Mapping spaces and automorphism groups of toric noncommutative spaces

We develop a sheaf theory approach to toric noncommutative geometry which allows us to formalize the concept of mapping spaces between two toric noncommutative spaces. As an application, we study the ‘internalized’ automorphism group of a toric noncommutative space and show that its Lie algebra has an elementary description in terms of braided derivations.


Introduction and summary
Toric noncommutative spaces are among the most studied and best understood examples in noncommutative geometry. Their function algebras A carry a coaction of a torus Hopf algebra H , whose cotriangular structure dictates the commutation relations in A. Famous examples are given by the noncommutative tori [27], the Connes-Landi spheres [15] and related deformed spaces [14]. More broadly, toric noncommutative spaces can be regarded as special examples of noncommutative spaces that are obtained by Drinfeld twist (or 2-cocycle) deformations of algebras carrying a Hopf algebra (co)action, see, e.g., [4][5][6] and references therein. For an algebraic geometry perspective on toric noncommutative varieties, see [12].
Noncommutative differential geometry on toric noncommutative spaces, and more generally on noncommutative spaces obtained by Drinfeld twist deformations, is far developed and well understood. Vector bundles (i.e., bimodules over A) have been studied in [4], where also a theory of noncommutative connections on bimodules was developed. These results were later formalized within the powerful framework of closed braided monoidal categories and therefore generalized to certain nonassociative spaces (obtained by cochain twist deformations) in [5,6]. Examples of noncommutative principal bundles (i.e., Hopf-Galois extensions [9,10]) in this framework were studied in [20], and these constructions were subsequently abstracted and generalized in [1]. In applications to noncommutative gauge theory, moduli spaces of instantons on toric noncommutative spaces were analyzed in [7,8,11,13], while analogous moduli spaces of self-dual strings in higher noncommutative gauge theory were considered by [25].
Despite all this recent progress in understanding the geometry of toric noncommutative spaces, there is one very essential concept missing: Given two toric noncommutative spaces, say X and Y , we would like to have a 'space of maps' Y X from X to Y . The problem with such mapping spaces is that they will in general be 'infinite-dimensional,' just like the space of maps between two finite-dimensional manifolds is generically an infinite-dimensional manifold. In this paper, we propose a framework where such 'infinite-dimensional' toric noncommutative spaces may be formalized and which in particular allows us to describe the space of maps between any two toric noncommutative spaces. Our approach makes use of sheaf theory: Denoting by H S the category of toric noncommutative spaces, we show that there is a natural site structure on H S which generalizes the well-known Zariski site of algebraic geometry to the toric noncommutative setting. The category of generalized toric noncommutative spaces is then given by the sheaf topos H G := Sh( H S ), and we show that there is a fully faithful embedding H S → H G which allows us to equivalently regard toric noncommutative spaces as living in this bigger category. The advantage of the bigger category H G is that it enjoys very good categorical properties; in particular, it admits all exponential objects. We can therefore make sense of the 'space of maps' Y X as a generalized toric noncommutative space in H G , i.e., as a sheaf on the site H S . As an application, we study the 'internalized' automorphism group Aut(X ) of a toric noncommutative space X , which is a certain subobject in H G of the selfmapping space X X . Using synthetic geometry techniques, we are able to compute the Lie algebra of Aut(X ) and we show that it can be identified with the braided derivations considered in [4,6]. Hence, our concept of automorphism groups 'integrates' braided derivations to finite (internalized) automorphisms, which is an open problem in toric noncommutative geometry that cannot be solved by more elementary techniques.
Besides giving rise to a very rich concept of 'internalized' automorphism groups of toric noncommutative spaces, there are many other applications and problems which can be addressed with our sheaf theory approach to toric noncommutative geometry. For example, the mapping spaces Y X may be used to describe the spaces of field configurations for noncommutative sigma models, see, e.g., [16][17][18]24]. Due to the fact that the mapping space Y X captures many more maps than the set of morphisms Hom(X, Y ) (compare with Example 5.3 in the main text), this will lead to a much richer structure of noncommutative sigma models than those discussed previously. Another immediate application is to noncommutative principal bundles: It was observed in [10] that the definition of a good notion of gauge transformations for noncommutative Hopf-Galois extensions is somewhat problematic, because there are in general not enough algebra automorphisms of the total space algebra. To the best of our knowledge, this problem has not yet been solved. Using our novel sheaf theory techniques, we can give a natural definition of an 'internalized' gauge group for toric noncommutative principal bundles P → X by carving out a subobject in H G of the 'internalized' automorphism group Aut(P) of the total space which consists of all maps that preserve the structure group action and the base space.
The outline of the remainder of this paper is as follows: In Sect. 2, we recall some preliminary results concerning cotriangular torus Hopf algebras H and their comodules, which form symmetric monoidal categories H M . In Sect. 3, we study algebra objects in H M whose commutation relations are controlled by the cotriangular structure on H . We establish a category of finitely presented algebra objects H A fp , which contains noncommutative tori, Connes-Landi spheres and related examples, and study its categorical properties, including coproducts, pushouts and localizations. The category of toric noncommutative spaces H S is then given by the opposite category of H A fp , and we show in Sect. 4 that H S can be equipped with the structure of a site. In Sect. 5, we introduce and study the sheaf topos H G whose objects are sheaves on H S which we interpret as generalized toric noncommutative spaces. We show that the Yoneda embedding factorizes through H G (i.e., that our site is subcanonical) and hence obtain a fully faithful embedding H S → H G of toric noncommutative spaces into generalized toric noncommutative spaces. An explicit description of the exponential objects Y X in H G is given, which in particular allows us to formalize and study the mapping space between two toric noncommutative spaces. Using a simple example, it is shown in which sense the mapping spaces Y X are richer than the morphism sets Hom(X, Y ) (cf. Example 5.3). In Sect. 6, we apply these techniques to define an 'internalized' automorphism group Aut(X ) of a toric noncommutative space X , which arises as a certain subobject in H G of the self-mapping space X X . It is important to stress that Aut(X ) is in general not representable, i.e., it has no elementary description in terms of a Hopf algebra, and hence, it is a truly generalized toric noncommutative space described by a sheaf on H S . The Lie algebra of Aut(X ) is computed in Sect. 7 by using techniques from synthetic (differential) geometry [19,21,26]. We then show in Sect. 8 that the Lie algebra of Aut(X ) can be identified with the braided derivations of the function algebra of X . Hence, in contrast to Aut(X ), its Lie algebra of infinitesimal automorphisms has an elementary description. This identification is rather technical, and it relies on a fully faithful embedding H M dec → Mod K ( H G ) of a certain full subcategory (called decomposables) of the category of left H -comodules H M into the category of K -module objects in the sheaf topos H G , where K denotes the line object in this topos; the technical details are presented in "Appendix."

Hopf algebra preliminaries
In this paper, all vector spaces will be over a fixed field K and the tensor product of vector spaces will be denoted simply by ⊗.
The Hopf algebra H := O(T n ) of functions on the algebraic n-torus T n is defined as follows: As a vector space, H is spanned by the basis on which we define a (commutative and associative) product and unit by The (cocommutative and coassociative) coproduct, counit and antipode in H are given by We choose a cotriangular structure on H , i.e., a linear map R: H ⊗ H → K satisfying (1) R g (2) ⊗ h (2) , (2.4c) for all f, g, h ∈ H , where we have used Sweedler notation (h) = h (1) ⊗ h (2) (with summation understood) for the coproduct in H . The quasi-commutativity condition g (1) h (1) R(h (2) ⊗ g (2) ) = R(h (1) ⊗ g (1) ) h (2) g (2) , for all g, h ∈ H , is automatically fulfilled because H is commutative and cocommutative. For example, if K = C is the field of complex numbers, we may take the usual cotriangular structure defined by where is an antisymmetric real n×n-matrix, which plays the role of deformation parameters for the theory. Let us denote by H M the category of left H -comodules.
We follow the usual abuse of notation and denote objects (V, ρ V ) in H M simply by V without displaying the coaction explicitly. We further use a Sweedler-like notation A morphism L: V → W in H M is a linear map preserving the left H -coactions, i.e., or in the Sweedler-like notation The category H M is a monoidal category with tensor product of two objects V and W given by the tensor product V ⊗ W of vector spaces equipped with the left H -coaction The monoidal unit in H M is given by the one-dimensional vector space K with trivial left H -coaction K → H ⊗K, c → 1 H ⊗c. The monoidal category H M is symmetric with commutativity constraint (2.10) for any two objects V and W in H M .

Algebra objects
We are interested in spaces whose algebras of functions are described by certain algebra objects in the symmetric monoidal category H M . An algebra object in H M is an object A in H M together with two H M -morphisms μ A : A ⊗ A → A (product) and η A : K → A (unit) such that the diagrams in H M commute. Because H M is symmetric, we may additionally demand that the product μ A is compatible with the commutativity constraints in H M , i.e., the diagram in H M commutes. This amounts to demanding the commutation relations for all a, a ∈ A, where we have abbreviated the product by μ A (a ⊗ a ) = a a ; in the following, we shall also use the compact notation 1 A := η A (1) ∈ A for the unit element in A, or sometimes just 1. Such algebras are not commutative in the ordinary sense once we choose a nontrivial cotriangular structure as, for example, in (2.5), see also Example 3.5. Let us introduce the category of algebras of interest.  (3.4) with the convention V ⊗0 := K. Then, T V is a left H -comodule when equipped with the coaction ρ T V : Moreover, T V is an algebra object in H M when equipped with the product (3.6) and the unit η T V : K → T V given by The algebra object T V does not satisfy the commutativity constraint (3.2); hence, it is not an object of the category H A . We may enforce the commutativity constraint by taking the quotient of T V by the two-sided ideal  between the morphism sets, for any object V in H M and any object A in H A . This is easy to see from the fact that any H A -morphism κ: Free(V ) → A is uniquely specified by its restriction to the vector space V = V ⊗1 ⊆ Free(V ) of generators and hence by an H M -morphism V → Forget(A). From a geometric perspective, the free H A -algebras Free(V ) describe the function algebras on toric noncommutative planes. In order to capture a larger class of toric noncommutative spaces, we introduce a suitable concept of ideals for H A -algebras.

Definition 3.2 Let
A be an object in H A . An H A -ideal I of A is a two-sided ideal I ⊆ A of the algebra underlying A which is stable under the left H -coaction, i.e., the coaction ρ A induces a linear map ρ A : I → H ⊗ I .

Lemma 3.3 If A is an object in H A and I is an H A -ideal of A, the quotient A/I is an object in H A when equipped with the induced coaction, product and unit.
This lemma allows us to construct a variety of H A -algebras by taking quotients of free H A -algebras by suitable H A -ideals. We are particularly interested in the case where the object V in H M that underlies the free H A -algebra Free(V ) is finite-dimensional; geometrically, this corresponds to a finite-dimensional toric noncommutative plane. We shall introduce a convenient notation for this case: First, notice that the one-dimensional left H -comodules over the torus Hopf algebra H = O(T n ) can be characterized by a label m ∈ Z n . The corresponding left H -coactions are given by (3.12) We shall use the notation K m := (K, ρ m ) for these objects in H M . The coproduct K m K m of two such objects is given by the vector space K ⊕ K K 2 together with the component-wise coaction, i.e., The free H A -algebra corresponding to a finite coproduct of objects K m i , for i = 1, . . . , N , in H M will be used frequently in this paper. Hence, we introduce the compact notation (3.14) By construction, the H A -algebras F m 1 ,...,m N are generated by N elements x i ∈ F m 1 ,...,m N whose transformation property under the left H -coaction is given by ρ F m 1 ,...,m N (x i ) = t m i ⊗ x i and whose commutation relations read as We denote by H A fp the full subcategory of H A whose objects are all finitely presented H A -algebras.
Example 3.5 Let us consider the case K = C and R given by (2.5). Take the free H A -algebra generated by x i and x * i , for i = 1, . . . , N , with left H -coaction specified by for some m i ∈ Z n ; in the notation above, we consider the free H A -algebra F m 1 ,...,m N ,−m 1 ,...,−m N and denote the last N generators by x * i := x N +i , for i = 1, . . . , N . The algebra of (the algebraic version of) the 2N −1dimensional Connes-Landi sphere is obtained by taking the quotient with respect to the H A -ideal which implements the unit sphere relation. The algebra of the N -dimensional noncommutative torus is obtained by taking the quotient with respect to the H A -ideal To obtain also the even dimensional Connes-Landi spheres, we consider the free , and take the quotient with respect to the H A -ideal All these examples are * -algebras with involution defined by x i → x * i and x * 2N +1 = x 2N +1 . An example which is not a * -algebra is the free H A -algebra F m , for some m = 0 in Z n , which we may interpret as the algebra of (anti)holomorphic polynomials on C.
We will now study some properties of the categories H A and H A fp that will be used in the following. First, let us notice that the category H A has (finite) coproducts: Given two objects A and B in H A , their coproduct A B is the object in H A whose underlying left H -comodule is A ⊗ B [with coaction ρ A B := ρ A⊗B given in (2.9)] and whose product μ A B and unit η A B are characterized by The canonical inclusion H A -morphisms ι 1 : A → A B and ι 2 : B → A B are given by for all a ∈ A and b ∈ B. The coproduct A B of two finitely presented H A -algebras A and B is finitely presented: Consequently, the category H A fp has finite coproducts.
In addition to coproducts, we also need pushouts in H A and H A fp , which are given by colimits of the form Such pushouts exist and can be constructed as follows: Consider first the case where we work in the category H A . We define The dashed H A -morphisms in (3.22) are given by It is easy to confirm that A C B defined above is a pushout of the diagram (3.22). Moreover, the pushout of finitely presented H A -algebras is finitely presented: for all generators x i and elements c ∈ C, in order to show that the H A -ideal I is equivalently generated by κ( Consequently, the category H A fp has pushouts. We also need the localization of H A -algebras A with respect to a single Hcoinvariant element s ∈ A, i.e., ρ A (s) = 1 H ⊗s. Localization amounts to constructing and that satisfies the following universal property: is a localization of A with respect to the H -coinvariant element s ∈ A. The inverse of s (s) = [s ⊗ 1 F 0 ] exists, and it is given by the new generator (3.19) and use the fact that x and s are coinvariants. Given now any . Finally, the localization of finitely presented H A -algebras is finitely presented:

Toric noncommutative spaces
From a geometric perspective, it is useful to interpret an object A in H A fp as the 'algebra of functions' on a toric noncommutative space X A . Similarly, a morphism κ: A → B in H A fp is interpreted as the 'pullback' of a map f : X B → X A between toric noncommutative spaces, where due to contravariance of pullbacks the direction of the arrow is reversed when going from algebras to spaces. We shall use the more intuitive notation κ = f * : A → B for the H A fp -morphism corresponding to f : X B → X A . This can be made precise with

Definition 4.1 The category of toric noncommutative spaces
is the opposite of the category H A fp . Objects in H S will be denoted by symbols like X A , where A is an object in H A fp . Morphisms in H S will be denoted by symbols like f : X B → X A , and they are (by definition) in bijection with H A fp -morphisms f * : A → B.
As the category H A fp has (finite) coproducts and pushouts, which we have denoted by A B and A C B, its opposite category H S has (finite) products and pullbacks. Given two objects X A and X B in H S , their product is given by together with the canonical projection H S -morphisms where ι 1 and ι 2 are the canonical inclusion H A fp -morphisms for the coproduct in H A fp [cf. (3.20)]. Pullbacks in H S are given by for κ = f * : C → A and ζ = g * : C → B [cf. (3.22)]. The dashed arrows in (4.3) are specified by their corresponding H A fp -morphisms in (3.24). We next introduce a suitable notion of covering for toric noncommutative spaces, which is motivated by the well-known Zariski covering families in commutative algebraic geometry.
Example 4.3 Recall from Example 3.5 that the algebra of functions on the 2Ndimensional Connes-Landi sphere is given by As the last generator x 2N +1 is H -coinvariant, we can define the two H -coinvariant elements Then, s 1 + s 2 = 1, and hence, we obtain an H S -Zariski covering family define an H S -Zariski covering family.
By universality of the pushout and localization, the pushout diagram for It is an elementary computation to confirm that the dashed arrow in this diagram is an isomorphism by using the explicit formulas for the pushout (3.23) and localization (3.28). As a consequence, is the localization with respect to the two H -coinvariant elements s i , s j ∈ A. The dashed arrows in (4.10) are specified by the canonical H A fp -morphisms Proof This follows immediately from the proof of Proposition 4.4.

Remark 4.6
For later convenience, we shall introduce the notation for the morphisms of this pullback diagram.

Generalized toric noncommutative spaces
The category H S of toric noncommutative spaces has the problem that it does not generally admit exponential objects X B X A , i.e., objects which describe the 'mapping space' from X A to X B . A similar problem is well known from differential geometry, where the mapping space between two finite-dimensional manifolds in general is not a finite-dimensional manifold. There is, however, a canonical procedure for extending the category H S to a bigger category that admits exponential objects. We review this procedure in our case of interest.
The desired extension of H S is given by the category is an equalizer in Set. We have used Corollary 4.5 to express the pullback of two covering morphisms by op was defined as the opposite category of H A fp , it is sometimes convenient to regard a sheaf on H S as a covariant functor Y : H A fp → Set. In this notation, the sheaf condition (5.2) looks like We will interchangeably use these equivalent points of view. The morphisms in H G are natural transformations between functors, i.e., presheaf morphisms. We shall interpret H G as a category of generalized toric noncommutative spaces. To justify this interpretation, we will show that there is a fully faithful embedding H S → H G of the category of toric noncommutative spaces into the new category. As a first step, we use the (fully faithful) Yoneda embedding H S → PSh( H S ) in order to embed H S into the category of presheaves on H S . The Yoneda embedding is given by the functor which assigns to any object X A in H S the presheaf given by the functor and to any H S -morphism f : X A → X B the natural transformation f : for all objects X C in H S .

Because the Hom-functor HomH
is an equalizer in H A fp .
Using the explicit characterization of localizations [cf. (3.28)], let us take a generic element where here there is no sum over the index i but an implicit sum of the form which we suppress. This is an element in the desired equalizer if and only if Remark 5.2 Heuristically, Proposition 5.1 implies that the theory of toric noncommutative spaces X A together with their morphisms can be equivalently described within the category H G . The sheaf X A specified by (5.4) is interpreted as the 'functor of points' of the toric noncommutative space X A . In this interpretation, (5.4) tells us all possible ways in which any other toric noncommutative space X B may be mapped into X A , which captures the geometric structure of X A . A generic object Y in H G (which we call a generalized toric noncommutative space) has a similar interpretation: The set Y (X B ) tells us all possible ways in which X B is mapped into Y . This is formalized by Yoneda's Lemma for any object X B in H S and any object Y in H G .
The advantage of the sheaf category H G of generalized toric noncommutative spaces over the original category H S of toric noncommutative spaces is that it has very good categorical properties, which are summarized in the notion of a Grothendieck topos, see, e.g., [22]. Most important for us are the facts that H G has all (small) limits and all exponential objects. Limits in H G are computed object-wise, i.e., as in presheaf categories. In particular, the product of two objects Y, Z in H G is the sheaf specified by the functor Y × Z : H S op → Set that acts on objects as where on the right-hand side × is the Cartesian product in Set, and on morphisms f : where on the right-hand side { * } is the terminal object in Set, i.e., a singleton set, and in the obvious way on morphisms. The fully faithful embedding H S → H G of Proposition 5.1 is limit preserving. In particular, we have for all objects X A , X B in H S and the terminal object X K in H S . Here K is the H A fp -algebra with trivial left H -coaction c → 1 H ⊗ c, i.e., the initial object in H A fp . The exponential objects in H G are constructed as follows: Given two objects Y, Z in H G , the exponential object Z Y is the sheaf specified by the functor Z Y : H S op → Set that acts on objects as and on morphisms f : Given two ordinary toric noncommutative spaces X A and X B , i.e., objects in H S , we can form the exponential object X B X A in the category of generalized toric noncommutative spaces H G . The interpretation of X B X A is as the 'space of maps' from X A to X B . In the present situation, the explicit description (5.13) of exponential objects may be simplified via In the first step, we have used (5.13), in the second step (5.12), and the third step is due to Yoneda's Lemma. Hence, can be expressed in terms of H A fp -morphisms.

Example 5.3
To illustrate the differences between the exponential objects X B X A in H G and the Hom-sets HomH S (X A , X B ), let us consider the simplest example where A = B = F m , for some m = 0 in Z n . In this case, the Hom-set is given by For the initial object A = K in H A fp , we recover the Hom-set Let us now take A T = F −m,m /(y * y − 1) to be the toric noncommutative circle, see Example 3.5. We write y −1 := y * in A T and recall that the left H -coaction is given by ρ F −m,m (y) = t −m ⊗ y and ρ F −m,m (y −1 ) = t m ⊗ y −1 . We then obtain an isomorphism of sets because elements a ∈ F m , i.e., polynomials a = j c j x j , for c j ∈ K, are in bijection with H A fp -morphisms κ: By construction, each summand on the right-hand side has left H -coaction given by the canonical projection H S -morphisms of the product. Under the Yoneda bijections that is specified by the natural transformation with components for all H S -morphisms g, h: X B × X A → X A . The diagonal H S -morphism diag X B : X B → X B × X B is defined as usual via universality of products by The object X A X A together with the identity (6.1) and composition H G -morphisms (6.2) is a monoid object in H G : It is straightforward to verify that the H G -diagrams commute. Notice that X A X A is not a group object in H G because, loosely speaking, generic maps do not have an inverse. We may, however, construct the 'subobject of invertible maps' (in a suitable sense to be detailed below) of the monoid object X A X A , which then becomes a group object in H G called the automorphism group Aut(X A ) of X A . Let us apply the fully faithful functor Sh( H S ) → PSh( H S ) (which assigns to sheaves their underlying presheaves) on the monoid object (X A X A , •, e) in H G = Sh( H S ) to obtain the monoid object (X A X A , •, e) in PSh( H S ), denoted with abuse of notation by the same symbol. This monoid object in PSh( H S ) may be equivalently regarded as a functor H S op → Monoid with values in the category of ordinary Setvalued monoids (i.e., monoid objects in the category Set). The functor assigns to any object X B in H S the monoid and to any H S -morphism f : X B → X C the monoid morphism For any object X B in H S , we define Aut(X A )(X B ) to be the subset of elements Because the inverse of an element in a monoid (if it exists) is always unique, it follows that any element g ∈ Aut(X A )(X B ) has a unique inverse g −1 ∈ Aut(X A )(X B ), and that the inverse of g −1 is g. The monoid structure on X A X A (X B ) induces to Aut(X A )(X B ), because the inverse of e X B is e X B itself and the inverse of the map that assigns the inverse, we obtain for any object X B in H S a group The monoid morphism X A X A ( f ) in (6.5) induces a group morphism which we denote by is an equalizer in Set. Recalling that Aut(X A ) is the subpresheaf of the sheaf X A X A specified by the invertibility conditions (6.6), an element in i Aut(X A )(X B[s −1 i ] ) can be represented by an element (6.10) such that each g i has an inverse g −1 . (6.11) This element is in the desired equalizer if and only if for all i, j, where we used the compact notation f i; j introduced in (4.12). Because X A X A is a sheaf, we can represent i g i by the element g ∈ X A X A (X B ) that is i.e., that there exists X A X A ( f j;i ) are monoid morphisms, both sides of the equality (6.12) are invertible and the inverse is given by Using again the property that X A X A is a sheaf, we can represent i g −1 i by the element g ∈ X A X A (X B ) that is uniquely specified by X A X A ( f i )(g) = g −1 i , for all i. It is now easy to check thatg is the inverse of g: Using once more the property that X A and similarly X A , for all i. Because X A X A is a sheaf, this impliesg • X B g = g • X Bg = e X B and hence thatg = g −1 .

Lie algebras of automorphism groups
The category H G of generalized toric noncommutative spaces has a distinguished object K := X F 0 , where F 0 is the free H A fp -algebra with one coinvariant generator x, i.e., x → 1 H ⊗ x. We call K the line object as it describes the toric noncommutative line. The line object K is a ring object in H G : The sum H G -morphism +: K × K → K is induced (via going opposite and the Yoneda embedding) by the H A fp -morphism The (additive) zero element is the H G -morphism 0: { * } → K induced by the H A fp -morphism F 0 → K, x → 0 ∈ K, and the (multiplicative) unit element is the H G -morphism 1: It is straightforward, but slightly tedious, to confirm that these structures make K into a ring object in H G .
Remark 7.1 Regarding the line object as a functor K : H S op → Set, it assigns to an object X B in H S the set The H A fp -algebra structure on B, B induces a (commutative) ring structure on B 0 , B 0 and f * preserves this ring structure. Hence, we have obtained a functor H S op → CRing with values in the category of commutative rings (in Set), which is an equivalent way to describe the ring object structure on K introduced above.
Heuristically, D describes the infinitesimal neighborhood of 0 in K , i.e., D is an infinitesimally short line, so short that functions on D (which are described by F 0 /(x 2 )) are polynomials of degree 1.
Following the ideas of synthetic (differential) geometry [19,21,26], we may use D to define the tangent bundle of a generalized toric noncommutative space.
The components of the projection then read as We shall now study in more detail the tangent bundle of the automorphism group of some object X A in H S . We are particularly interested in the fiber T e Aut(X A ) of this bundle over the identity e: { * } → Aut(X A ), because it defines the Lie algebra of Aut(X A ). The fiber T e Aut(X A ) is defined as the pullback Aut(X A ) (7.6) in H G . In particular, T e Aut(X A ) is an object in H G . Using the perspective explained in Remark 7.3, we obtain for all objects X B in H S . The pullback (7.6) then introduces a further condition for all objects X B in H S . Using (6.6) and (5.16), it follows that any g ∈ A satisfying the invertibility condition imposed in (6.6). For our purposes, it is more convenient to equivalently regard g as an H A fp -morphism g: and B is the usual flip map because the left H -coaction on F 0 /(x 2 ) is trivial). Because any element in is of the form c 0 + c 1 x, for some c 0 , c 1 ∈ K, we obtain two H M -morphisms g 0 , g 1 : A → B A which are characterized uniquely by for all a ∈ A. Since g: A → F 0 /(x 2 ) B A is an H A fp -morphism, it follows that g 0 : A → B A is an H A fp -morphism and that g 1 : A → B A satisfies the condition for all a, a ∈ A. From (7.8), it follows that g ∈ T e Aut(X A )(X B ) if and only if g 0 = ι 2 : A → B A, a → 1 B ⊗ a is the canonical inclusion for the coproduct. Then, (7.10) simplifies to for all a, a ∈ A. Notice that (7.11) is the Leibniz rule for the A-bimodule structure on B A that is induced by the H A fp -algebra structure on B A and the inclusion H A fp -morphism ι 2 : A → B A.

Lemma 7.4 Let g: A → F 0 /(x 2 ) B A be any H
A fp -morphism such that g 0 = ι 2 : A → B A in the notation of (7.9). Then, the H A fp -morphismg: Proof From the hypothesis, (6.2) and x 2 = 0, it follows that  A, B A),

Corollary 7.5 The presheaf underlying T e Aut(X A ) is isomorphic to H Der(A, − A) via the natural isomorphism with components
where g 1 is defined according to (7.9). Hence, H Der(A, − A) is a sheaf, i.e., an object in H G , and T e Aut(X A ) is also isomorphic to H Der (A, − A) in H G .
Proof Since g is uniquely specified by g 0 , g 1 [via (7.9)] and g 0 = ι 2 for all g ∈ T e Aut(X A )(X B ), it follows that (7.14) is injective. Surjectivity of (7.14) follows from Lemma 7.4. Naturality of (7.14) is obvious. Because  (A, B A) specified by the Leibniz rule (7.11). Because the Leibniz rule is a linear condition, it follows that H Der (A, B A) is closed under taking sums and additive inverses, and that it contains the zero map. From (7.13), one immediately sees that this Abelian group structure is natural with respect to H S -morphisms f : X B → X B ; hence, H Der(A, − A) is an Abelian group object in H G . The B 0 -module structure is defined by setting for all a ∈ A. In order to verify that (b · v) ∈ H Der (A, B A), i.e., that it is Hequivariant and satisfies the Leibniz rule (7.11), it is essential to use the fact that b is coinvariant, . From (7.13), one immediately sees that this B 0 -module structure is natural with respect to H S -morphisms f : X B → X B , i.e.,  A, B A). Let us set A straightforward but slightly lengthy computation (using the Leibniz rule (7.11) for v and w) shows that [v, w] X B satisfies the Leibniz rule; hence, it is an element in H Der(A, B A). Antisymmetry of [ −, − ] X B follows immediately from the definition, and the Jacobi identity is shown by direct computation. Moreover, B 0 -linearity of the Lie bracket, i.e.,

Braided derivations
The Lie algebra object H Der(A, − A) constructed in Proposition 7.6 is (isomorphic to) the Lie algebra of the automorphism group Aut(X A ). Hence, we may interpret it as the Lie algebra of infinitesimal automorphisms of the toric noncommutative space X A with function H A fp -algebra A. Another (a priori unrelated) way to think about the infinitesimal automorphisms of X A is to consider the Lie algebra der(A) of braided derivations of A, see [2][3][4][5][6]. In this section, we show that these two points of view are equivalent. We briefly introduce the concept of braided derivations of H A fp -algebras A. Let us first consider the case where A = F m 1 ,...,m N is the free H A -algebra with N generators ...,m N , for j = 1, . . . , N , be the linear map defined by We denote elements L ∈ der(F m 1 ,...,m N ) by L = j L j ∂ j , where L j ∈ F m 1 ,...,m N , because the H -coaction then takes the convenient form ρ der (F m 1 ,...,m N )  The evaluation H M -morphism is similar to that in the case of free H A -algebras and is given by Notice that ev is well defined because of the conditions imposed in (8.6). The braided Leibniz rule (8.5) also holds in the case of finitely presented H A -algebras. Proof Using the braided Leibniz rule (8.5), we can compute the right-hand side of (8.8) and obtain For any object X B in H S , we define a map  (8.12) where the last step follows from (3.19) and (2.4). This shows that the image of ξ X B lies in H Der(A, B A), as we have asserted in (8.11). The maps ξ X B are clearly B 0 -module morphisms with respect to the B 0 -module structure on H Der(A, B A) introduced in (7.15) and that on j (der(A))(X B ) introduced in (9.10), and they are natural with respect to H S -morphisms f : X B → X C . Hence, we have defined a morphism ξ : j (der(A)) −→ H Der(A, − A) (8.13) in the category Mod K ( H G ) of K -module objects in H G . The main result of this section is for all generators x i and all a, a ∈ F m 1 ,...,m N . There is an isomorphism given by the assignment v → j v j ∂ j . Because A and B are decomposable, we obtain a chain of isomorphisms for all a ∈ A, where in the last two steps we used the properties (2.4) of the cotriangular structure R and the fact that b ⊗ L ∈ (B ⊗ der(A)) 0 is coinvariant. Going now along the lower path of the diagram, we obtain (8.19b) for all a ∈ A, where we used the definition of the Lie bracket [−, −] X B given in (7.17). These two expressions coincide because, using without loss of generality b ⊗ L ∈ B m ⊗ der(A) −m and b ⊗ L ∈ B m ⊗ der(A) −m , the second term in (8.19b) can be rearranged as for all m ∈ Z n . Notice that V 0 is the vector space of coinvariants and that V m ⊆ V are H M -subobjects, for all m.
is an isomorphism. We denote by H M dec the full subcategory of decomposables.
and it is easy to see that it is an isomorphism. For the case where A = F m 1 ,...,m N /I is finitely presented, we use the property that F m 1 ,...,m N is decomposable and hence so is the H A -ideal I ⊆ F m 1 ,...,m N . Consequently, the quotient A = F m 1 ,...,m N /I is decomposable as well.

Corollary 9.4 Let A be an object in H A fp . Then, der(A) is decomposable.
Proof Recalling the definition of der(A) in (8.6), the claim follows from the fact that A is decomposable (cf. Lemma 9.3), and Lemma 9.2(b) and (c).

Embedding of H M into H G
We first define a functor is an equalizer in Set. This follows from the same argument that we have used in the second paragraph of the proof of Proposition 5.1.
For any object X B in H S , the set j (V )(X B ) = (B ⊗ V ) 0 is a B 0 -module with Abelian group structure induced by the vector space structure of B ⊗ V and B 0 -action given by (9.10) These structures are natural with respect to H S -morphisms f : X B → X C , i.e., for all c ∈ C 0 , c ∈ C and v ∈ V ; hence, they endow j (V ) with the structure of a K -module object in H G . For any H M -morphism L: V → W the H G -morphism j (L): j (V ) → j (W ) is compatible with this K -module object structure, i.e., (9.8) is a B 0 -module morphism, for all objects X B in H S . We have therefore obtained (9.14) For the special case where B = F m is the free H A -algebra with one generator with coaction x → t m ⊗ x, we use B k∈Z ≥0 K k m to simplify this expression further to (9.15) where the coproducts here are in the category of vector spaces. Using this explicit characterization, we can establish the main result of "Appendix." are B 0 -module morphisms, for all objects X B in H S , such that for any H S -morphism f : X B → X C the diagram commutes.
We first show that η is uniquely determined by the components η X Fm , for all free H A -algebras F m with one generator. Using (9.14), we find that η X B is specified by its action on elements of the form b ⊗ v ∈ B n ⊗ V −n , for all n. Given any such element, we define an H A fp -morphism f * : F n → B by sending x → b. (Notice that the morphism f * depends on the chosen element b ⊗ v.) Then, the commutative diagram (9.18) implies that η X B (b ⊗ v) = ( f * ⊗ id W )(η X Fn (x ⊗ v)); hence, the value of η X B at b ⊗ v is fixed by η X Fn . As b ⊗ v was arbitrary, we find that η is uniquely determined by the components {η X Fm : m ∈ Z n }.
In the next step, we show that the components {η X Fm : m ∈ Z n } are uniquely determined by an H M -morphism L: V → W . Consider the H A fp -morphism f * : F m → F m defined by x → c x, where c ∈ K is an arbitrary constant. Using (9.15) and the commutative diagram (9.18) corresponding to this morphism, we obtain a commutative diagram The vertical arrows map elements v ∈ V −k m to c k v ∈ k∈Z ≥0 V −k m (and similarly for w ∈ W −k m ), where the power in c k depends on the term in the coproduct. Hence, by F m 0 -linearity of η X Fm (which in particular implies K-linearity), we find that η X Fm decomposes into K-linear maps and hence by the assumption that V and W are decomposable also a unique H Mmorphism L: V → W .