r,s =1 q ki δ ir δ ks ( g rm · g sp ) q mp δ mj δ pl = q ki q jl ( g ij · g kl ) , (2.10)where q ij := exp (cid:0) i2 θ ij (cid:1) . Introducing coefficients(2.11) Q ij ; kl = q ki q jl = q − ik q jl , Q ij ; kl = q ki q jl we write the commutation rule for the deformed product as(2.12) g ij × θ g kl = Q ij ; kl g kl × θ g ij . As usual, the coproduct ∆ ∨ and the counit ε ∨ are left unchanged. On the other hand,the commutativity of the generators H i ’s implies, as in § S F θ ∨ ( g ij ) = S ∨ ( g ij ) is unaltered as well. Definition 2.13.
The noncommutative Hopf algebra F θn = ( F n , × θ , ∆ ∨ , ε ∨ , S ∨ ) is calledthe algebraic torus deformation quantum group of GL( n ) . It is dual to a noncommutativevariety denoted GL θ ( n ) . A proper definition of the variety GL θ ( n ) involves the notion of noncommutative de-terminant; we will return to this point in detail in § Remark 2.14.
This formalism may also be adapted to define noncommutative rectangular d × n matrix algebras, with d < n , as the C -subalgebra of F θn generated by g ij with i ≤ d .There is a C -algebra retraction of F θn onto this subalgebra whose kernel is generated by g ij with i > d , and hence the subalgebra is isomorphic to F θn / h g ij i i>d . In the sequel we will drop the product notation × θ for simplicity. The Hopf algebra F θn is dually paired with H nθ under the same pairing which links the untwisted algebras. The left H nθ -module structure of F θn is given by (1.9) and is easily computed to get E ij ⊲ g kl = g (1) kl (cid:10) E ij , g (2) kl (cid:11) = n X m =1 g km (cid:10) E ij , g ml (cid:11) = n X m =1 g km g ml ( E ij )= n X m =1 g km δ mi δ jl = δ jl g ki . Quantum determinants.
The coordinate algebra of the noncommutative varietyGL θ ( n ) should be properly defined as the Ore localization of the noncommutative algebragenerated by arbitrary matrix units with respect to an invertible and permutable elementdet θ , the determinant element. If we consider the elements at the crossings of rows i, j and columns k, l of a given matrix, then the determinant of this 2 × g ik g jl − g jk g il . In order to get a well-defined element of F θn , we put infront of every monomial in the matrix elements g ij a suitable element of the deformationmatrix. For example, in front of g ik g jl we write Q jl ; ik , so that the determinant of theminor above is Q jl ; ik g ik g jl − Q il ; jk g jk g il . This is well-defined because if we choose towrite the determinant using a different ordering of the monomials, then we get the sameelement of F θn thanks to the relations (2.12) which imply Q jl ; ik g ik g jl = Q ik ; jl g jl g ik . For a generic n × n matrix we can define the determinant by adapting the usual Laplaceexpansion in minors, with respect to either rows or columns, or the Leibniz formula whichexpresses it as a linear combination of products Q i g i σ ( i ) or Q i g σ ( i ) i as σ runs throughthe symmetric group S n weighted by its sign. Using the above rule for the coefficientsin front of every monomial to pull out a factor Q lj ; ki for every pair g ki g lj appearing in Q i g i σ ( i ) , we definedet θ := X σ ∈ S n sgn( σ ) (cid:16) n − Y j =1 n − j Y i =1 Q i + j σ ( i + j ) ; j σ ( j ) (cid:17) g σ (1) · · · g n σ ( n ) = X σ ∈ S n sgn( σ ) (cid:16) n − Y j =1 n − j Y i =1 Q σ ( i + j ) i + j ; σ ( j ) j (cid:17) g σ (1) 1 · · · g σ ( n ) n . (2.15)This element corresponds to a mapping of S n into the braid group B n on n strands, aswe shall see below.The formula (2.15) may be rewritten in a more succinct way by using the fact that theclassical Leibniz formula can be expressed in terms of the totally antisymmetric Levi–Civita symbol ǫ as ǫ i ··· i n g i · · · g n i n = 1 n ! ǫ j ··· j n ǫ i ··· i n g j i · · · g j n i n . In the noncommutative case, we introduce a θ -deformed Levi–Civita symbol ǫ θ whichsatisfies braided antisymmetry rules. Since the row and column indices in (2.11) and LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 15 (2.12) behave differently, we actually require two different symbols ǫ ( r ) θ , which refers torow indices, and ǫ ( c ) θ , which refers to column indices. In this way we may absorb the Q -dependent coefficients of (2.15), consistently with the braided antisymmetry. Explicitly, ǫ i ··· i n ( c ) θ = sgn( i · · · i n ) n − Y k =1 n − k Y s =1 Q s + k i s + k ; k i k ,ǫ j ··· j n ( r ) θ = sgn( j · · · j n ) n − Y k =1 n − k Y s =1 Q j s + k s + k ; j k k . They obey the alternating rules ǫ j ··· j α ··· j β ··· j n ( c ) θ = − q j β j α ǫ j ··· j β ··· j α ··· j n ( c ) θ ,ǫ i ··· i α ··· i β ··· i n ( r ) θ = − q i α i β ǫ i ··· i β ··· i α ··· i n ( r ) θ . (2.16)For example, for n = 2 we have ǫ
12 ( c ) θ = 1 and ǫ
21 ( c ) θ = − q , and the sole braidedantisymmetry relation ǫ
12 ( c ) θ = − q ǫ
21 ( c ) θ is satisfied. Similarly, we have ǫ
12 ( r ) θ = 1 and ǫ
21 ( r ) θ = − q − . In this sense ǫ ( r ) θ may be thought of as the inverse of the symbol ǫ ( c ) θ .Clearly, we are referring to the ordered multi-index J = (12). In computing minors withunordered indices, like J = (21), we get the extra sign from the permutation. Definition 2.17.
The quantum determinant is the element of F θn given by (2.18) det θ = 1 n ! ǫ i ··· i n ( r ) θ ǫ j ··· j n ( c ) θ g i j · · · g i n j n . Theorem 2.19.
The element det θ is a T -eigenvector which is left and right permutablein F θn . Proof : The first statement follows from an elementary calculation using the coproduct∆ θ ( H i ) of § C × ) n -action H i ⊲ g kl = δ il g kl . For the second statement, note thatsince every monomial occuring in det θ is of the form Q i g i σ ( i ) for some permutation σ in S n , every row and column index appears exactly once. By (2.12), commuting a genericelement g kl from right to left in such a monomial picks up the coefficient n Y i =1 Q i σ ( i ) ; kl = n Y i =1 q ki q σ ( i ) l . It follows that (det θ ) g kl = (cid:16) n Y i =1 Q ii ; kl (cid:17) g kl (det θ )for all k, l = 1 , . . . , n , and hence (det θ ) F θn = F θn (det θ ). (cid:4) Corollary 2.20.
The set of non-negative powers of det θ is a left and right denominatorset in F θn . Corollary 2.21.
The element det θ is central in F θn if and only if n X k =1 θ ki = n X k =1 θ kj (mod 2 π ) for all i, j = 1 , . . . , n . Although our deformation of the general linear group lies in the class of deformationsconsidered in [3], our definition of quantum determinant is different, though it satisfies thesame formal properties. The element (2.18) originates from the braiding of the categoryof Hopf-module algebras described in § H nθ dual to F θn . The θ -deformed exterior algebra of degree d for anelement V in the category H nθ M of H nθ -module algebras is defined as(2.22) V dθ V := V ⊗ d (cid:14) (cid:10) v ⊗ v + Ψ θ ( v ⊗ v ) (cid:11) v ,v ∈ V , where Ψ θ := Ψ F θ = τ ◦ F − θ is the braiding morphism of the category. For θ = 0 werecover the usual flip operator Ψ = τ and the exterior algebra V d V . For θ = 0 we obtaina braided skew-symmetric algebra V dθ V , which is spanned by the collection of minors oforder d ≤ n in elements of V when n is the number of generators of V . For this, considertwo multi-indices I = ( i · · · i d ) and J = ( j · · · j d ) which label the rows and columns of agiven minor, and define the determinant Λ IJ of this sub-matrix as(2.23) Λ IJ = 1 d ! X ǫ i ··· i d ( r ) θ ǫ j ··· j d ( c ) θ g i j · · · g i d j d where the symbols ǫ θ satisfy alternating rules derived from (2.22). Here the H nθ -modulestructure of GL( n ) ∼ = GL( V ) is induced from the H nθ -module structure of V and of its dual V ∗ . When this H nθ -module structure induces the noncommutative product (2.12) amongthe entries of elements of GL( V ), the alternating properties of the deformed Levi–Civitasymbols coincide with those of (2.16).In the classical case, there is a Laplace expansion for the above determinant in terms oflower order minors. If I is a row multi-index, J a column multi-index with | I | = | J | = d wewrite I α = I \ { i α } = ( i α , . . . , i αd − ) and J α = J \ { j α } = ( j α , . . . , j αd − ) for α ∈ (1 , . . . , d ).The classical Laplace expansion with respect to the k -th row of the determinant Λ I ; J isthen written as:(2.24) Λ IJ = d X α =1 ǫ i k ∪ I k ǫ j α ∪ J α g kα Λ I k ; J α . In the deformed case, we need to take into account the Q -coefficients associated to each g kα standing in front of Λ I k ; J α . Since Λ I k ; J α is a product of elements g i kβ j αβ ′ with i kβ ∈ I k and j αβ ′ ∈ J α and the coefficient does not depend on the order of the elements, we haveas noncommutative version of (2.24) the following(2.25) Λ IJ = d X α =1 d − Y β =1 ( − k + α Q i kβ j αβ ; kα g kα Λ I k J α = d X α =1 d − Y β =1 ( − k + α Q kα ; i kβ j αβ Λ I k J α g kα . Similarly, if we expand with respect to the k -th column we have(2.26) Λ IJ = d X α =1 d − Y β =1 ( − k + α Q i kβ j αβ ; αk g αk Λ I α J k = d X α =1 d − Y β =1 ( − k + α Q αk ; i kβ j αβ Λ I α J k g αk . Remark 2.27.
Our definition (2.22) of exterior algebra is equivalent to the standarddefinition of an exterior algebra in a braided monoidal category [41] (see also [29, § ),written in the symmetric case. In this construction, one takes the quotient of the tensoralgebra by the kernel of the antisymmetrizer. A slightly different, but somewhat simpler,definition involves the quotient by the ideal generated by the kernel of the antisymmetrizer LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 17 in degree two, which coincides with the morphism id − Ψ θ [29, p. 512] . This agrees withour definition (2.22), since we work in a symmetric category with Ψ θ = id , and so thekernel of the antisymmetrizer id − Ψ θ coincides with the image of the symmetrizer id + Ψ θ . For later use we work out the explicit commutation rules between any two d × d and d ′ × d ′ minors Λ IJ and Λ I ′ J ′ for the case V = F θn , regarded as the coordinate algebra A (GL θ ( n ))of the noncommutative variety GL θ ( n ), with | I | = | J | = d and | I ′ | = | J ′ | = d ′ . One hasΛ IJ Λ I ′ J ′ = ǫ i ··· i d ( r ) θ ǫ j ··· j d ( c ) θ ǫ i ′ ··· i ′ d ′ ( r ) θ ǫ j ′ ··· j ′ d ′ ( c ) θ (cid:0) g i j · · · g i d j d (cid:1) (cid:0) g i ′ j ′ · · · g i ′ d ′ j ′ d ′ (cid:1) = (cid:16) d Y α =1 d ′ Y α ′ =1 Q i α j α ; i ′ α ′ j ′ α ′ (cid:17) × ǫ i ··· i d ( r ) θ ǫ j ··· j d ( c ) θ ǫ i ′ ··· i ′ d ′ ( r ) θ ǫ j ′ ··· j ′ d ′ ( c ) θ (cid:0) g i ′ j ′ · · · g i ′ d ′ j ′ d ′ (cid:1) (cid:0) g i j · · · g i d j d (cid:1) = (cid:16) d Y α =1 d ′ Y α ′ =1 Q i α j α ; i ′ α ′ j ′ α ′ (cid:17) Λ I ′ J ′ Λ IJ . Introducing the coefficient R IJ ; I ′ J ′ = d Y α =1 d ′ Y α ′ =1 Q i α j α ; i ′ α ′ j ′ α ′ (2.28)we have the commutation relations(2.29) Λ IJ Λ I ′ J ′ = R IJ ; I ′ J ′ Λ I ′ J ′ Λ IJ . In particular, this shows that the minors of order d generate a subalgebra.Another useful identity concerns how minors behave when we choose two multi-indiceswhich differ only by transposition on a pair of indices. Consider a pair of multi-indicesof the form J = ( j · · · j α · · · j β · · · j d ) and J t αβ = ( j · · · j β · · · j α · · · j d ). From (2.23) oneobtains the alternating relations(2.30) Λ IJ = ( − | β − α | Λ IJ tαβ , which can be further generalized to arbitrary permutations.3. Noncommutative toric varieties
The strategy of (toric) isospectral deformations is that once we have a noncommutativedeformation of the torus we can deform every space acted upon by it. For Riemannianmanifolds the isospectral condition means restricting to isometric actions. Using thealgebraic torus T ∼ = ( C × ) n and its deformation constructed in § Noncommutative deformations of toric varieties.
Toric varieties X may bedescribed in several equivalent ways. As complex varieties they come with an open em-bedding of an algebraic torus, which is dense in X . In this picture their geometry isencoded by combinatorial data, a fan, that describes the way in which ( C × ) n acts on X . As symplectic manifolds they come with a hamiltonian action of a real torus. Thecorresponding moment map, whose image is a convex polytope, provides the information about the structure of X . Noncommutative deformations of toric varieties in the sym-plectic framework are defined in [8]. In this paper we will use the fan picture. For a moreexhaustive introduction to toric varieties, along with further definitions and terminology,see e.g. [16, 17, 22]. Definition 3.1. A toric variety X of dimension n is an irreducible algebraic variety over C which contains ( C × ) n as a Zariski open subset and the regular action of ( C × ) n on itselfextends to an action on the whole of X . Basic examples are the affine planes C n , the projective spaces CP n , and the weightedprojective spaces CP n [ a , a , . . . , a n ]. Additional examples comes from cones (of suitabletype) and families of them as we now show. We denote by L R = L ⊗ Z R ∼ = R n the realvector space obtained from a lattice L . Its dual vector space is L ∗ R = L ∗ ⊗ Z R ∼ = ( R n ) ∗ . Definition 3.2. A rational polyhedral cone σ ⊂ L R is a cone σ = R + v ⊕ · · · ⊕ R + v s generated by finitely many elements v , . . . , v s ∈ L . It is strongly convex if it does notcontain any real line, σ ∩ ( − σ ) = 0 . Definition 3.3.
For every rational polyhedral cone σ ⊂ L R we define the dual cone σ ∨ = (cid:8) m ∈ L ∗ R (cid:12)(cid:12) h m, u i ≥ ∀ u ∈ σ (cid:9) . Then, the set σ ∨ ∩ L ∗ is a finitely generated semigroup under addition (Gordan’s Lemma).Given a rational polyhedral cone σ which is in addition strongly convex, one constructsa normal affine toric variety U [ σ ]. We sketch the main points of the construction; for moredetails one may refer for instance to [17, § I.6]. Note that in general σ ∨ is not stronglyconvex (even if σ is), so that if ( m , . . . , m l ) are the generators of the initely generatedsemigroup σ ∨ ∩ L ∗ one has that l ≥ n . To each of the generators m a = P i ( m a ) i e ∗ i there is associated a Laurent monomial in C [ t ± , . . . , t ± n ] by the assignment m a t m a = t ( m a ) · · · t ( m a ) n n . The product between two such elements is given by the correspondingsum of characters, t m a · t m b := t m a + m b . Thus the generators of σ ∨ ∩ L ∗ span a subalgebra of C [ t ± , . . . , t ± n ] which we denote by C [ σ ]. The affine toric variety U [ σ ] is defined to be thespectrum of C [ σ ], i.e. C [ σ ] is the coordinate algebra of U [ σ ]. The variety U [ σ ] is shownto be normal (i.e. C [ σ ] is integrally closed) and of dimension n . These are all normalaffine varieties that are also toric, that is, any such a variety is isomorphic to U [ σ ] forsome strongly convex rational polyhedral cone σ . Note that the inclusion 0 ֒ → σ inducesan embedding of the torus T = U [0] as a dense open subset of U [ σ ].The variety U [ σ ] may also be described as an embedding in the complex plane C l . If σ ∨ ∩ L ∗ has l generators, consider the polynomial algebra C [ x , . . . , x l ] (one variable x a for each m a ). Recall that the generators m a are l rational vectors in L ∗ R , so there are atleast l − n linear relations among them. Then we may quotient the algebra C [ x , . . . , x l ]by the ideal generated by these relations among the vectors m a , realized as multiplicativerelations among the variables x a . If we denote the subspace generated by these relationsas R [ m a ] ⊂ C [ x , . . . , x l ], then we get a realization of U [ σ ] as the spectrum of the quotientalgebra C [ σ ] = C [ x , . . . , x l ] / h R [ m a ] i .We obtain generic toric varieties by gluing together affine toric varieties. This has acorresponding picture in terms of cones. Definition 3.4.
Given a cone σ ⊂ L R , a face τ ⊂ σ is a subset of the form τ = σ ∩ m ⊥ for some m ∈ σ ∨ , where m ⊥ := { u ∈ L R | h m, u i = 0 } . LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 19
Definition 3.5. A fan Σ ⊂ L R is a non-empty finite collection of strongly convex rationalpolyhedral cones in L R satisfying the following conditions: (1) If σ ∈ Σ and τ is a face of σ , then τ ∈ Σ ; and (2) If σ, τ ∈ Σ , then the intersection σ ∩ τ is a face of both σ and τ . To a fan Σ in L R we associate a toric variety X = X [Σ]. The cones σ ∈ Σ correspondto the open affine subvarieties U [ σ ] ⊂ X [Σ], and U [ σ ] and U [ τ ] are glued together alongtheir common open subset U [ σ ∩ τ ] = U [ σ ] ∩ U [ τ ]. Various properties of X [Σ], such assmoothness and compactness, may be stated entirely in terms of the fan structure Σ (seee.g. [22] for details).Our definition of noncommutative toric varieties will involve a multi-parameter defor-mation X [Σ] → X θ [Σ] which makes use of the same fan structure Σ, deforming only theproduct structure of the coordinate algebra of every strongly convex rational polyhedralcone of Σ. We have already defined the quantum Laurent algebra C θ [ t ± , . . . , t ± n ], thecoordinate algebra of the noncommutative algebraic torus ( C × θ ) n . Since the undeformedtorus ( C × ) n is densely contained in every toric variety X [Σ] = S σ ∈ Σ U [ σ ], we expect tohave morphisms between the noncommutative algebras corresponding to the noncommu-tative varieties X θ [Σ] and C θ [ t ± , . . . , t ± n ].We begin by defining noncommutative affine toric varieties. They are associated toa strongly convex rational polyhedral cone σ ⊂ L R , just as in the commutative case.However, now we use the complex skew-symmetric matrix θ to define a noncommutativeproduct in the algebra C [ σ ], according to the group character relation given by χ p ⋆ θ χ q = exp (cid:0) i2 X ij p i θ ij q j (cid:1) χ p + q . Thus if ( m , . . . , m l ) are the generators of the semigroup σ ∨ ∩ L ∗ and t m a are the as-sociated Laurent monomials, then the algebra C θ [ σ ] is defined to be the subalgebra of C θ [ t ± , . . . , t ± n ] generated by { t m a } with product t m a ⋆ θ t m b := exp (cid:0) i2 X ij ( m a ) i θ ij ( m b ) j (cid:1) t m a + m b . This may be regarded as a deformation of the algebra generated by the characters, but,we stress once again, not of their group structure. It is for this reason that we will describenoncommutative toric varieties by using the same fan of the corresponding commutativevarieties. The noncommutative affine variety corresponding to the algebra C θ [ σ ] is denoted U θ [ σ ]. It is a multi-parameter deformation of U [ σ ]. Proposition 3.6.
The action of the torus T on ( C × θ ) n restricts to a faithful torus action Φ on U θ [ σ ] , which is dually a map Φ : T → Aut( C θ [ σ ]) . Proof : On generators of the algebra C θ [ σ ] of the form t m a = t ( m a ) · · · t ( m a ) n n with m a ∈ σ ∨ ∩ L ∗ and a = 1 , . . . , l , the action of τ = ( τ , . . . , τ n ) ∈ T is given byΦ τ ( t m a ) = n Y i =1 τ i t ( m a ) i i The corresponding infinitesimal action of the torus generator H i is then H i ⊲ t m a = ( m a ) i t m a , i.e. multiplication by the coefficient ( m a ) i , the i -th component of m a . If the action isnot faithful, there is at least one index i with corresponding generator H i acting trivially and for this i one would have ( m a ) i = 0 for every a , i.e. the generators of the dual conewould have vanishing i -th component. But this would mean that every vector along the i -th component has negative pairing with elements of the cone σ , which contradicts theassumption that σ is strongly convex. (cid:4) This toric action really parallels the undeformed situation: strongly convex cones σ represent the affine toric varieties U [ σ ] that are glued together to get the full toric variety X ; and in each of them the torus is embedded and acts freely (the usual extension of theaction of the torus on itself). In other words, the U [ σ ]’s are open affine toric subvarietiesof X , so they carry a faithful action of the torus.Recall that the L ∗ -grading gives precisely the eigenspace decompositions of algebraicobjects, dual to T -invariant geometric objects. In particular, since the torus T acts on C θ [ σ ] by C -algebra automorphisms for each σ ∈ Σ, the algebra C θ [ σ ] is spanned by T -eigenvectors for which the corresponding eigenvalues are rational. This yields a vectorspace decomposition(3.7) C θ [ σ ] = M p ∈ L ∗ C θ [ σ ] p , where C θ [ σ ] p denotes the eigenspace of C θ [ σ ] labelled by the character p ∈ L ∗ , and C θ [ σ ] p ⋆ θ C θ [ σ ] q ⊂ C θ [ σ ] p + q for all p, q ∈ L ∗ , since T acts by automorphisms. Thus we geta grading of C θ [ σ ] by the free abelian group of characters L ∗ , such that the homogeneouselements are the T -eigenvectors in C θ [ σ ].We have seen how affine toric varieties may also be regarded as subvarieties of complexplanes C l , via the quotient algebra C [ σ ] = C [ x , . . . , x l ] / h R [ m a ] i . An analogous realizationis possible for noncommutative affine toric varieties. Remembering that in general l ≥ n ,the noncommutative deformation of the polynomial algebra C [ x , . . . , x l ] is obtained fromthe multiplicative relations between the monomials t m a . If we denote ˇ θ ab := ( m a ) i θ ij ( m b ) j with a, b = 1 , . . . , l , i, j = 1 , . . . , n and ˇ q ab = exp( i2 ˇ θ ab ), then the relation between Laurentmonomials becomes(3.8) t m a ⋆ ˇ θ t m b := ˇ q ab t m a + m b . As a consequence, the generators of the algebra of the affine variety obeyˇ q ba x a ⋆ ˇ θ x b = ˇ q ab x b ⋆ ˇ θ x a or equivalently(3.9) x a ⋆ ˇ θ x b = (cid:0) ˇ q ab (cid:1) x b ⋆ ˇ θ x a . The relations (3.9) define the l -dimensional noncommutative complex plane with coor-dinate algebra C ˇ θ [ x , . . . , x l ], which is a special instance of the general class of quantumaffine spaces considered by Manin [38].The l − n linear relations among the generators of the dual cone { m a } are now expressedin the character algebra. These relations can always be brought to the form l X a =1 ( p s,a − r s,a ) m a = 0 , LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 21 for s = 1 , . . . , l − n , with non-negative integer coefficients p s,a , r s,a . For each s , one obtainsfrom (3.8) the additional relation(3.10) x p s, ⋆ ˇ θ · · · ⋆ ˇ θ x p s,l l = (cid:16) Y ≤ a
Toric isospectral deformations can be shown to be strict deformation quan-tizations in the sense of Rieffel [39] . It is an open question if our deformation, which maybe thought of as generated by C n instead of Rieffel’s R n , satisfies a similar property. In the remainder of this section we will work out some explicit examples of noncom-mutative deformations of toric varieties. We set q ij := exp (cid:0) i2 θ ij (cid:1) for i < j . It maybe regarded as a form q ∈ V T ∼ = ( C × ) n ( n − / with q ij = q ( e ∗ i , e ∗ j ) = h e ∗ i , Θ( e ∗ j ) i , orequivalently as a map q ∈ Hom Z ( V L ∗ , C × ). When n = 2 we write q := exp (cid:0) i2 θ (cid:1) with θ = θ = − θ ∈ C . In the following we omit the star product symbol ⋆ θ from thenotation for brevity.3.2. Algebraic Moyal plane and D -modules. Besides T itself, the simplest toric vari-ety is the n -dimensional complex plane C n . It contains an embedding of the commutativetorus ( C × ) n ֒ → C n given by the log map t i z i = log t i , i = 1 , . . . , n . Then the toric action on C n is λ i ⊲ z j = z j + δ ij log λ j for ( λ , . . . , λ n ) ∈ ( C × ) n . Passingto the multi-parameter deformation ( C × θ ) n of the torus defined by the quantum Laurentalgebra C θ [ t ± , . . . , t ± n ], the elements z i obey the commutator relations[ z i , z j ] = i θ ij . The corresponding algebra of complex polynomial functions C θ [ z , . . . , z n ] is dual to anoncommutative affine variety that we call the algebraic Moyal plane C nθ . This algebracan be identified with the d -th Weyl algebra D ( C d ) of polynomial differential operatorson the complex space C d , with d = ⌊ n ⌋ , whose projective modules furnish basic examplesof D -modules. Note that C nθ and C nθ ′ are isomorphic if and only if the matrices θ and θ ′ have the same rank.3.3. Noncommutative projective plane.
The fan Σ CP ⊂ Z of CP contains threeone-dimensional cones τ i = R + v i , i = 1 , ,
3, with vectors v = (1 , v = (0 ,
1) and v = ( − , − CP are generated by pairs of these as σ i = R + v i +1 ⊕ R + v i +2 , i = 1 , , U [ σ i ] generate an open cover of X [Σ] = CP .There are no relations among the generators of the subalgebras C θ [ σ i ] ⊂ C θ ( t , t ), aseach dual cone σ ∨ i is strongly convex. For example, the semigroup σ ∨ ∩ Z is generatedby m = (1 ,
0) and m = (0 , θ = θ := θ for the deformation matrix, and thealgebra C θ [ σ ] = C θ [ x , x ] is generated by x a = t m a = t a , a = 1 ,
2, with the relation x x = q x x (3.13)where q := q . The other two cones σ i for i = 1 , θ = θ and that C θ [ σ i ] is generatedby elements x , x satisfying the relations (3.13). All three varieties U θ [ σ i ] ∼ = C θ are thuscopies of the two-dimensional complex Moyal plane.To glue the noncommutative affine toric varieties together, consider for example theface τ = σ ∩ σ . The semigroup τ ∨ ∩ Z is generated by m = (1 , m = (0 ,
1) and m = − m . The generators of the subalgebra C θ [ τ ] = C θ [ t , t , t − ] are the elements y = t , y = t and y = t − with the relations y y = q y y , y y = q − y y , y y = 1 = y y , (3.14)which we may identify as the algebra dual to a noncommutative projective line CP θ . Thealgebra morphisms C θ [ σ ] → C θ [ τ ] and C θ [ σ ] → C θ [ τ ] are both natural inclusions of LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 23 subalgebras, and in this manner there is a natural algebra automorphism C θ [ σ ] → C θ [ σ ].The other faces are similarly treated.3.4. Noncommutative orbifold.
We can also deform singular toric varieties in ourformalism. For illustration, let us consider the quotient singularity C / Z , where thecyclic group Z acts as C ∋ ( z , z ) ( − z , − z ). The fan Σ ⊂ Z consists of a singlecone σ = R + v ⊕ R + v , where v = (1 ,
0) and v = (1 , σ is generatedby m = (2 , − m = (0 ,
1) and m = (1 , C θ [ t ± , t ± ] Z of thenoncommutative affine variety X θ [Σ] = U θ [ σ ] is thus generated by x = t t − , y = t and z = t with the relations x y = q y x , x z = q z x , y z = q − z y , x y − q z = 0 . The blow-up of the quotient singularity C / Z is obtained by adding the vector v =(1 ,
1) to the fan Σ above. There are now two maximal cones σ + = R + v ⊕ R + v and σ − = R + v ⊕ R + v , with dual semigroups generated by m ± = ± e ∗ and m ± = e ∗ ∓ e ∗ ,respectively. The coordinate algebras of the noncommutative affine toric varieties U θ [ σ ± ]are generated respectively by elements u ± = t ± , v ± = t ∓ t subject to the relations u ± v ± = q ± v ± u ± , and hence U θ [ σ ± ] ∼ = C θ . The intersection τ = σ + ∩ σ − = R + v is generated by m = (1 , m = (1 , −
1) and m = ( − , C θ [ τ ] are thus y = t , y = t − t and y = t t − with the relations (3.14).3.5. Noncommutative conifold.
The threefold ordinary double point, or conifold sin-gularity, is defined by the locus of the equation x y − z w = 0 in C . Its fan Σ ⊂ Z consistsof a single maximal cone σ generated by w = e , w = e , w = e + e and w = e + e ,where e i ( i = 1 , ,
3) are the standard generators of Z . The dual cone σ ∨ is generatedby m = e , m = e , m = e and m = e + e − e , so that m + m = m + m . Thegenerators of the coordinate algebra of the noncommutative conifold X θ [Σ] = U θ [ σ ] arethus the elements x = t , y = t , z = t and w = t t t − subject to the relations x y = q y x , x z = q z x , x w = q q − w x ,y z = q z y , y w = q − q − w y , z w = q q w z , and x y − q q − q − z w = 0 . The crepant resolution of the conifold singularity is the non-singular toric Calabi–Yauthreefold whose fan Σ ⊂ Z is defined by the vectors v = e + e + e , v = e + e , v = e and v = e + e , and the maximal cones σ = R + v ⊕ R + v ⊕ R + v and σ = R + v ⊕ R + v ⊕ R + v . So for example σ is generated by m = e ∗ , m = e ∗ − e ∗ and m = e ∗ − e ∗ , thus C θ [ σ ] is generated by x = t , y = t − t and z = t t − with therelations x y = q y x , x z = q − q − z x , y z = q − z y . The other maximal cone is treated similarly, and the gluing morphism is similar to thatof the quotient singularity blow-up of § Sheaves on noncommutative toric varieties
In this section we develop a sheaf theory on noncommutative toric varieties, follow-ing [25]. The idea is that the “topology” of the noncommutative space X θ = X θ [Σ] isgiven by the cones in the fan Σ (the toric open sets in the topology of X θ ). The assign-ment σ C θ [ σ ] of the noncommutative algebra C θ [ σ ] to every cone σ ∈ Σ is viewed asthe structure sheaf O X θ of the noncommutative toric variety X θ .4.1. Quasi-coherent sheaves.
We begin with recalling the following result, of which weomit the elementary proof.
Lemma 4.1.
For each cone σ ∈ Σ , the algebra C θ [ σ ] is a noetherian domain. We use the category
Open ( X θ ) of toric open sets to define the category of sheaves onthe variety X θ = X θ [Σ]. We call a set of inclusions ( σ i ֒ → σ ) i ∈ I of cones a covering if σ = S i ∈ I σ i . Then Open ( X θ ) always contains a sufficiently fine open cover. The category Open ( X θ ) with the data of coverings forms a Grothendieck topology on X θ . Proposition 4.2.
The map σ C θ [ σ ] defines a sheaf of C -algebras O X θ on Open ( X θ ) . Proof : Let ( σ i ֒ → σ ) i ∈ I be a covering, i.e. σ = S i ∈ I σ i . Then C θ [ σ ] = T i ∈ I C θ [ σ i ], wherethe intersection is well-defined since each algebra C θ [ σ i ] is contained in C θ [ t ± , . . . , t ± n ].Thus, as in (3.11), the sequence0 −→ C θ [ σ ] p −→ Y i ∈ I C θ [ σ i ] q −→ Y i,j ∈ I C θ [ σ i ∩ σ j ](4.3)is exact, and the result follows. (cid:4) We now define mod ( X θ ) to be the category of sheaves of right O X θ -modules on Open ( X θ ).If Σ consists of a single cone σ , i.e. X θ [Σ] = U θ [ σ ] is an affine variety, then mod (cid:0) U θ [ σ ] (cid:1) ∼ = mod (cid:0) C θ [ σ ] (cid:1) (4.4)coincides with the category of right C θ [ σ ]-modules. We denote by f M the sheaf associatedto a module M under the isomorphism (4.4). A sheaf of right O X θ -modules is called quasi-coherent if its restriction to each affine open set U θ [ σ ] is of the form f M for someright C θ [ σ ]-module M . It is called coherent if M is finitely-generated.Let coh ( X θ ) denote the category of quasi-coherent sheaves of right O X θ -modules. Givena cone σ in Σ, we write coh ( σ ) for the category of right C θ [ σ ]-modules. There are restric-tion functors j • σ : coh ( X θ ) −→ coh ( σ )(4.5)for each open inclusion j σ : U [ σ ] ֒ → X [Σ]. Let tor ( σ ) be the full Serre subcategory of coh ( X θ ) generated by objects E such that j • σ ( E ) = 0. In [25, Prop. 4.3] the followingfundamental result is proven. Proposition 4.6.
Let σ be a cone in Σ . Then the restriction functor (4.5) is exact, andthere is a natural equivalence of categories coh ( X θ ) (cid:14) tor ( σ ) ∼ = coh ( σ ) . LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 25
Each cone σ in the fan Σ gives a toric open set of X θ [Σ]. We will use Proposition 4.6 toreduce geometric problems in the category coh ( X θ ) to algebraic problems in the algebra C θ [ σ ] via the localization functors j • σ . This gives an explicit description of the quotientcategory. The objects of coh ( σ ) are the same as those of coh ( X θ ), and we write E σ forthe object in coh ( σ ) corresponding to a sheaf E . The morphisms are given byHom coh ( σ ) ( E σ , F σ ) = lim −→ E ′ Hom coh ( X θ ) ( E ′ , F ) , where the inductive limit is taken over all subsheaves E ′ ⊂ E with j • σ ( E/E ′ ) = 0.For any pair of sheaves E, F ∈ coh ( X θ ), let Ext p ( E, F ) be the p -th derived functor ofthe Hom-functor Hom( E, F ) = Hom coh ( X θ ) ( E, F ). For a sheaf E ∈ coh ( X θ ), we define H p ( X θ , E ) := Ext p ( O X θ , E ) . Definition 4.7. (1)
A coherent sheaf E ∈ coh ( X θ ) is called locally free or a bundle ifeach E σ , σ ∈ Σ corresponds to a free module C θ [ σ ] ⊕ r for some r ∈ N . The integer r is called the rank of E . (2) A coherent sheaf E ∈ coh ( X θ ) is called torsion free if each E σ , σ ∈ Σ has nofinite-dimensional submodules, or equivalently if it admits an embedding E ֒ → E into a locally free sheaf E . The rank of E is the rank of E minus the rank of E /E . Equivariant sheaves.
Recall from § σ ∈ Σ there is a grading(3.7) of the algebra C θ [ σ ] by the free abelian group of characters L ∗ , the homogeneouselements in the decomposition being identified with the eigenvectors of the T -action on C θ [ σ ]. To get a similar eigenspace decomposition on right C θ [ σ ]-modules, we need to liftthe T -action. We denote with mod H θ (cid:0) C θ [ σ ] (cid:1) the subcategory of the category mod ( C θ [ σ ])made of left T -equivariant right C θ [ σ ]-modules. There is a left action of the Hopf algebra H θ on elements M ∈ mod H θ (cid:0) C θ [ σ ] (cid:1) which is compatible with the H θ -action on C θ [ σ ].This means that h ⊲ ( M · a ) = ( h (1) ⊲ M ) · ( h (2) ⊲ a ) for h ∈ H θ , a right C θ [ σ ]-module M ,and a ∈ C θ [ σ ] (with the usual notation ∆( h ) = h (1) ⊗ h (2) for the coproduct). Objectsof mod H θ (cid:0) C θ [ σ ] (cid:1) admit then an L ∗ -graded T -eigenspace decomposition M = L p ∈ L ∗ M p such that M p · C θ [ σ ] q ⊂ M p + q for all p, q ∈ L ∗ , and t m a ⊲ M p ⊂ M m a + p for all p ∈ L ∗ andfor m a ∈ σ ∨ ∩ L ∗ . This also means that the category of right C θ [ σ ]-modules carryinga compatible left H θ -action is naturally a braided monoidal category of left H θ -modules.Via the braiding morphism Ψ θ , we can deform the category H M as described in § mod H (cid:0) C [ σ ] (cid:1) and mod H θ (cid:0) C θ [ σ ] (cid:1) .This construction extends to give a left H θ -action on the category coh ( X θ ) and T -equivariant sheaves on Open ( X θ ), i.e. the subcategory coh H θ ( X θ ) of coherent sheaves E ∈ coh ( X θ ) with a compatible T -action, which decompose as direct sums E = M p ∈ L ∗ E p of T -eigensheaves E p of O X θ -modules. If E is locally free, then each summand E p isalso locally free. There is a functorial equivalence between the categories coh H ( X ) and coh H θ ( X θ ). The equivalence holds since we have an Ore domain, but not in general (see e.g. [31, Ex. 10.19B]).
Invariant subschemes and ideal sheaves.
In applications to instanton countingproblems, which will be presented in [12], one is faced with the task of classifying the fixedpoints of the natural torus action on the category coh ( X θ ) obtained by lifting the actionof T on X θ as described in § X [Σ] is a disjoint union over the orbits O σ of the T -action on X , which are in bijective correspondence with the cones σ ∈ Σ. Onehas dim C ( σ ) + dim C ( O σ ) = n , and O σ ⊂ O τ if and only if τ is a face of σ . In particular,the fixed points of the torus action, i.e. the closed T -orbits, correspond to the maximalcones in the fan Σ, while O = T . We will now show that these orbits are somewhat moreeasily classified in the noncommutative case, in the sense that they arise as the generic T -invariant subvarieties in X θ .In analogy with the classical setting, we have the notion of a “noncommutative scheme”. Definition 4.8. A closed subscheme of X θ is a full subcategory Y θ ⊆ coh ( X θ ) whoseinclusion functor i • has a right-adjoint i ! and a left-adjoint i • . Definition 4.9. An ideal sheaf on Open ( X θ ) is a coherent sheaf I ∈ coh ( X θ ) whoserestriction to each affine open set U θ [ σ ] is a two-sided ideal I σ of the algebra C θ [ σ ] . For each cone σ ∈ Σ, it follows from Lemma 4.1 that every torsion free module ofrank one in coh ( σ ) = mod ( C θ [ σ ]) is isomorphic to a right ideal of C θ [ σ ]. Hence an idealsheaf I ∈ coh ( X θ ) can be regarded as a torsion free sheaf of rank one on Open ( X θ ). (Theconverse does not hold globally: there are torsion free sheaves of rank one on X θ whichare not isomorphic to ideal sheaves.) Moreover, the category of sheaves of right O X θ /I -modules determines a closed subscheme Y θ of X θ . The following result describes to whatextent this correspondence fails to be a bijection (generalizing thus the commutative case;see e.g. [15, § Theorem 4.10.
There is a bijective correspondence between closed subschemes of X θ andideal sheaves I on Open ( X θ ) such that I σ ⋆ θ C θ [ σ ∩ σ ′ ] = I σ ′ ⋆ θ C θ [ σ ∩ σ ′ ] on overlaps U θ [ σ ∩ σ ′ ] . Proof : Let i • be the inclusion of a subcategory in coh ( X θ ) corresponding to a closedsubscheme Y θ , with left adjoint functor i • . Then the map Y θ → Y θ , M i • i • ( M ) issurjective. Fix a cone σ ∈ Σ, and suppose that M ∈ tor ( σ ), i.e. j • σ ( M ) = 0. Sincethe restriction functor j • σ is exact, the map j • σ ( M ) → j • σ i • i • ( M ) is also surjective, andhence by Proposition 4.6 the functor i • i • acts on the category coh ( σ ). It follows [25,Prop. 4.5] that C θ [ σ ] → i • i • ( C θ [ σ ]) is a surjective bimodule morphism, whose kernelis the desired two-sided ideal I σ . Conversely, given an ideal sheaf I on Open ( X θ ) withthe stated property, we define the functor i • by mapping the module M over C θ [ σ ] to M/M ⋆ θ I σ ∈ mod ( C θ [ σ ] /I σ ). (cid:4) If σ is a cone in the fan Σ, and τ ∈ Σ is a face of σ , define I σ ( τ ) to be the kernel of thealgebra morphism C θ [ σ ] → C θ [ τ ]. Then I σ ( τ ) = M m/ ∈ τ ∨ ∩ L ∗ C χ m (4.11)is an ideal in C θ [ σ ], and hence each face τ ⊂ σ canonically determines a closed subschemeof X θ . The cone point of a strongly convex cone σ is a distinguished torus fixed point of U [ σ ]. It follows that for any given face τ ֒ → σ , there is a natural morphism C θ [ σ ] → C θ [ τ ]dual to inclusion of an orbit closure. LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 27
Definition 4.12.
A closed subscheme Y θ is irreducible if each inclusion of a full subcat-egory Y θ ⊂ W θ ∪ Z θ implies Y θ ⊂ W θ or Y θ ⊂ Z θ , where W θ , Z θ are closed subschemes of X θ and W θ ∪ Z θ is the full subcategory of coh ( X θ ) whose objects M are extensions −→ ω −→ M −→ ζ −→ , of objects ω and ζ of W θ and Z θ respectively. The union operation ∪ in Definition 4.12 corresponds to the product of ideals in eachalgebra C θ [ σ ], σ ∈ Σ [25, Prop. 4.5] for the correspondence in Theorem 4.10. It followsthat irreducible subschemes give prime ideals on each open affine set U θ [ σ ] under thecorrespondence of this theorem. For a subset S ⊂ L ∗ R , we denote S ⊥ := (cid:8) v ∈ L R (cid:12)(cid:12) h u, v i = 0 ∀ u ∈ S (cid:9) , and for a C -algebra A we denote by Spec( A ) the spectrum of A , i.e. the set of primeideals equipped with the Zariski topology.Recall from Definition 3.3 that σ ∨ denotes the cone dual to σ . The following charac-terization of the irreducible subschemes of X θ is proven in [25, Thm. 6.8]. Proposition 4.13.
There is a natural bijection between the set of irreducible subschemesof X θ (Σ) and the disjoint union F σ ∈ Σ Spec (cid:0) C θ [( σ ⊥ ) ∨ ] (cid:1) . For θ sufficiently generic, the only subschemes of X θ are dual to closed T -orbits andto all points of one-dimensional torus orbits [25, § J is any ideal of the algebra C θ [ σ ] for σ ∈ Σ, the intersection, T t ∈ T t ⊲ J , ofthe T -orbit of J is the largest torus invariant ideal of C θ [ σ ] contained in J . In particular,it is a T -invariant prime ideal for every J ∈ Spec( C θ [ σ ]). The T -strata partition the spaceof prime ideals Spec( C θ [ σ ]) into a disjoint union over T -invariant prime ideals. Proposition 4.14.
For each cone σ ∈ Σ and for every T -invariant prime ideal I in C θ [ σ ] , the T -stratum { J ∈ Spec( C θ [ σ ]) | T t ∈ T t ⊲ J = I } is a single T -orbit. Proof : This follows by Lemma 4.1 and [24, Thm. 6.8], which imply that the torus T actstransitively on the T -strata of prime ideals in C θ [ σ ]. (cid:4) Proposition 4.15.
There is a natural bijection between the sets of T -equivariant idealsheaves on Open ( X θ ) , satisfying the conditions of Theorem 4.10, and L ∗ -graded sub-schemes of X θ [Σ] . Proof : Let Y θ be a closed subscheme of X θ , defined by an ideal sheaf I according toTheorem 4.10. Then Y θ is invariant under the torus action if and only if the action of T on the category coh ( X θ ) induces an action on Y θ . Suppose first that X θ [Σ] = U θ [ σ ] isaffine. Then this invariance is equivalent to the requirement that there is a commutativediagram C θ [ σ ] × T Φ / / (cid:15) (cid:15) C θ [ σ ] (cid:15) (cid:15) I σ × T Φ | Iσ × T / / I σ where Φ is the right covariant action of T on C θ [ σ ] constructed in Proposition 3.6, I σ is a two-sided ideal in C θ [ σ ], and the vertical morphisms are restrictions. This is true if and only if Φ τ ( I σ ) ⊂ I σ , for all τ ∈ T . It follows that if P a α a t m a is in I σ , with m a ∈ σ ∨ ∩ L ∗ ⊂ L ∗ for a = 1 , . . . , l , then the transformed P a α a Φ τ ( t m a ) is also in I σ ,and so α a t m a ∈ I σ for every a = 1 , . . . , l . Thus I σ is an L ∗ -graded ideal of C θ [ σ ]. If wenow write I σ = L p ∈ S C χ p for some subset S ⊂ σ ∨ ∩ L ∗ , then the condition for I σ to bean ideal in C θ [ σ ] is equivalent to the requirement that for all m a ∈ σ ∨ ∩ L ∗ and p ∈ S ,one has m a + p ∈ S . Hence I σ is T -equivariant. The global statement for general X θ [Σ]now follows by gluing these equivalences together. (cid:4) Remark 4.16.
For σ ∈ Σ , the T -invariant ideal I σ of the algebra C θ [ σ ] appearing in theproof of Proposition 4.15 is generated by elements of the form t m a for m a ∈ σ ∨ ∩ L ∗ , i.e. I σ is a monomial ideal . Moreover, I σ is prime if and only if ( σ ∨ ∩ L ∗ ) \ S is a sub-semigroupof σ ∨ ∩ L ∗ . It follows that the irreducible invariant subschemes of U θ [ σ ] are in bijectivecorrespondence with the faces τ of σ , such that the corresponding monomial ideal is givenby (4.11). For fixed σ ∈ Σ, let L σ = L ∩ σ and let p : L → L ( σ ) := L/L σ be the canonicalprojection. Then L ( σ ) ∗ = L ∗ ∩ σ ⊥ . The homomorphism Θ : L ∗ → T naturally restrictsto the sublattice L ( σ ) ∗ ⊂ L ∗ . Let p R = p ⊗ R . Then the collection of cones p R ( τ ),where τ ∈ Σ is a cone for which σ is a face of τ , form a fan Σ( σ ) in L ( σ ) ⊗ Z R . Set V θ ( σ ) = X θ [Σ( σ )]. By Theorem 4.10, the projection Σ → Σ( σ ) shows that V θ ( σ ) definesa closed subscheme of X θ = X θ [Σ]. Example 4.17.
Suppose that σ is the maximal cone of Σ generated by the basis e , . . . , e n of the lattice L ∼ = Z n , with dual basis e ∗ , . . . , e ∗ n . Then the corresponding noncommutativeaffine variety is the algebraic Moyal plane U θ [ σ ] ∼ = C nθ , i.e. C θ [ σ ] = C θ [ t , . . . , t n ] where t i = t e ∗ i . Let τ be a face of σ generated by { e i } i ∈ N for some subset N ⊂ { , . . . , n } . Then V θ ( τ ) is defined by the monomial ideal h t i i i ∈ N in C θ [ t , . . . , t n ] . K¨ahler differential forms.
We will now construct sheaves of noncommutative dif-ferential forms. We start by recalling some definitions and properties of K¨ahler differen-tials. We then show how the general construction behaves under a Drinfel’d twist usingthe braided monoidal category theory of § X θ = X θ [Σ].The general framework we need from the theory of K¨ahler differentials describes deriva-tions of a unital C -algebra ( A, µ ) into an A -bimodule M , i.e. C -linear maps D : A → M obeying the Leibniz rule D ( a b ) = ( Da ) b + a ( Db ) for every a, b ∈ A .The universal algebra of derivations over A is realized by the A -bimoduleΩ A,un = I A := ker( µ : A ⊗ A → A )which is a two-sided ideal of the algebra A ⊗ A generated by elements of the form a ⊗ − ⊗ a with a ∈ A , and differential given by d a := a ⊗ − ⊗ a . The universal property means thatevery derivation D : A → M factors through Ω A,un by a unique morphism of A -bimodules φ D : Ω A,un → M with D = φ D ◦ d. The morphism φ D is defined by φ D (cid:0) a (d a ) a (cid:1) := a D ( a ) a . (4.18)The construction of Ω A,un respects the inclusion of subalgebras, i.e. Ω A ′ ,un = ker( µ | A ′ ⊗ A ′ ) =ker( µ ) ∩ ( A ′ ⊗ A ′ ) for any subalgebra A ′ ⊂ A . LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 29
For the K¨ahler differential forms one is interested (see e.g. [34, § A -bimodule M (i.e. a m = m a for all a ∈ A and m ∈ M ). Sincefor all a, a ∈ A one has a (d a ) − (d a ) a = ( a ⊗ − ⊗ a ) ( a ⊗ − ⊗ a ) ∈ I A , (4.19)the A -bimodule of symmetric differential forms is I A /I A =: Ω A , which can be shown tobe universal.We will begin by defining bimodules Ω θ [ σ ] of noncommutative K¨ahler differentials onnoncommutative affine varieties for each cone σ ∈ Σ, and then show that the assignment σ Ω θ [ σ ] defines a sheaf on Open ( X θ ). Each affine open set U θ [ σ ] of a noncommutativetoric variety X θ [Σ] has noncommutative coordinate algebra C θ [ σ ] which is a Drinfel’dtwist deformation of the classical coordinate algebra, coming from the algebraic torusaction. The construction of K¨ahler differential forms on noncommutative affine toricvarieties follows from the general theory of K¨ahler differentials for twisted Hopf-modulealgebras, and the natural setting for the construction is the functorial framework of § A is an object in the braided monoidal category H M , the A -bimodule ofuniversal one-forms Ω A,un is naturally an H -module algebra with H -action h ⊲ d a := d( h ⊲ a ) . This is the universal covariant differential calculus, in the sense of Woronowicz [41], andit has a natural noncommutative deformation in the category H F M of twisted Hopf-module algebras. If A F is a twisted Hopf-module algebra defined by a Drinfel’d twistelement F ∈ H ⊗ H as in Theorem 1.4, then the bimodule Ω A F ,un is defined as beforeto be the kernel of the multiplication map µ F = µ ◦ (cid:0) F − ⊲ (cid:1) : A F ⊗ A F → A F . Higherdegree differential forms may be introduced via the N -graded braided exterior algebra ofone-forms Ω • A F ,un = V • F Ω A F ,un := T (cid:0) Ω A F ,un (cid:1) (cid:14) (cid:10) ω ⊗ η + Ψ F ( ω ⊗ η ) (cid:11) ω,η ∈ Ω AF ,un , where T (Ω A F ,un ) = L n ≥ (Ω A F ,un ) ⊗ AF n is the tensor algebra of covariant twisted diffe-rential one-forms with (Ω A F ,un ) := A F , and Ψ F is the braiding morphism on the category H F M defined as in Proposition 1.18 with the twist deformed R -matrix R F . This algebracoincides with the twist deformation of the Hopf-module algebra Ω • A,un , with the actionof the twist F extended to the whole of T (Ω A,un ) by
F ⊲ ( ω ⊗ · · · ⊗ ω n ) = (cid:0) F (1) ⊲ ( ω ⊗ · · · ⊗ ω k ) (cid:1) ⊗ (cid:0) F (2) ⊲ ( ω k +1 ⊗ · · · ⊗ ω n ) (cid:1) . The choice of k here is irrelevant thanks to the associativity of the tensor product, and F (1) and F (2) act by iterating the formula (1.2) for covariant actions on H -module algebras.The A F -bimodule structure of Ω A F ,un is then deformed according to the deformationof the associative product in A F as a ◮ F (d a ) ◭ F a := a ⋆ F ( a ⊗ − ⊗ a ) ⋆ F a . It agrees with the usual deformation induced in the category, a ◮ F (d a ) = α (cid:0) F − ⊲ ( a ⊗ d a ) (cid:1) , (d a ) ◭ F a = α (cid:0) F − ⊲ (d a ⊗ a ) (cid:1) , where α : A ⊗ Ω A,un ⊗ A → A denotes the action of A on Ω A,un . Then the differentiald of the untwisted differential calculus is still a derivation of the deformed product ⋆ F ,as expected by general twisting theory [37]. It naturally extends to the braided exterioralgebra Ω • A F ,un as a graded derivation of degree one by definingd( γ ⊗ γ ) := (d γ ) ⊗ γ + ( − deg( γ ) γ ⊗ (d γ )for homogeneous differential forms γ , γ ∈ Ω • A F ,un .The notion of symmetric bimodule has a braided analog by demanding that the leftand right module morphisms λ F : A F ⊗ Ω A F ,un → Ω A F and ρ F : Ω A F ,un ⊗ A F → Ω A F arerelated by the braiding morphism of H F M . Definition 4.20.
Let A F be an H F -module algebra, and let Ψ = Ψ F be the braidingmorphism of Proposition 1.18. An A F -bimodule M in the category H F M is said to be braided symmetric if one of the following two conditions is satisfied: (1) λ F = ρ F ◦ Ψ A F ,M ; or (2) ρ F = λ F ◦ Ψ M,A F . The two conditions in Definition 4.20 are not equivalent unless the category itself issymmetric, i.e. Ψ = id. This is the case, for example, for Drinfel’d twists of triangularHopf algebras such as the ones we are dealing with in this paper. In the non-symmetriccase they are not compatible with each other, so there are two distinct and inequivalentnotions of braided symmetric bimodule structure that one can choose from.We want to show that a natural quotient I A F /I A F is the universal braided symmet-ric A F -bimodule for braided commutative algebras in (twisted) braided monoidal cate-gories H F M , with universality understood in the same sense as the untwisted A -bimoduleΩ A . Then we can define noncommutative differential forms via the usual deformation inthe category of Hopf-module algebras, and this definition is compatible with the construc-tion of universal differential forms in braided monoidal categories. Proposition 4.21.
Let A be a commutative H -module algebra, and F a Drinfel’d twistelement for a triangular Hopf algebra H . Let I A F = ker( µ F : A F ⊗ A F → A F ) , and con-sider the quotient Ω A F = I A F /I A F . Then (Ω A F , d) is the universal algebra of derivationsover A F with values in a braided symmetric A F -bimodule. Proof : We will prove this by direct computation for the twisted Hopf algebra of § H M and H F M discussed in § A θ := A F θ , etc. Given a simple tensor a ⊗ ω ∈ A θ ⊗ Ω A θ with a ∈ A θ and ω the class of w ⊗ − ⊗ w , w ∈ A θ , we will comparethe quantity ( λ θ − ρ θ ◦ Ψ A θ , Ω Aθ )( a ⊗ ω ) with ( a ⊗ − ⊗ a ) ⋆ θ ( w ⊗ − ⊗ w ) ∈ I A θ .On the one hand, one computes( a ⊗ − ⊗ a ) ⋆ θ ( w ⊗ − ⊗ w ) = a ⋆ θ w ⊗ − a ⊗ w + 1 ⊗ a ⋆ θ w − ∞ X n =0 i n n ! θ i j · · · θ i n j n (cid:0) H j · · · H j n ⊲ w (cid:1) ⊗ (cid:0) H i · · · H i n ⊲ a (cid:1) . On the other hand, one has λ θ ( a ⊗ ω ) = a ⋆ θ ( w ⊗ − ⊗ w ) = a ⋆ θ w ⊗ − a ⊗ w , LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 31 while ρ θ ◦ Ψ A θ , Ω Aθ ( a ⊗ ω ) = ∞ X n =0 i n n ! θ i j · · · θ i n j n (cid:16)(cid:0) H j · · · H j n ⊲ w (cid:1) ⊗ (cid:0) H i · · · H i n ⊲ a (cid:1) − ⊗ (cid:0) H j · · · H j n ⊲ w (cid:1) ⋆ θ (cid:0) H i · · · H i n ⊲ a (cid:1)(cid:17) . It remains to show that the second formal power series in this last equation is equal to1 ⊗ a ⋆ θ w . This follows from the equality a ⋆ θ a = µ ( F θ ⊲ ( a ⊗ a )) [11, Lem. 1.16].Universality follows by the same argument of the undeformed case, i.e. by the formula(4.18) now understood in the twisted setting. (cid:4) We can now apply this construction of K¨ahler differentials for noncommutative alge-bras with product induced by a Drinfel’d twist to each affine open set in a toric variety X [Σ]. Starting from a strongly convex rational polyhedral cone σ ∈ Σ, we form thenoncommutative coordinate algebra C θ [ σ ] as in § C θ [ σ ]-bimodule ofK¨ahler differentials Ω θ [ σ ] = Ω C θ [ σ ] as above. To show that this construction defines asheaf of noncommutative differential forms on a generic noncommutative toric variety X θ ,as we did for the structure sheaf O X θ in Proposition 4.2, we have to show that these localdefinitions glue together in such a way that they satisfy the sheaf axioms. Proposition 4.22.
The noncommutative differential forms σ Ω θ [ σ ] define a coherentsheaf of O X θ -bimodule algebras Ω X θ on Open ( X θ ) . Proof : We will show that for each affine covering ( σ i ֒ → σ ) i ∈ I there is an exact sequence(4.23) 0 −→ Ω θ [ σ ] −→ Y i ∈ I Ω θ [ σ i ] −→ Y i,j ∈ I Ω θ [ σ i ∩ σ j ] . Exactness of (4.23) is proved by using the exactness of the corresponding sequence (4.3)of coordinate algebras. For brevity, we use the shorthand notation A i = C θ [ σ i ] , A = C θ [ σ ] = \ i ∈ I A i , A ij = C θ [ σ i ∩ σ j ] , and let µ A denote the product map of A . Let I A = ker( µ A ) with canonical inclusiondenoted by ı A : I A → A ⊗ A .Consider the commutative diagram of sequences0 / / A p / / Q i ∈ I A i q / / Q i,j ∈ I A ij / / A ⊗ A µ A O O p / / Q i ∈ I A i ⊗ A iµ Ai O O q / / Q i,j ∈ I A ij ⊗ A ijµ Aij O O / / I Aı A O O p / / Q i ∈ I I A i ı Ai O O q / / Q i,j ∈ I I A ij ı Aij O O O O O O O O where p = p ⊗ p , p = p | I A and similarly for q , q . All columns are exact. The exactnessof the middle row thus follows from the exactness of the top row. Then the exactness ofthe bottom row is proven with standard homological algebra. The map p is injective dueto the injectivity of the maps ı A , p and ı A i , because if there exists 0 = ω ∈ I A such that p ( ω ) = 0 then p ( ı A ( ω )) = 0 but p ( ı A ( ω )) = ı A i ( p ( ω )) = 0. The composition q ◦ p iszero, since if there exists ω ∈ I A such that q ( p ( ω )) = 0 then further composing with ı A ij gives a non-zero element in Q i,j ∈ I A ij ⊗ A ij , while q ( p ( ı A ( ω ))) = 0. Finally, we showthat im( p ) = ker( q ). Let β ∈ ker( q ) and consider its lift b = ı A i ( β ). One has q ( b ) = 0since ı A ij ( q ( β )) = 0, so there exists b ′ ∈ A ⊗ A such that p ( b ′ ) = b . But p ( µ A ( b ′ )) = 0since µ A i ( b ) = 0, so there exists β ′ ∈ I A such that ı A ( β ′ ) = b ′ and p ( β ′ ) = β . Thiscompletes the proof for universal differential forms (the third row).For braided-symmetric differential forms, we further consider the commutative diagram0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / I A ¯ p / / A (cid:15) (cid:15) Q i ∈ I I A i ¯ q / / Ai (cid:15) (cid:15) Q i,j ∈ I I A ij Aij (cid:15) (cid:15) / / I A p / / π A (cid:15) (cid:15) Q i ∈ I I A i q / / π Ai (cid:15) (cid:15) Q i,j ∈ I I A ij π Aij (cid:15) (cid:15) / / Ω A e p / / (cid:15) (cid:15) Q i ∈ I Ω A i e q / / (cid:15) (cid:15) Q i,j ∈ I Ω A ij (cid:15) (cid:15) A is the inclusion I A ֒ → I A and π A is the projection I A → I A /I A , while we set¯ p = p | I A , e p = p | I A /I A and similarly for ¯ q , e q . Again all columns are exact, and theexactness of the bottom row follows from the exactness of the top and middle rows, as onecan check directly by using the same homological algebra we employed above. It followsthat the noncommutative differential forms define a sheaf Ω X θ on Open ( X θ ).The fact that this sheaf is coherent follows from the construction of Ω X θ . Since theconstruction of K¨ahler differentials commutes with the localization functors j • σ of § §
3] and [35, Thm. 1.2.1]), for each affine open set U θ [ σ ] there is an isomorphism ofsheaves j • σ (cid:0) Ω X θ (cid:1) ∼ = Ω θ [ σ ] over U θ [ σ ]. For any finitely generated algebra A the A -bimoduleof K¨ahler differentials Ω A is a finitely generated module over A , since if a , . . . , a n are thegenerators of A then Ω A is generated by d a , . . . , d a n as an A -bimodule. (cid:4) Noncommutative projective varieties
In this section we will specialize to the noncommutative projective spaces X θ = CP nθ .The example n = 2 was treated in detail in § CP nθ provided by the noncommutative affineopen sets U θ [ σ ]. Moreover, they may be used to define noncommutative deformations of LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 33 projective varieties via restriction from CP nθ . In the remainder of this paper we will omitthe star product symbols ⋆ θ for brevity.5.1. Noncommutative projective spaces CP nθ . The construction in § CP gen-eralizes straightforwardly to the higher-dimensional projective spaces CP n , n >
2, re-garded as a toric variety X [Σ] generated by a fan Σ of the lattice L ∼ = Z n of characters ofthe torus T = L ⊗ Z C × ∼ = ( C × ) n . Choose a basis e , . . . , e n of L . Set v i = e i for i = 1 , . . . , n and v n +1 = − e − · · · − e n , which generate the one-dimensional cones τ i = R + v i of Σ. The n + 1 maximal cones of Σ are labelled by the missing generator and are given by σ i = R + v i +1 ⊕ · · · ⊕ R + v i + n , i = 1 , . . . , n + 1 , with indices understood mod n + 1 and σ i ∩ σ i + k = R + v i + k +1 ⊕ · · · ⊕ R + v i + n a maximalcone of CP n − k ֒ → CP n . There are of course many other overlaps, and hence cones, in thisinstance.Again there are no relations and C [ σ ] = C [ x , . . . , x n ] for each maximal cone.(1) The generators of the semigroup σ ∨ n +1 ∩ L ∗ are m i = e ∗ i for i = 1 , . . . , n . Thesubalgebra C θ [ σ n +1 ] ⊂ C θ [ t ± , . . . , t ± n ] is generated by the elements x i = t m i = t i subject to the relations x i x j = q ij x j x i , i < j , (5.1) and hence U θ [ σ n +1 ] ∼ = C nθ .(2) For 1 ≤ k ≤ n , the semigroup σ ∨ k ∩ L ∗ is generated by m i = e ∗ i − e ∗ k for i = k and m k = − e ∗ k . The subalgebra C θ [ σ k ] in this case is generated by elements x i = t i t − k , i = k and x k = t − k with relations x i x k = q ki x k x i , i = k ,x i x j = q ij q ik q jk x j x i , k = i < j . (5.2)The faces can be treated analogously to the n = 2 case.5.2. Homogeneous coordinate algebras.
We now show that there is a noncommu-tative homogeneous coordinate algebra for the noncommutative projective spaces CP nθ ,with a local description given by noncommutative Ore localization which is equivalentto that of the noncommutative affine open sets U θ [ σ ]. The construction depends on anembedding ( C × θ ) n ֒ → ( C × ˜ θ ) ˜ n , with ˜ n > n and ˜ θ suitably defined. Explicit computationsare simplified by considering the embedding ( C × θ ) n → ( C × ˜ θ ) n +1 with˜ θ = (cid:18) θ
00 0 (cid:19) . The corresponding algebraic Moyal plane C n +1˜ θ is defined by the graded polynomial algebra C ˜ θ [ w , . . . , w n +1 ] in n + 1 generators w i , i = 1 , . . . , n + 1 of degree 1 with the quadraticrelations w n +1 w i = w i w n +1 , i = 1 , . . . , n ,w i w j = q ij w j w i , i, j = 1 , . . . , n . (5.3)This algebra is called the homogeneous coordinate algebra A = A ( CP nθ ) of the non-commutative toric variety CP nθ . It is a special instance of the noncommutative weightedprojective spaces defined in [5, § n = 2, it is the same as the noncommutative variety P q, ~ =0 defined in [27, § A is by the usual polynomial degree and one has A = ∞ M k =0 A k , with A = C and A k = L i + ··· + i n +1 = k C w i · · · w i n +1 n +1 for k >
0. The algebra A is made intoa right comodule algebra over the Hopf algebra F ˜ θn +1 via the natural action of GL( n + 1).The ( C × ) n torus action can be recovered by restriction with respect to the embedding of( C × ) n in ( C × ) n +1 described above.It is straightforward to verify that each monomial w i generates a left (and right) de-nominator set in A . Let A [ w − i ] be the left Ore localization of A with respect to w i .Since w i is homogeneous of degree 1, the algebra A [ w − i ] is also N -graded. Elements ofdegree 0 in A [ w − i ] form a subalgebra which we denote by A [ w − i ] . It is not difficult toprove that for each maximal cone σ i ∈ Σ, i = 1 , . . . , n + 1, there is a natural T -equivariantisomorphism of noncommutative algebras C θ [ σ i ] ∼ = A (cid:2) w − i (cid:3) .If I ⊂ A is a graded two-sided ideal generated by a set of homogeneous polynomials f , . . . , f m ∈ C ˜ θ [ w , . . . , w n +1 ], then the quotient algebra A I := A /I is identified as thecoordinate algebra of a noncommutative projective variety . The projection π I : A → A I can be regarded as the dual of a closed embedding given by X θ ( I ) ֒ → CP nθ , identifiedwith the common zero locus in C n +1˜ θ given by the set of relations { f = 0 , . . . , f m = 0 } .Its homogeneous coordinate algebra π I ( C ˜ θ [ w , . . . , w n +1 ]) has relations (5.3) together with f = 0 , . . . , f m = 0. It is also graded, A I = L k ≥ ( A I ) k , with ( A I ) = C and dim C ( A I ) k < ∞ for all k ≥
0. The torus action on A naturally restricts to A I . What is constructedhere could be taken as an example of a noncommutative polarization of a given X θ ( I ).Note that a variety is projective if and only if its deformation is, in the sense that X θ =0 ( I )is projective if and only if X θ ( I ) is projective. This follows from the fact that, once wefix θ , we get a canonical deformation of every algebra acted upon by ( C × ) n , the inverseprocess given by setting θ = 0.In the remainder of this section we will look at some explicit examples, which amongother things will illustrate that in general certain additional algebraic constraints mustbe imposed on the noncommutative ambient space CP nθ .5.3. Noncommutative grassmannians G r θ ( d ; n ) . Using our noncommutative defor-mation of the general linear group GL( n ) from § G r( d ; n ) ∼ = G r( d ; V ), d ≤ n of d -dimensionalsubspaces of an n -dimensional complex vector space V . For this, we will derive a suitablenoncommutative version of Pl¨ucker equations in A ( CP N Θ ) for N = (cid:0) nd (cid:1) −
1, yielding anoncommutative projective variety G r θ ( d ; n ) whose homogeneous coordinate algebra isa graded quadratic algebra with (2.22) as the space of generators. The Drinfel’d twistvia the n × n skew-symmetric complex matrix θ induces constraints on the form of the N × N matrix Θ which realizes the noncommutativity relations in the projective space inwhich we embed the grassmannian. We will find these constraints, whence showing thatin general it is not possible to go in the opposite direction, i.e. there are noncommutativeprojective spaces CP N Θ which do not admit any such embedding due to the form of theirdeformation matrix Θ. LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 35
There is a rich literature on quantum or noncommutative deformations of grassman-nians (see e.g. [32, 40, 23, 21, 27]), mostly relying on q -deformations of matrices, so ournoncommutative relations are somewhat different and easier to deal with. This is becausein our construction the minors of a noncommutative matrix still close to a noncommutativealgebra and in § CP N Θ . Here we shall follow [32] to define thenoncommutative deformation of Pl¨ucker equations, or Young symmetry relations, whichis an approach to noncommutative grassmannians based on quasideterminants [23].Classically, the Pl¨ucker embedding Pl : G r( d ; n ) ∼ = G r( d ; V ) → P ( V d V ) ∼ = CP N , withdim C ( V ) = n and N = (cid:0) nd (cid:1) −
1, is defined as follows: a d × n matrix Λ of maximal rank,representing an element in G r( d, n ) by associating to Λ the subspace of V spanned by therows of Λ, is mapped into the (cid:0) nd (cid:1) -tuple ( . . . , Λ J , . . . ) where each component is a d × d minor of Λ. In the notation of § I = (1 2 · · · d ) so welabel minors by the column multi-index J alone. Pl¨ucker equations in CP N express thecondition on points of the projective space to belong to the image of this embedding.Each Pl¨ucker coordinate can be viewed as a section of a certain ample line bundle over G r( d ; n ), and the collection of such sections defines an embedding of G r( d ; n ) into CP N .Let us fix some notation. For 1 ≤ r ≤ d , denote with I = ( i · · · i d + r ) a ( d + r ) multi-index, with J a ( d − r ) multi-index, and with Ξ = ( i ξ · · · i ξ r ) an r multi-index. Then by I \ Ξ we mean the multi-index ( i · · · ˆ i ξ · · · ˆ i ξ r · · · i d + r ) with the hats indicating omittedindices, and by A ∪ B the multi-index ( a · · · a k b · · · b s ) when | A | = k and | B | = s .Finally, we will use the short-hand notation ǫ A = ǫ a ··· a k . One way to express the Pl¨uckerequations is through the following result [32]. Proposition 5.4.
A point x ∈ CP N ∼ = P ( V d V ) belongs to the image of the Pl¨ucker map Pl( G r( d ; V )) if and only if for all ≤ r ≤ d , and for all choices of multi-indices I and J ,the homogeneous coordinates of x , expressed as d × d minors Λ K of d × n matrices, satisfy (5.5) X Ξ ⊂ I : | Ξ | = r ǫ ( I \ Ξ) ∪ Ξ Λ I \ Ξ Λ Ξ ∪ J = 0 . As a way of exemplification, we show how to prove the classical Plucker relations (5.5)for r = 1 using the Laplace expansion (2.24). We have | I | = d + 1, | J | = d − D = (1 , . . . , d ); we have (when the row are labeled by D we omit it) d +1 X α =1 ǫ I α ∪ i α Λ I α Λ i α ∪ J = d +1 X α =1 ( − ( d +1 − α ) Λ I α (cid:16) d X β =1 ( − (1+ β ) g βi α Λ D β J (cid:17) = d X β =1 ( − (1+ β ) ( − d (cid:16) d +1 X α =1 ( − (1 − α ) Λ I α g βi α (cid:17) Λ D β J = d X β =1 ( − (1+ β + d ) Λ ( β ∪ D ) I Λ D β J = 0where in the first line we expand Λ i α ∪ J with respect to the first column i α , in the secondline we recognize the expression in parenthesis as the expansion of Λ ( β ∪ D ) I along the firstrow β . The last expression is zero since every term in the sum vanishes: for all β ∈ D one has Λ ( β ∪ D ) I = 0, being the determinant of a matrix with two identical rows. Note that each equation (5.5) is quadratic in the homogeneous coordinates of the projec-tive space and has as many terms as the number of submulti-indices of I with cardinality r .The total number of equations is quite large as there is one for each choice of the integer r ,and of the multi-indices I and J . One shows [32, Prop. 13] that all relations with r ≥ r = 1.Let us now turn to the noncommutative setting. In § θ ( n ) ∼ = GL θ ( V ), where V is an H nθ -module of dimension n . An element of the homogeneous coordinate algebra of thenoncommutative grassmannian G r θ ( d ; n ) ∼ = G r θ ( d ; V ) is defined as an element in P ( V dθ V ),obtained by taking the θ -deformed exterior product of d rows of a matrix in A (GL θ ( V ))(and quotienting by the appropriate equivalence relation). The Pl¨ucker maps still makesense. We take a noncommutative d × n matrix representing an element of A ( G r θ ( d ; n ))and send it into the (cid:0) nd (cid:1) -tuple of its minors. Then we need to find the noncommutativityrelations between the minors, seen now as homogeneous coordinates in A ( CP N Θ ) with N = (cid:0) nd (cid:1) −
1, as well as noncommutative Pl¨ucker relations between them.From (2.29) with | J | = | J ′ | = d representing two different minors we have(5.6) Λ J Λ J ′ = (cid:16) d Y α,β =1 q j α j ′ β (cid:17) Λ J ′ Λ J . This implies that the N × N noncommutativity matrix Θ of the projective space con-taining the embedding of G r θ ( d ; n ) is completely determined (mod 2 π ) from the n × n noncommutativity matrix θ of the grassmannian as(5.7) Θ JJ ′ = d X α,β =1 θ j α j ′ β . These relations mean that while given θ there is always one and only one noncommutativeprojective space CP N Θ in which the grassmannian G r θ ( d ; n ) embeds, the converse is ingeneral not true. One can always find a noncommutative projective space for which thereis no compatible noncommutativity matrix θ parametrizing a grassmannian G r θ ( d ; n )which would embed into it. The necessary and sufficient conditions for such an embeddingto exist are given by (5.7). Note that if we instead chose to use ordered column multi-indices, we would again obtain noncommutative relations among the minors which agreewith those in CP N Θ , now with a minus sign on the right-hand side of (5.7).Given the noncommutative relations between generators of the projective space, thenext step is to exhibit noncommutative Pl¨ucker relations. They generate an ideal in thehomogeneous coordinate algebra A ( CP N Θ ) of the projective space, and we will define thenoncommutative quotient algebra to be the homogeneous coordinate algebra A ( G r θ ( d ; n ))of the (embedding of the) noncommutative grassmannian.The noncommutative version of (5.5) is obtained by using the noncommutative Laplaceexpansions (2.25) and (2.26). Indeed, we have the following: Proposition 5.8.
Noncommutative minors of order d in GL θ ( n ) obey Pl¨ucker relations(5.9) d +1 X γ =1 d Y α =1 d − Y β =1 ( − ( γ +1) q i γ i γα q i γ j β Λ I γ Λ i γ ∪ J = 0 , for every choice of multi-indices I, J such that | I | = d + 1 and | J | = d − . LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 37
Proof : We expand Λ i γ ∪ J with respect to its first column i γ ; by (2.26) we haveΛ i γ ∪ J = d X ρ =1 d − Y β =1 ( − (1+ ρ ) Q d ρβ j β ; ρi γ g ρi γ Λ D ρ ; J so that substituting in (5.9) we get(5.10) d +1 X γ =1 d X ρ =1 d Y α =1 d − Y β =1 ( − (1+ γ ) ( − (1+ ρ ) q i γ i γα q ρd ρβ Λ I γ g ρi γ Λ D ρ ; J . In the expression above we can recognize the expansion of Λ ρ ∪ D ; I with respect to the firstrow ρ , which according to (2.25) is d +1 X γ =1 d Y α =1 ( − ( γ +1) Q ρi γ ; αi γα Λ I γ g ρi γ and which is zero for every ρ = (1 , . . . , d ). With this identification we can read (5.10) as d X ρ =1 d Y µ =1 d − Y β =1 ( − (1+ ρ ) q ρd ρβ q ρµ Λ ρ ∪ D ; I Λ D ρ ; J = 0 . (cid:4) By these definitions, one has G r θ (1; n ) = ( CP n − θ ) ∗ . Since dim C ( G r( d ; n )) = d ( n − d ),the n × n matrix θ , which deforms the maximal torus of GL( n ), should be expressedin terms of the ( C × ) d ( n − d ) -action on the grassmannian through a suitable embedding,analogous to those described in § § Remark 5.11.
The Pl¨ucker relations (5.9) are the generalization of the classical ones (5.5) for the the case r = 1 . We are unable to state Pl¨ucker relations for arbitrary r norto prove that the general case can be reduced to the case r = 1 , as in the undeformedsituation, though this is true for every example we have worked out. For q -deformationsconsidered in [32] , this is implied by Prop. 13 there.
The classical Pl¨ucker relations (5.5) contain trivial identities when I ∩ J = ∅ , togetherwith “true” Pl¨ucker equations. The same situation arises in the noncommutative case,but now the “trivial” identities encode the noncommutativity and alternating relations ofthe noncommutative minors. In fact, in certain instances it seems that starting from (5.9),one can derive all relations necessary to describe the noncommutative Grassmann variety,i.e. the “true” Pl¨ucker equations as well as the noncommutativity relations between thegenerators of A ( CP N Θ ) in (5.6) and the alternating property (2.30). Again we will returnto this point in more generality below.5.4. Noncommutative flag varieties F l θ ( d , . . . , d r ; n ) . We will now generalize the con-structions of § n -dimensional complex vectorspace V and a sequence of positive integers γ = ( γ , . . . , γ r +1 ) with 1 ≤ r ≤ n − n , i.e. a Young diagram, we consider an increasing chain of nested vectorsubspaces of V , 0 = V V V · · · V r +1 = V , such that γ i = dim C ( V i ) − dim C ( V i − ) for i = 1 , . . . , r + 1. The corresponding flagvariety F l( γ ; V ) ∼ = F l( γ ; n ) is the moduli space of chains (or “flags”) associated to the sequence γ = ( γ , . . . , γ r +1 ). Two typical examples are the complete flag varieties withpartition γ = (1 , . . . ,
1) ( n times), i.e. the sequences of subspaces where dim C ( V i ) = i for i = 1 , . . . , n , and the grassmannians G r( d ; n ) which are here recovered from the two-termpartitions γ = ( d, n − d ).By choosing a basis in V , the flag varieties F l( γ ; V ) can also be represented as spacesof equivalence classes of matrices in the reductive algebraic group GL( n ). We representa chain of subspaces by a matrix whose rows are the basis vectors of each subspace, andnotice that the part of GL( n ) which acts trivially on such a representation is given by blockupper (or lower) triangular matrices, with r +1 diagonal blocks of dimensions γ , . . . , γ r +1 .These matrices form a subgroup of GL( n ) denoted P γ . It is a parabolic group, and the flagvariety may be realized as the homogeneous space F l( γ ; n ) = GL( n ) /P γ with associatedprincipal bundle P γ ֒ → GL( n ) −→ F l( γ ; n ) . (5.12)The Borel subgroup of GL( n ) is the parabolic group P γ associated with γ = (1 , . . . , B n . Since B n is the minimal parabolic subgroup of GL( n ), each flagvariety F l( γ ; n ) is the total space of a canonical fibration over the corresponding completeflag variety with fibre P γ /B n given by P γ (cid:14) B n ֒ → GL( n ) (cid:14) P γ π −→ GL( n ) (cid:14) B n . We shall describe the Pl¨ucker embedding of flag varieties into projective spaces, in asimilar way as in the case of grassmannians. This involves the minors of the n × n matrixrepresenting each flag. Set d i = P a ≤ i γ a = dim C ( V i ) for i = 1 , . . . , r + 1. Given a point in F l( γ ; n ) represented by an equivalence class [ A ] in GL( n ) /P γ , there is a natural Pl¨uckermap Pl i : F l( γ ; n ) → CP N i , with N i = (cid:0) nd i (cid:1) − i , where the image is the (cid:0) nd i (cid:1) -tupleof all minors of A obtained from the first d i rows. Hence each minor is labelled by amulti-index representing the d i columns involved while the rows are always given by thestandard ordered multi-index (1 2 · · · d i ). Assembling all of these maps together we get aPl¨ucker embedding(5.13) Pl : F l( γ ; n ) −→ CP ( γ ; n ) := CP N × · · · × CP N r , where the last factor corresponding to i = r + 1 gives a trivial contribution since N r +1 = (cid:0) nn (cid:1) − CP ( γ ; n ) is described by a set of quadraticequations called the Young symmetry relations. With the same notation, a generalizationof Proposition 5.4 to flag varieties is given by the following result [33]. Proposition 5.14.
Given a partition γ of n and the Pl¨ucker map Pl in (5.13), a point x in CP ( γ ; n ) belongs to the image Pl( F l( γ ; n )) if and only if for all choices of multi-indicesgiven by I = ( i · · · i d + s ) and J = ( j · · · j d ′ − s ) , as subsets of (1 2 · · · n ) for all s ≥ andfor all d, d ′ ∈ { d i } i =1 ,...,r +1 with d ≥ d ′ , the homogeneous coordinates of x , expressed as d i × d i minors of n × n matrices now of variable size, satisfy the Young symmetry relations (5.15) X Ξ ⊆ I : | Ξ | = s ǫ ( I \ Ξ) ∪ Ξ Λ I \ Ξ Λ Ξ ∪ J = 0 . We are now ready to construct a noncommutative deformation of flag varieties, general-izing what we did in § LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 39 matrices with noncommuting entries is the same as in (2.23). We now need to handle non-commutative minors of different size, with each size describing a projective space in thecartesian product CP ( γ ; n ), and apply a noncommutative version of the Young symme-try relations (5.15) instead of (5.5). The relations (5.5) essentially describe the relationsamong minors of fixed size, so they describe the image of the Pl¨ucker embedding in eachprojective space copy (with appropriate dimension) inside CP ( γ ; n ). What (5.15) adds isto express relations between minors of different size, i.e. relations between coordinates ofdifferent factors in CP ( γ ; n ).In this case we use the more general noncommutative relations (2.29) between d × d and d ′ × d ′ minors of different size, i.e. with multi-indices of different lengths | I | = | J | = d and | I ′ | = | J ′ | = d ′ . The noncommutative Young symmetry relations are again derivedfrom the Laplace expansion of the minors in (2.25) and (2.26). In particular the r = 1case of the classical relations (5.15) is proved in a way similar to (5.9). Proposition 5.16.
Noncommutative minors of order d and d ′ in GL θ ( n ) obey Youngsymmetry relations(5.17) d +1 X γ =1 d Y µ =1 d ′ − Y ν =1 ( − ( γ +1) q i γ i γµ q i γ j ν Λ I γ Λ i γ ∪ J = 0 , for every choice of multi-indices I and J with | I | = d + 1 and | J | = d ′ − . In this setting the coordinate algebra of the noncommutative flag variety F l θ ( γ ; n ) = F l θ ( d , . . . , d r ; n ) is the quotient of the homogeneous coordinate algebra of CP Θ ( γ ; n ) bythe ideal generated by the noncommutative Young symmetry relations (5.17). As we didfor noncommutative grassmannians, it is useful to distinguish between the different kindsof equations that are generated by the noncommutative Young symmetry relations. Wewill divide them into three classes, called alternating equations, structure equations, andPl¨ucker equations.By alternating equations we mean relations like (2.30), i.e. the behaviour of a minorunder interchange of two indices inside the multi-index which parametrizes it. These equa-tions are in principle contained in the definition of noncommutative minors, and once wehave decided to parametrize coordinates in the projective spaces which are targets for ourPl¨ucker map by ordered multi-indices, they are not to be interpreted as relations betweencoordinates of these projective spaces. However, in Proposition 5.14 it is convenient toconsider unordered multi-indices I and J , since even when J is ordered the multi-index i γ ∪ J is in general not ordered, so the Young symmetry relations automatically generateequations with unordered multi-indices. This increases the number of equations in theYoung symmetry relations, as it increases the number of ways in which one can choose I and J , exactly by adding relations of alternating type. These are the ones in which I γ and i γ ∪ J differ only by permutations. This can only happen when d = d ′ , and thealternating relations are a particular class of equations where only two terms in the sumsurvive. Thus by including unordered multi-indices, alternating relations arise as a subsetof the Young symmetry relations.By structure equations we mean the class of equations where only two terms repre-senting distinct noncommuting coordinates in A ( CP Θ ( γ ; n )) survive. They specify thenoncommutativity of the target space of the Pl¨ucker embedding. In § matrix Θ of CP N Θ has to satisfy the constraints (5.7). It is natural to now ask if thesestructure equations could have been completely deduced from the noncommutative Youngsymmetry relations, or if they have to be put in by hand when defining the noncommu-tative projective space of the Pl¨ucker embedding. Some straightforward combinatorialconsiderations show that only a small part of the structure equations for CP Θ ( γ ; n ) are asubset of the Young symmetry relations, and all other noncommutativity relations mustbe introduced independently. Proposition 5.18.
The only structure equations contained in the noncommutative Youngsymmetry relations are those within a single factor of the algebra A ( CP Θ ( γ ; n )) involvingminors whose multi-indices differ in only one index. Proof : We look at the conditions needed for an equation of the Young symmetry relations(5.15) to reduce to a two-term equation. Each equation has d + 1 terms. To reduce thisnumber to 2 and get a structure equation, I and J must contain some common indicesso that when J takes indices from I we get a repetition of indices in i γ ∪ J , and thecorresponding term in the equation vanishes. Denote by k the number of shared indices,i.e. | I ∩ J | = k . The constraints are k ≤ d ′ − d ≥ d ′ . The number of survivingterms in each equation is d + 1 − k and hence the condition we want is d + 1 − k = 2. Thisimplies that structure equations only arise for noncommutative minors of equal size d = d ′ (i.e. inside a single factor of A ( CP Θ ( γ ; n ))), and it is not possible to recover any of thestructure equations between minors of different size (i.e. between coordinates of differentnoncommutative projective space factors in CP Θ ( γ ; n )). For fixed d = d ′ , these constraintsalso show that k = d − d ′ −
1. So to obtain structure equations, J must be a subset of I obtained by removing two indices, i.e. the two minors involved differ only by one index. (cid:4) The remaining relations involving more than two terms are called
Pl¨ucker equations .They are quadratic in the coordinate algebra generators of the noncommutative projectivespaces, and are the ones which genuinely describe the image of the Pl¨ucker embedding,i.e. the projection given by A ( CP Θ ( γ ; n )) → A ( F l θ ( γ ; n )) which realizes F l θ ( γ ; n ) as anoncommutative quadric in CP Θ ( γ ; n ). By (2.29) and Proposition 5.14, there are canonicalinclusions of homogeneous coordinate algebras p i : A (cid:0) F l θ ( d , . . . , ˆ d i , . . . , d r ; n ) (cid:1) −→ A (cid:0) F l θ ( d , . . . , d r ; n ) (cid:1) of noncommutative flag varieties for each i = 1 , . . . , r . For generic n , this leads to aweb of multiple noncommutative fibrations, which are classically obtained by truncat-ing flags in the obvious way. Furthermore, the additional relations coming from (2.29)are naturally compatible with the structure of the braided tensor product of algebras A ( G r θ ( d ; n )) b ⊗ θ · · · b ⊗ θ A ( G r θ ( d r ; n )) induced by the braiding morphism Ψ θ on the cate-gory H nθ M of H nθ -module algebras as explained in § A ( F l θ ( d , . . . , d r ; n )) may be realized as the quotient algebra of this braidedtensor product by the additional relations arising from (5.15), and there is a naturalalgebra surjection A (cid:0) G r θ ( d ; n ) (cid:1) b ⊗ θ · · · b ⊗ θ A (cid:0) G r θ ( d r ; n ) (cid:1) −→ A (cid:0) F l θ ( d , . . . , d r ; n ) (cid:1) . Geometry of noncommutative projective varieties
We now develop a more thorough noncommutative sheaf theory and, with the alter-native description of § LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 41 of projective varieties. In this way noncommutative projective varieties inherit algebraicand geometric properties from CP nθ by restriction. These properties are described below.6.1. Cohomology of CP nθ . We start by summarizing the pertinent cohomological prop-erties of the homogeneous coordinate algebras A . We write mod ( A ) for the category ofall finitely-generated right A -modules. Since most of the results hold for generic valuesof θ , and hence they are similar to the commutative case, we often omit the proofs (seee.g. [7, 9] for some details).The algebra A = A ( CP nθ ) is a quadratic algebra whose Koszul dual A ! is generated byelements ˇ w i , i = 1 , . . . , n + 1, with the relationsˇ w i = 0 , i = 1 , . . . , n + 1 , ˇ w i ˇ w n +1 + ˇ w n +1 ˇ w i = 0 , i = 1 , . . . , n , ˇ w i ˇ w j + q ij ˇ w j ˇ w i = 0 , i, j = 1 , . . . , n . (6.1)The dual algebra A ! = L k ≥ A ! k is a deformation of the exterior algebra of A ∗ , gradedagain by polynomial degree. It is a special case of the graded DG-algebras defined in [5, § H ˜ θ M of H ˜ θ -modules, there are isomorphisms A ! k ∼ = V k ˜ θ A ∗ . There is a canonical identification ( A ! ) ! = A . In a way similar to the commutative case,one defines the right Koszul complex K • ( A ) (as well as the left one). One use we makeof the Koszul complex is in establishing crucial “smoothness” properties of our algebras.Considering a minimal free resolution of the trivial right A -module A = C ,0 −→ E d ⊗ A −→ · · · −→ E ⊗ A −→ A −→ A −→ , (6.2)with E = A and E = R ⊂ A ⊗ A the space of quadratic relations (5.3), the integer d is the “global homological dimension” gl-dim( A ) of the algebra A [4]; it is shown to befinite for the case at hand.By applying the functor Hom mod ( A ) ( A , − ) to the chain complex of (free) right A -modulesin (6.2), one obtains a cochain complex of left A -modules whose cohomology is denotedby Ext • mod ( A ) ( A , A ). One can show that Ext k mod ( A ) ( A , A ) = δ k,d C , and this means thatthe algebra A is “Gorenstein” and that the cochain complex as well defines a minimalprojective resolution of the trivial left A -module A . Together with (6.2) this implies theisomorphisms E ∗ k = Ext k mod ( A ) ( A , A ) ∼ = E d − k of vector spaces for k = 0 , , . . . , d . Thus the Gorenstein property is a variant of Poincar´eduality for the noncommutative toric variety CP nθ .It is also not hard to prove that the homogeneous coordinate algebra A = A ( CP nθ ) is anoetherian domain, a Koszul algebra, and that A ! is a Frobenius algebra of index n + 1.Algebras of finite global homological dimension with the Gorenstein property are calledregular [19]. The following result is a corollary of [5, Prop. 2.6]. Proposition 6.3.
The quadratic algebra A is a regular algebra of global homological di-mension d = gl-dim( A ) = n + 1 . Proof : This follows similarly to [6, Prop. 7.2.3]. As mentioned, the global homologicaldimension of A equals the length of the minimal projective resolution for A = C . Since the Koszul complex is exact, it provides such a minimal resolution, and the global ho-mological dimension coincides with the number of non-trivial graded components of thealgebra A ! , each of which can be identified as A ! k ∼ = Ext k mod ( A ) ( A , A ) . The dual algebra A ! provides a Frobenius resolution, thus the only non-trivial cohomologygroup is Ext n +1 mod ( A ) ( A , A ) ∼ = A ! n +1 ⊗ A , and the Gorenstein property follows. (cid:4) For an algebra A of polynomial growth (which is the case for the homogeneous coordi-nate algebra A = A ( CP nθ )), one has also the notion of Gel’fand–Kirillov dimensionGK-dim( A ) := lim inf k →∞ n α ∈ R (cid:12)(cid:12)(cid:12) dim C (cid:16) k L l =0 A l (cid:17) ≤ k α o . This is finite for A = A ( CP nθ ). Indeed dim C ( A k ) = p n +1 ( k ) is the number of partitions of k into n +1 parts, and it is a classic result [20] that the function p n +1 ( k ) grows asymptoticallylike n +1)! (cid:0) k − n (cid:1) . Then the Stirling expansion shows that the dimension of A k grows like k n for k ≫
0, so that the Gel’fand–Kirillov dimension is n + 1. Combining this with theGorenstein properties of Proposition 6.3 we see that A = A ( CP nθ ) is regular in the senseof Artin–Schelter [1].6.2. Sheaves on CP nθ . By Propositions 4.2 and 4.6, together with the results of § Open ( CP nθ ) can be identified with objects ofthe module category mod ( A ), with A = A ( CP nθ ). Let gr ( A ) be the category of finitely-generated graded right A -modules M = L k ≥ M k and degree zero morphisms, and let tor ( A ) be the full Serre subcategory of gr ( A ) consisting of finite-dimensional graded A -modules M , i.e. M k = 0 for k ≫
0. Henceforth, we will identify the category of coherentsheaves on
Open ( CP nθ ) with the abelian quotient category gr ( A ) / tor ( A ), and denote itby coh ( CP nθ ). Let π : gr ( A ) → coh ( CP nθ ) be the canonical projection functor. Underthis correspondence, the structure sheaf O CP nθ is the image π ( A ) of the homogeneouscoordinate algebra itself, regarded as a free right A -module of rank one. If E = π ( M )where M ∈ gr ( A ) is a graded right A -module, then M [ w − i ] = ( M ⊗ A A [ w − i ]) is a right C θ [ σ i ]-module for each i = 1 , . . . , n + 1.On the category gr ( A ) there is a natural autoequivalence defined by the degree shiftfunctor M M (1), where M ( l ) is the l -th shift of the graded module M = L k ≥ M k with degree k component M ( l ) k = M l + k . For each k ∈ Z we define the sheaf O CP nθ ( k ) := π (cid:0) A ( k ) (cid:1) . For any sheaf E = π ( M ) we write E ( k ) for the sheaf π ( M ( k )) in coh ( CP nθ ). Conversely,given a sheaf E ∈ coh ( CP nθ ), the vector space M = Γ( E ) := ∞ M k =0 Hom (cid:0) O CP nθ ( − k ) , E (cid:1) is a graded right A -module with π ( M ) = E (with the A -module structure given in generalby [2, eq. (4.0.3)]).As in [5, § Open ( CP nθ ) have the following basic cohomological properties. LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 43
Proposition 6.4.
Every sheaf E ∈ coh ( CP nθ ) enjoys the following properties: (1) Ampleness: There exists an epimorphism s M i =1 O CP nθ ( − k i ) −→ E −→ for some positive integers k , . . . , k s , and there exists a positive integer k suchthat H p ( CP nθ , E ( k )) = 0 for all k ≥ k and p > ; (2) χ -condition: dim C ( H p ( CP nθ , E )) < ∞ for all p ≥ ; and (3) Serre duality: There are natural isomorphisms of complex vector spaces H p (cid:0) CP nθ , E (cid:1) ∼ = Ext n − p (cid:0) E , O CP nθ ( − n − (cid:1) ∗ where ( − ) ∗ denotes the C -dual. Proof : This follows from the regularity properties of the algebra A derived in § (cid:4) The following result is a special case of [5, Prop. 2.7].
Proposition 6.5. (1)
There are isomorphisms H p (cid:0) CP nθ , O CP nθ ( k ) (cid:1) = A k for p = 0 , k ≥ , A ∗− k − n − for p = n , k ≤ − n − , otherwise . (2) The cohomological dimension of the category coh ( CP nθ ) is equal to n , i.e. one has H p ( CP nθ , E ) = 0 for all E ∈ coh ( CP nθ ) and p > n . Proof : This follows from the regularity properties of the homogeneous coordinate algebra A derived in § (cid:4) Let gr L ( A ) be the abelian category of finitely-generated graded left A -modules. Wewill denote by π L : gr L ( A ) → coh L ( CP nθ ) := gr L ( A ) / tor L ( A ) the corresponding quotientprojection. For any sheaf E ∈ coh ( CP nθ ), the graded space H om (cid:0) E , O CP nθ (cid:1) = π L (cid:16) ∞ L k =0 Hom (cid:0)
E , O CP nθ ( k ) (cid:1) (cid:17) has a natural left A -module structure (see [27, § § coh L ( CP nθ ). It is called the dual sheaf of E andis denoted E ∨ . The internal Hom-functor H om ( − , O CP nθ ) is left exact on coh ( CP nθ ) → coh L ( CP nθ ) and has corresponding right derived functors E xt p ( − , O CP nθ ) given by E xt p (cid:0) E , O CP nθ (cid:1) = π L (cid:16) ∞ L k =0 Ext p (cid:0) E , O CP nθ ( k ) (cid:1) (cid:17) for p ≥
0. Since A is a noetherian regular algebra, the functor E xt p ( − , O CP nθ ) gives ananti-equivalence between the derived categories of coh ( CP nθ ) and coh L ( CP nθ ) (see [42, § § p ( E, F ) ∼ = Ext pL ( F ∨ , E ∨ ) := Ext p coh L ( CP nθ ) ( F ∨ , E ∨ )for any p ≥ E, F ∈ coh ( CP nθ ). For a sheaf F ∈ coh ( CP nθ ), there is a functorial isomorphism H L (cid:0) CP nθ , H om ( F, O CP nθ ) (cid:1) ∼ = Hom (cid:0) F , O CP nθ (cid:1) , and also a functorial spectral sequence E p,q = H pL (cid:0) CP nθ , E xt q ( F, O CP nθ ) (cid:1) = ⇒ Ext • (cid:0) F , O CP nθ (cid:1) . The sheaves O CP nθ ( k ), k ∈ Z are locally free, with H om ( O CP nθ ( k ) , O CP nθ ( l )) = O CP nθ ( l − k ) assheaves of bimodules. More generally, bundles over noncommutative projective varietiesmay be characterized as follows. Proposition 6.6.
Let E ∈ coh ( CP nθ ) and M = Γ( E ) ∈ gr ( A ) . Then the following state-ments are equivalent: (1) E is a locally free sheaf; (2) E xt p ( E , O CP nθ ) = 0 for all p > ; and (3) M [ w − i ] is projective in coh ( σ i ) for each i = 1 , . . . , n + 1 . Proof : This is a consequence of Proposition 4.6 and Definition 4.7, together with thefunctorial equivalence of § θ = 0 [27]. If E is locally free, then its restrictions E σ i are direct sums of shifts of C θ [ σ i ],with C θ [ σ i ]( k ) := (cid:0) A ( k ) ⊗ A A [ w − i ] (cid:1) . Since Ext p gr ( A ) ( A ( l ) , A ( k )) = 0 for k > l and p >
0, it follows from the χ -conditionof Proposition 6.4 that L k ≥ Ext p ( E, O CP nθ ( k )) is finite-dimensional, and hence one has E xt p ( E , O CP nθ ) = 0 for all p >
0. Conversely, by Serre duality of Proposition 6.4 one has E xt p ( E , O CP nθ ) ∼ = π L (cid:16) ∞ L k =0 H n − p (cid:0) CP nθ , E ( − k − n − (cid:1) ∗ (cid:17) , where the group H n − p ( CP nθ , E ( − k − n − n − p ( O CP nθ ( k + n + 1) , E ).Hence if E xt p ( E , O CP nθ ) = 0 for p >
0, then by the χ -condition Ext s ( O CP nθ ( k + n + 1) , E ) = 0for all 0 ≤ s < n and k ≫
0. That E is locally free now follows again by localization andthe corresponding result in the category gr ( A ). Finally, if M is projective, then the func-tor Hom gr ( A ) ( M, − ) is exact, and hence Ext p gr ( A ) ( M, A ( k )) = 0 for all p > k ≥ (cid:4) Example 6.7.
For noncommutative projective varieties we can provide an equivalentglobal description of the sheaves of differential forms, constructed in § U θ [ σ ] correspond to localizations of the homo-geneous coordinate algebra A = C ˜ θ [ w , . . . , w n +1 ] of CP nθ , and the construction of K¨ahlerdifferentials commutes with Ore localization (see e.g. [15, § and [35, Thm. 1.2.1] ). Thebimodule of K¨ahler differentials Ω A = I A /I A is defined as in § µ A : A ⊗ A → A . Using the constructions of § Ω A is isomorphic to the free A -bimodule A ⊕ ( n +1) . On the other hand, since A is aKoszul algebra one can define the left (resp. right) A -module K p ( A ) as the cohomology ofthe left (resp. right) Koszul complex of A in § p -th term [27, Def. 4.8] .For p = 1 , the module K ( A ) sits in the exact sequence −→ K ( A ) −→ (cid:0) A !1 (cid:1) ∗ ⊗ A d −→ A ε −→ C −→ LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 45 so that K ( A ) = ker(d) . But here the differential d is exactly µ A . It follows that there isa natural identification Ω p A ,un ∼ = K p ( A ) , and so the Koszul description of the sheaves ofdifferential forms coincides with that in terms of K¨ahler differentials in these cases. Tautological bundles on G r θ ( d ; n ) . We give some explicit examples of locallyfree sheaves on the noncommutative Grassmann varieties of § § S is the vector bundle over G r( d ; V ) such that the fibre over each point [Λ] ∈ G r( d ; V ) isthe d -plane V Λ ⊂ V defined by Λ itself. It sits inside the Euler sequence0 −→ S −→ G r( d ; V ) × V −→ Q −→ , (6.8)where Q is the quotient sub-bundle. To describe the embedding of S in the trivial bundle G r( d ; V ) × V , we note that, when dim C ( V ) = n , a section of G r( d ; V ) × V is an n -dimensional vector w = n X i =1 w i (Λ) ⊗ v i ∈ A (cid:0) G r( d ; V ) (cid:1) ⊗ V (6.9)of functions w i (Λ) on G r( d ; V ), where { v i } ni =1 is any basis for V . This defines a sectionof S if and only if for each Λ the vector (6.9) belongs to V Λ .In that case, if we add the vector w to the d × n matrix Λ as the ( d + 1)-th row, thusgenerating a ( d + 1) × n matrix, then all the minors of order d + 1 are zero. Denote by J = ( j · · · j d +1 ) an ordered ( d + 1) multi-index with j < j < · · · < j d +1 , and by J α theorder d multi-index with j α removed. Then, as before, Λ J α is the minor of order d in Λobtained from the columns labelled by J α = ( j α , . . . , j αd ). By expanding the minors withrespect to the ( d + 1)-th row w , the requisite condition can be expressed as the equations(6.10) d +1 X α =1 ǫ ( J α ) ∪ j α Λ J α w j α = 0for every ordered ( d + 1) multi-index J . A section of the trivial bundle (6.9) is a sectionof S if and only if it satisfies (6.10). This is a local description since we have to choose a d × n matrix Λ to represent a point in G r( d ; V ), and our condition (6.10) is written usingthe data of this local representative.To pass to the noncommutative coordinate algebra A ( G r θ ( d ; n )), we use the Laplaceexansion in (2.25). Then S θ is defined to be the subsheaf of elements of the free mod-ule ( w (Λ) , . . . , w n (Λ)) ∈ A ( G r θ ( d ; n )) ⊕ n over the noncommutative grassmannian whichsatisfy the equations(6.11) d +1 X α =1 d Y β =1 q j α j αβ ! ( − α Λ J α w j α = 0for every ordered ( d +1) multi-index J , where the minors of order d obey the relations (5.6).We can use the Pl¨ucker map to regard the noncommutative minors Λ J α as homogeneouscoordinates in P ( V dθ V ). Then the quotient by the graded two-sided ideal generatedby the set of homogeneous relations (6.11) defines the projection from the free module P ( V dθ V ) ⊗ V → S θ . In this case we have to consider the restriction of (6.11) to thoseelements Λ which also satisfy the Young symmetry relations (5.9). This gives the sheaf S θ the natural structure of a graded A ( G r θ ( d ; n ))-bimodule. Proposition 6.12.
The sheaf S θ is locally free on Open ( G r θ ( d ; n )) . Proof : The geometric description of the embedding of S in G r( d ; V ) × V by a projectoramounts to taking a section (6.9) and projecting the vector w over the d -plane V Λ for each[Λ] ∈ G r( d ; V ). To obtain a well-defined projector, we choose an inner product h− , −i Λ onthe complex vector space V such that the vectors v , . . . , v d which span V Λ are orthonor-mal. Then the projection of a vector w ∈ V over V Λ is given by p Λ ( w ) = P i h w, v i i Λ v i .This yields a unique idempotent p : A ( G r( d ; V )) ⊗ V → A ( G r( d ; V )) ⊗ V which maps w (Λ) in (6.9) to p Λ ( w (Λ)), with p = p , trace equal to d , and im( p ) = S . The matrixrepresentation of p Λ is given by the n × n matrix Λ ⊤ Λ, where for Λ we choose a ma-trix representative whose d rows are the orthonormal generators of the plane V Λ so thatΛ Λ ⊤ = 1 and (Λ ⊤ Λ) (Λ ⊤ Λ) = Λ ⊤ Λ. The extension to the noncommutative setting onlyrequires using noncommuting entries in Λ with noncommutative relations in the coordi-nate algebra F θn of GL θ ( n ), given in § (cid:4) Example 6.13.
For d = 1 , it is easy to see that the equations (6.11) are solved bytaking w j (Λ) = Λ j to be the generators of the homogeneous coordinate algebra A ( CP n − θ ) ,and one has a canonical isomorphism of bimodules S θ ∼ = O CP n − θ (1) . Alternatively, useProposition 6.12 to get im( p ) ∼ = A ( CP n − θ ) . Differential forms on G r θ ( d ; n ) . There is also a useful alternative description ofthe bundle of K¨ahler differentials Ω G r θ ( d ; n ) . In the classical case, the tangent bundle over G r( d ; V ) is represented in terms of the Euler sequence (6.8) as the morphism bundleHom( S , Q ) ∼ = S ∨ ⊗ Q , whose fibre spaces are given by T [Λ] G r( d ; V ) = Hom C ( V Λ , V /V Λ ).This description can be transported to the noncommutative setting via the followingcharacterization. Lemma 6.14.
The total space of the cotangent bundle over the grassmannian G r( d ; n ) isthe base of the principal fibration L d,n − d := GL( d ) × GL( n − d ) ֒ → GL( n ) −→ T ∗ G r( d ; n ) . Proof : Let E denote the principal P d,n − d -bundle given in (5.12) for γ = ( d, n − d ).Let g and p be the Lie algebras of GL( n ) and P d,n − d , respectively. Then the cotangentbundle can be represented by T ∗ G r( d ; n ) = E × Ad ∗ ( P d,n − d ) ( g / p ) ∗ . If P d,n − d is embeddedin GL( n ) as the subgroup of upper triangular matrices, then a ∈ g / p is represented bya (strictly) block upper triangular matrix. Embed L d,n − d in GL( n ) as the subgroup ofblock diagonal matrices. Then L d,n − d is the reductive Levi subgroup of P d,n − d and thereis a Levi decomposition P d,n − d = R d,n − d ⋉ Ad( L d,n − d ) L d,n − d , where R d,n − d is the unipotentradical of the parabolic group P d,n − d which is the additive subgroup of GL( n ) representedby block upper d × ( n − d ) matrices with respect to this embedding. On GL( n ) /L d,n − d there is still the proper and free left action of R d,n − d , and the quotient is our grassmannian R d,n − d (cid:15) GL( n ) (cid:14) L d,n − d = GL( n ) (cid:14) P d,n − d = G r( d ; n ) . We claim that this principal R d,n − d -bundle F → G r( d ; n ) is isomorphic to the cotangentbundle. For this, we define a bundle map T ∗ G r( d ; n ) → F , such that on the fibre overthe equivalence class of the identity of GL( n ) in GL( n ) /P d,n − d there is an isomorphism P d,n − d × Ad ∗ ( P d,n − d ) ( g / p ) ∗ → R d,n − d . With respect to the block embeddings described LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 47 above, this is given by (cid:16) (cid:18)
M A N (cid:19) , a (cid:17) (cid:18) M a N − (cid:19) . Since the two fibrations have the same base space, the bundle map reduces to a morphismbetween the fibre spaces. Since the base space is homogeneous with respect to the actionof GL( n ), the isomorphism on a generic fibre is the conjugation by GL( n ) of the isomor-phism over the identity constructed above. (cid:4) We will use Lemma 6.14 to provide a purely algebraic description of the cotangentbundle in terms of coinvariant elements in the Hopf algebra F n of GL( n ) with respect to thecoaction induced from the subgroup L d,n − d . Then we will deform this construction usinga Drinfel’d twist, obtaining an alternative description of the bundle of noncommutativeK¨ahler differentials Ω G r θ ( d ; n ) . The algebraic version of the inclusion L d,n − d ֒ → GL( n ) is asurjective algebra homomorphism π ( L d,n ) from F n to the Hopf subalgebra L d,n dual to thesubgroup L d,n − d . As in § F n = Fun(GL( n )) by g ij with i, j = 1 , . . . , n . The generators of L d,n = Fun( L d,n − d ) are denoted l ij with 1 ≤ i, j ≤ d and d + 1 ≤ i, j ≤ n . Then the projection homomorphism π ( L d,n ) : F n → L d,n is given by(6.15) π ( L d,n ) ( g ij ) = (cid:26) l ij , ≤ i, j ≤ d and d + 1 ≤ i, j ≤ n , , otherwise . The left coaction L d,n Φ : F n → L d,n ⊗ F n dual to the right multiplicative action of L d,n − d on GL( n ) is the unital algebra morphism given by L d,n Φ := (cid:0) π ( L d,n ) ⊗ (cid:1) ∆ ∨ , or explicitly(6.16) L d,n Φ( g ) = (cid:0) π ( L d,n ) ⊗ (cid:1) ∆ ∨ ( g ) = π ( L d,n ) ( g (1) ) ⊗ g (2) . The subalgebra of left coinvariants, defined in the usual way by co − L d,n F n = (cid:8) g ∈ F n (cid:12)(cid:12) L d,n Φ( g ) = 1 ⊗ g (cid:9) , gives the algebraic description of the base of the fibration GL( n ) /L d,n − d , i.e. the cotangentbundle T ∗ G r( d ; n ). We use the general strategy to find coinvariants through projectormaps [29, Ch. 13]. Proposition 6.17.
A set of generators for co − L d,n F n is given by elements (6.18) η ij := d X k =1 S ∨ ( g ik ) g kj , ≤ i, j ≤ n and (6.19) η ⊥ ij := n X k = d +1 S ∨ ( g ik ) g kj , ≤ i, j ≤ n . Proof : By direct computation one has L d,n Φ (cid:16) d X k =1 S ∨ ( g ik ) g kj (cid:17) = (cid:0) π ( L d,n ) ⊗ (cid:1) (cid:16) d X k =1 n X m,p =1 (cid:0) S ∨ ( g pk ) g km (cid:1) ⊗ (cid:0) S ∨ ( g ip ) g mj (cid:1) (cid:17) = d X k,m,p =1 (cid:0) S ∨ ( l pk ) l km (cid:1) ⊗ (cid:0) S ∨ ( g ip ) g mj (cid:1) = d X m,p =1 δ pm ⊗ (cid:0) S ∨ ( g ip ) g mj (cid:1) = 1 ⊗ (cid:16) d X p =1 S ∨ ( g ip ) g pj (cid:17) . The coinvariance of the second set of generators follows easily from n X k =1 S ∨ ( g ik ) g kj = δ ij , since the coinvariants generate a vector space. (cid:4) The generators η ij and η ⊥ ij = δ ij − η ij are not independent, but are characterized by aset of relations. They can be regarded as entries of n × n matrices, yielding an algebraicdescription of the vector bundle with associated principal bundle given in Lemma 6.14. Proposition 6.20.
The generators η ij (resp. η ⊥ ij ) for i, j = 1 , . . . , n are the entries of anidempotent matrix η (resp. η ⊥ ) with trace equal to d (resp. n − d ). Proof : Again by direct computation one has n X m =1 η im η mj = n X m =1 d X k,p =1 S ∨ ( g ik ) g km S ∨ ( g mp ) g pj = d X k,p =1 S ∨ ( g ik ) δ kp g pj = d X k =1 S ∨ ( g ik ) g kj = η ij . The trace condition is easily computed as n X m =1 η mm = n X m =1 d X k =1 S ∨ ( g mk ) g km = d X k =1 δ kk = d . The corresponding results for η ⊥ = 11 n × n − η now easily follow. (cid:4) Comparing with Proposition 6.12 and Lemma 6.14, it follows that we can interpret η as the matrix describing the finitely-generated projective A ( G r( d ; n ))-module S ∼ = η (cid:0) A ( G r( d ; n )) ⊕ n (cid:1) . Recall that there is a canonical isomorphism G r( d ; V ) ≈ −→ G r( n − d ; V ∗ )of grassmannians given by V Λ ( V /V Λ ) ∗ . Under this isomorphism, the universal quotientbundle Q on G r( d ; V ) corresponds to the dual of the tautological bundle S ⊥ of rank n − d on the variety G r( n − d ; V ∗ ). We may then identify S ⊥ = η ⊥ (cid:0) A ( G r( d ; n )) ⊕ n (cid:1) , and onehas the anticipated isomorphism co − L d,n F n ∼ = S ⊗ A ( G r( d ; n )) S ⊥ of A ( G r( d ; n ))-modules. LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 49
We now consider the Drinfel’d twist deformation F θn of the coordinate algebra of GL( n ),given in Definition 2.13. This deformation applies to the Hopf subalgebra L d,n as well.Since we are interested in toric ( C × ) d ( n − d ) deformations of the variety G r( d ; n ), we con-sider a deformation F θ ( d ) d ⊗ F θ ( n − d ) n − d of the Hopf algebra Fun( L d,n − d ) and use the subgroupinclusion described by the algebra homomorphism (6.15). Then, as explained in § n × n matrix θ is given by θ ij = θ ( d ) ij for the block 1 ≤ i, j ≤ d , θ ij = θ ( n − d ) ij for the block d + 1 ≤ i, j ≤ n , and θ ij = 0 otherwise. Hence the noncommutative Hopf algebra L θd,n isalso well-defined. The left coaction of L θd,n on F θn is the same as that of (6.16), since thetwist does not change the coproduct. In analogy with the undeformed case, we interpretthe algebra co − L θd,n F θn of left coinvariants as the algebra of the ( C × ) d ( n − d ) deformation ofthe cotangent manifold T ∗ G r( d ; n ) = GL( n ) /L d,n . This identification will be justifiedbelow. The algebra co − L θd,n F θn is generated by elements η ij introduced in (6.18) and byelements η ⊥ ij given in (6.19). Theorem 6.21.
The noncommutative product in co − L θd,n F θn is described by commutationrelations among generators η ij and η ⊥ ij given by η ij × θ η i ′ j ′ = K ij ; i ′ j ′ η i ′ j ′ × θ η ij ,η ⊥ ij × θ η ⊥ i ′ j ′ = K ij ; i ′ j ′ η ⊥ i ′ j ′ × θ η ⊥ ij ,η ij × θ η ⊥ i ′ j ′ = K ij ; i ′ j ′ η ⊥ i ′ j ′ × θ η ij , (6.22) where K ij ; i ′ j ′ = q ii ′ q j ′ i q i ′ j q jj ′ . (6.23) Proof : We compute the twisted relations between η ij directly from the definition (1.12).For this, we need the quantity (id ⊗ ∆ ∨ ) ∆ ∨ ( η ij ) = η (1) ij ⊗ η (2) ij ⊗ η (3) ij . Beginning with∆ ∨ ( η ij ) = d X k =1 n X m,p =1 (cid:0) S ∨ ( g pk ) ⊗ S ∨ ( g ip ) (cid:1) · (cid:0) g km ⊗ g mj (cid:1) = d X k =1 n X m,p =1 (cid:0) S ∨ ( g pk ) g km (cid:1) ⊗ (cid:0) S ∨ ( g ip ) g mj (cid:1) , we expand the second factor at the end to get η (1) ij ⊗ η (2) ij ⊗ η (3) ij = d X k =1 n X m,p,r,s =1 (cid:0) S ∨ ( g pk ) g km (cid:1) ⊗ (cid:0) S ∨ ( g rp ) g ms (cid:1) ⊗ (cid:0) S ∨ ( g ir ) g sj (cid:1) and similarly η (1) i ′ j ′ ⊗ η (2) i ′ j ′ ⊗ η (3) i ′ j ′ = d X k ′ =1 n X m ′ ,p ′ ,r ′ ,s ′ =1 (cid:0) S ∨ ( g p ′ k ′ ) g k ′ m ′ (cid:1) ⊗ (cid:0) S ∨ ( g r ′ p ′ ) g m ′ s ′ (cid:1) ⊗ (cid:0) S ∨ ( g i ′ r ′ ) g s ′ j ′ (cid:1) . Using these expressions we compute the three terms of the deformed product in (1.12).Starting with F θ (cid:0) η (1) ij ⊗ η (1) i ′ j ′ (cid:1) = (cid:10) F θ , η (1) ij ⊗ η (1) i ′ j ′ (cid:11) = (cid:10) exp (cid:0) − i2 θ ab H a ⊗ H b (cid:1) , η (1) ij ⊗ η (1) i ′ j ′ (cid:11) and looking at the first order term in θ we compute separately (cid:10) H a , η (1) ij (cid:11) = d X k =1 (cid:10) H a , S ∨ ( g pk ) g km (cid:11) = d X k =1 (cid:10) H a ⊗ ⊗ H a , S ∨ ( g pk ) ⊗ g km (cid:11) = d X k =1 (cid:16) − h H a , g pk i ε ∨ ( g km ) + ε ∨ (cid:0) S ∨ ( g pk ) (cid:1) h H a , g km i (cid:17) = d X k =1 (cid:0) − δ ap δ ak δ km + δ pk δ ak δ am (cid:1) = 0 , where we have used duality to transfer the antipode S ∨ from F θn to the enveloping algebra H nθ in the pairing. An identical calculation shows that h H b , η (1) i ′ j ′ i = 0. Only the zerothorder term gives a contribution, so that F θ (cid:0) η (1) ij ⊗ η (1) i ′ j ′ (cid:1) = (cid:10) ⊗ , η (1) ij ⊗ η (1) i ′ j ′ (cid:11) = d X k,k ′ =1 ε ∨ (cid:0) S ∨ ( g pk ) g km (cid:1) ε ∨ (cid:0) S ∨ ( g p ′ k ′ ) g k ′ m ′ (cid:1) = d X k,k ′ =1 δ pk δ mk δ p ′ k ′ δ m ′ k ′ . The third factor in (1.12) is given by F θ − (cid:0) η (3) ij ⊗ η (3) i ′ j ′ (cid:1) = (cid:10) F − θ , η (3) ij ⊗ η (3) i ′ j ′ (cid:11) = (cid:10) exp (cid:0) i2 θ bc H b ⊗ H c (cid:1) , η (3) ij ⊗ η (3) i ′ j ′ (cid:11) . Looking at the first order term in θ , we compute separately (cid:10) H b , η (3) ij (cid:11) = (cid:10) H b , S ∨ ( g ir ) g sj (cid:11) = (cid:10) H b ⊗ ⊗ H b , S ∨ ( g ir ) ⊗ g sj (cid:11) = − h H b , g ir i ε ∨ ( g sj ) + ε ∨ (cid:0) S ∨ ( g ir ) (cid:1) h H b , g sj i = − δ bi δ ri δ sj + δ bj δ ri δ sj . An identical calculation shows that h H c , η (3) i ′ j ′ i = − δ ci ′ δ r ′ i ′ δ s ′ j ′ + δ cj ′ δ r ′ i ′ δ s ′ j ′ . So the firstorder term is given by i2 θ bc ( − δ bi δ ri δ sj + δ bj δ ri δ sj ) ( − δ ci ′ δ r ′ i ′ δ s ′ j ′ + δ cj ′ δ r ′ i ′ δ s ′ j ′ )and summing over all orders we finally arrive at F θ − (cid:0) η (3) ij ⊗ η (3) i ′ j ′ (cid:1) = q ii ′ q j ′ i q i ′ j q jj ′ δ ri δ sj δ r ′ i ′ δ s ′ j ′ . We are now ready to write the deformed product between generators η ij as η ij × θ η i ′ j ′ = F θ (cid:0) η (1) ij ⊗ η (1) i ′ j ′ (cid:1) (cid:0) η (2) ij · η (2) i ′ j ′ (cid:1) F θ − (cid:0) η (3) ij ⊗ η (3) i ′ j ′ (cid:1) = n X m,p,r,s =1 n X m ′ ,p ′ ,r ′ ,s ′ =1 (cid:16) d X k,k ′ =1 δ pk δ mk δ p ′ k ′ δ m ′ k ′ (cid:17) × (cid:0) S ∨ ( g rp ) g ms S ∨ ( g r ′ p ′ ) g m ′ s ′ (cid:1) (cid:0) q ii ′ q j ′ i q i ′ j q jj ′ δ ri δ sj δ r ′ i ′ δ s ′ j ′ (cid:1) = q ii ′ q j ′ i q i ′ j q jj ′ η ij η i ′ j ′ . LGEBRAIC DEFORMATIONS OF TORIC VARIETIES I 51
Computing in exactly the same way the deformed product η i ′ j ′ × θ η ij and comparing thetwo expressions, we find the first set of relations in (6.22). The remaining relations followfrom η ⊥ ij = δ ij − η ij . (cid:4) The noncommutative relations (6.22) are not compatible with the constraints of Propo-sition 6.20. However, the new generatorsˆ η ij = q − ij η ij , ˆ η ⊥ ij = q − ij η ⊥ ij enjoy the same commutation relations (6.22) as well as the orthogonal projector rela-tions of Proposition 6.20. By Proposition 6.12, there is a natural isomorphism S θ ∼ =ˆ η (cid:0) A ( G r θ ( d ; n )) ⊕ n (cid:1) of bundles on Open ( G r θ ( d ; n )), and we define the orthogonal comple-ment of the tautological bundle S ⊥ θ := ˆ η ⊥ (cid:0) A ( G r θ ( d ; n )) ⊕ n (cid:1) . Note that the duality betweenthe bundles S θ and S ⊥ θ now also involves interchange of the block matrices θ ( d ) and θ ( n − d ) above. Denoting by V θ the trivial bimodule A ( G r θ ( d ; n )) ⊗ V , the noncommutative versionof the exact sequence (6.8) of bundles is then given by0 −→ (cid:0) S ⊥ θ (cid:1) ∨ (ˆ η ⊥ ) ∗ −−−→ V θ ˆ η −→ S θ −→ , (6.24)and it follows from Theorem 6.21 that the sheaf of noncommutative differential forms isisomorphic to the braided tensor product co − L θd,n F θn ∼ = S θ b ⊗ θ S ⊥ θ (6.25)as a bimodule algebra over A ( G r θ ( d ; n )) in the category H nθ M .The geometric meaning of the generators η ij and η ⊥ ij can be better understood bycomputing their transformation properties under the action of the torus T = ( C × ) d ( n − d ) . Proposition 6.26.
The noncommutative fibration co − L θd,n F θn is a T -equivariant bundlewith eigenbasis generated by η ij . Proof : We show that the generators η ij are T -eigenvectors with respect to the leftaction of ( C × ) d ( n − d ) induced by the algebra homomorphism (6.15) and the right coactionΦ L d,n : F θn → F θn ⊗ L θd,n given byΦ L d,n ( g ij ) = (cid:0) ⊗ π ( L d,n ) (cid:1) ∆ ∨ ( g ij ) = g (1) ij ⊗ π ( L d,n ) (cid:0) g (2) ij (cid:1) . (6.27)Let H a (resp. h a ), a = 1 , . . . , n be the toric generators in the enveloping algebra of GL( n )(resp. L d,n − d ). Dual to π ( L d,n ) , there is an injective algebra homomorphism ι ( L d,n ) betweenthe corresponding enveloping algebras such that ι ( L d,n ) ( h a ) = H a . Using results of § ι ( L d,n ) of the left action (1.9) of the enveloping algebra of T dually inducedby the the right coaction (6.27) of L θd,n on F θn is then given by H a ⊲ g ij = g (1) ij (cid:10) h a , π ( L d,n ) (cid:0) g (2) ij (cid:1)(cid:11) = n X k =1 g ik (cid:10) ι ( L d,n ) ( h a ) , g kj (cid:11) = n X k =1 g ik h H a , g kj i = δ aj g ij , (6.28) where we have used the duality between π ( L d,n ) and ι ( L d,n ) . Similarly, one computes H a ⊲ S ∨ ( g ij ) = (cid:10) h a , π ( L d,n ) (cid:0) S ∨ ( g ij ) (2) (cid:1)(cid:11) S ∨ ( g ij ) (1) = (cid:10) ι ( L d,n ) ( h a ) , S ∨ ( g ij ) (2) (cid:11) S ∨ ( g ij ) (1) = n X k =1 (cid:10) H a , S ∨ ( g ik ) (cid:11) S ∨ ( g kj )= − n X k =1 h H a , g ik i S ∨ ( g kj ) = − δ ai S ∨ ( g ij ) . (6.29)Using (6.28) and (6.29), the left action of H a on the left coinvariant generators η ij is thuscomputed to be H a ⊲ η ij = d X k =1 (cid:16)(cid:0) H a ⊲ S ∨ ( g ik ) (cid:1) g kj + S ∨ ( g ik ) ( H a ⊲ g kj ) (cid:17) = ( δ aj − δ ai ) η ij , (6.30)as required. (cid:4) By (6.30), we notice that the diagonal elements of the matrices η and η ⊥ are T -invariant.However, in contrast to the deformed products obtained by Drinfel’d twists of Hopf-module algebras (such as those defined in § Example 6.31.
For d = 1 , one has G r θ (1; n ) = ( CP n − θ ) ∗ with θ = θ ( n − , and theOre localization with respect to the embeddings above identifies the generators η ik with theelements n y k = n w − i w k generating the degree localized subalgebras as one readily checks using (6.22). The non-commutative affine subvarieties U θ [ σ i ] , i = 1 , . . . , n constructed from each maximal cone σ i in the fan Σ of CP n − are thus generated exactly by each row of the matrix η . ByExample 6.13 one has a natural isomorphism S θ ∼ = O CP n − θ (1) , and in a similar vein S ⊥ θ ∼ = O CP n − θ ( − . By tensoring the exact sequence (6.24) from the right with the lo-cally free sheaf S ∨ θ ∼ = O CP n − θ ( − , and by using (6.25) and dualizing, one finds the Eulersequence −→ co − L θ ,n F θn −→ V ∨ θ ( − −→ O CP n − θ −→ , analogous to that of [27, § . In the commutative case, this sequence is dual to thedescription of the tangent bundle in terms of the surjective bundle map O CP n − ⊗ V → O CP n − which evaluates global sections of the hyperplane bundle. The construction aboveprovides a geometrical interpretation for the sequence of Example 6.7 which describes thebundle of K¨ahler differentials Ω CP n − θ . References [1] M. Artin, W. Schelter:
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Max Planck Institute for Mathematics, Vivatgasse 7, 53111 Bonn, Germany.
Present address : Mathematisches Institut, Universit¨at M¨unster, Einsteinstrasse 62, 48149M¨unster, Germany
E-mail address : [email protected] Dipartimento di Matematica, Universit`a di Trieste, Via A. Valerio 12/1, I-34127 Tri-este, Italy and INFN, Sezione di Trieste, Trieste, Italy
E-mail address : [email protected] Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Ric-carton, Edinburgh EH14 4AS, U.K. and Maxwell Institute for Mathematical Sciences,Edinburgh, U.K.
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