Abstract
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.
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1 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 \({^{H}}\mathscr {S}\) the category of toric noncommutative spaces, we show that there is a natural site structure on \({^{H}}\mathscr {S}\) 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 \({^{H}}\mathscr {G}:= \mathrm {Sh}({^{H}}\mathscr {S})\), and we show that there is a fully faithful embedding \({^{H}}\mathscr {S}\rightarrow {^{H}}\mathscr {G}\) which allows us to equivalently regard toric noncommutative spaces as living in this bigger category. The advantage of the bigger category \({^{H}}\mathscr {G}\) 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 \({^{H}}\mathscr {G}\), i.e., as a sheaf on the site \({^{H}}\mathscr {S}\). As an application, we study the ‘internalized’ automorphism group \(\mathrm {Aut}(X)\) of a toric noncommutative space X, which is a certain subobject in \({^{H}}\mathscr {G}\) of the self-mapping space \(X^X\). Using synthetic geometry techniques, we are able to compute the Lie algebra of \(\mathrm {Aut}(X)\) 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 \(\mathrm {Hom}(X,Y)\) (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 \(P\rightarrow X\) by carving out a subobject in \({^{H}}\mathscr {G}\) of the ‘internalized’ automorphism group \(\mathrm {Aut}(P)\) 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 \({^{H}}\mathscr {M}\). In Sect. 3, we study algebra objects in \({^{H}}\mathscr {M}\) whose commutation relations are controlled by the cotriangular structure on H. We establish a category of finitely presented algebra objects \({^{H}}\mathscr {A}_{\mathrm {fp}}\), 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 \({^{H}}\mathscr {S}\) is then given by the opposite category of \({^{H}}\mathscr {A}_{\mathrm {fp}}\), and we show in Sect. 4 that \({^{H}}\mathscr {S}\) can be equipped with the structure of a site. In Sect. 5, we introduce and study the sheaf topos \({^{H}}\mathscr {G}\) whose objects are sheaves on \({^{H}}\mathscr {S}\) which we interpret as generalized toric noncommutative spaces. We show that the Yoneda embedding factorizes through \({^{H}}\mathscr {G}\) (i.e., that our site is subcanonical) and hence obtain a fully faithful embedding \({^{H}}\mathscr {S}\rightarrow {^{H}}\mathscr {G}\) of toric noncommutative spaces into generalized toric noncommutative spaces. An explicit description of the exponential objects \(Y^X\) in \({^{H}}\mathscr {G}\) 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 \(\mathrm {Hom}(X,Y)\) (cf. Example 5.3). In Sect. 6, we apply these techniques to define an ‘internalized’ automorphism group \(\mathrm {Aut}(X)\) of a toric noncommutative space X, which arises as a certain subobject in \({^{H}}\mathscr {G}\) of the self-mapping space \(X^X\). It is important to stress that \(\mathrm {Aut}(X)\) 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 \({^{H}}\mathscr {S}\). The Lie algebra of \(\mathrm {Aut}(X)\) 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 \(\mathrm {Aut}(X)\) can be identified with the braided derivations of the function algebra of X. Hence, in contrast to \(\mathrm {Aut}(X)\), its Lie algebra of infinitesimal automorphisms has an elementary description. This identification is rather technical, and it relies on a fully faithful embedding \({^{H}}\mathscr {M}_{\mathrm {dec}}\rightarrow \mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\) of a certain full subcategory (called decomposables) of the category of left H-comodules \({^{H}}\mathscr {M}\) into the category of \(\underline{K}\)-module objects in the sheaf topos \({^{H}}\mathscr {G}\), where \(\underline{K}\) denotes the line object in this topos; the technical details are presented in “Appendix.”
2 Hopf algebra preliminaries
In this paper, all vector spaces will be over a fixed field \(\mathbb {K}\) and the tensor product of vector spaces will be denoted simply by \(\otimes \).
The Hopf algebra \(H:= \mathcal {O}(\mathbb {T}^n)\) of functions on the algebraic n-torus \(\mathbb {T}^n\) 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 \(R{:}\,H\otimes H\rightarrow \mathbb {K}\) satisfying
for all \(f,g,h\in H\), where we have used Sweedler notation \(\Delta (h) = h_{(1)}\otimes h_{(2)}\) (with summation understood) for the coproduct in H. The quasi-commutativity condition \(g_{(1)}\,h_{(1)} \,R(h_{(2)}\otimes g_{(2)}) = R(h_{(1)}\otimes g_{(1)})\, h_{(2)}\, g_{(2)}\), for all \(g,h\in H\), is automatically fulfilled because H is commutative and cocommutative. For example, if \(\mathbb {K}=\mathbb {C}\) is the field of complex numbers, we may take the usual cotriangular structure defined by
where \(\Theta \) is an antisymmetric real \(n{\times } n\)-matrix, which plays the role of deformation parameters for the theory.
Let us denote by \({^{H}}\mathscr {M}\) the category of left H-comodules. An object in \({^{H}}\mathscr {M}\) is a pair \((V,\rho ^V)\), where V is a vector space and \(\rho ^{V}{:}\,V\rightarrow H\otimes V\) is a left H-coaction on V, i.e., a linear map satisfying
We follow the usual abuse of notation and denote objects \((V,\rho ^V)\) in \({^{H}}\mathscr {M}\) simply by V without displaying the coaction explicitly. We further use a Sweedler-like notation \(\rho ^V(v) = v_{(-1)}\otimes v_{(0)}\) (with summation understood) for the left H-coactions. Then, (2.6) reads as
A morphism \(L{:}\,V\rightarrow W\) in \({^{H}}\mathscr {M}\) is a linear map preserving the left H-coactions, i.e.,
or in the Sweedler-like notation
for all \(v\in V\).
The category \({^{H}}\mathscr {M}\) is a monoidal category with tensor product of two objects V and W given by the tensor product \(V\otimes W\) of vector spaces equipped with the left H-coaction
The monoidal unit in \({^{H}}\mathscr {M}\) is given by the one-dimensional vector space \(\mathbb {K}\) with trivial left H-coaction \(\mathbb {K}\rightarrow H\otimes \mathbb {K},~c\mapsto \mathbbm {1}_H^{}\otimes c\). The monoidal category \({^{H}}\mathscr {M}\) is symmetric with commutativity constraint
for any two objects V and W in \({^{H}}\mathscr {M}\).
3 Algebra objects
We are interested in spaces whose algebras of functions are described by certain algebra objects in the symmetric monoidal category \({^{H}}\mathscr {M}\). An algebra object in \({^{H}}\mathscr {M}\) is an object A in \({^{H}}\mathscr {M}\) together with two \({^{H}}\mathscr {M}\)-morphisms \(\mu _{A}^{}{:}\,A\otimes A\rightarrow A\) (product) and \(\eta _{A}^{}{:}\,\mathbb {K}\rightarrow A\) (unit) such that the diagrams
in \({^{H}}\mathscr {M}\) commute. Because \({^{H}}\mathscr {M}\) is symmetric, we may additionally demand that the product \(\mu _A^{}\) is compatible with the commutativity constraints in \({^{H}}\mathscr {M}\), i.e., the diagram
in \({^{H}}\mathscr {M}\) commutes. This amounts to demanding the commutation relations
for all \(a,a^\prime \in A\), where we have abbreviated the product by \(\mu _A^{}(a\otimes a^\prime ) = a\,a^\prime \); in the following, we shall also use the compact notation \(\mathbbm {1} _A^{}:= \eta _A^{}(1)\in A\) for the unit element in A, or sometimes just \(\mathbbm {1}\). 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.
Definition 3.1
The category \({^{H}}\mathscr {A}\) has as objects all algebra objects in \({^{H}}\mathscr {M}\) which satisfy the commutativity constraint (3.2). The morphisms between two objects are all \({^{H}}\mathscr {M}\)-morphisms \(\kappa {:}\,A\rightarrow B\) which preserve products and units, i.e., for which \(\mu _B^{}\circ \kappa \otimes \kappa = \kappa \circ \mu _A^{}\) and \(\kappa \circ \eta _A^{} = \eta _B^{}\).
There is the forgetful functor \(\mathrm {Forget}{:}\,{^{H}}\mathscr {A}\rightarrow {^{H}}\mathscr {M}\) which assigns to any object in \({^{H}}\mathscr {A}\) its underlying left H-comodule, i.e., \((A,\mu _A^{},\eta _A^{}) \mapsto A\). This functor has a left adjoint \(\mathrm {Free}{:}\,{^{H}}\mathscr {M}\rightarrow {^{H}}\mathscr {A}\) which describes the free \({^{H}}\mathscr {A}\)-algebra construction: Given any object V in \({^{H}}\mathscr {M}\), we consider the vector space
with the convention \(V^{\otimes 0} :=\mathbb {K}\). Then, \(\mathcal {T}V\) is a left H-comodule when equipped with the coaction \(\rho ^{\mathcal {T}V}{:}\,\mathcal {T}V \rightarrow H\otimes \mathcal {T}V\) specified by
Moreover, \(\mathcal {T}V\) is an algebra object in \({^{H}}\mathscr {M}\) when equipped with the product \(\mu _{\mathcal {T}V}^{}{:}\,\mathcal {T}V\otimes \mathcal {T}V \rightarrow \mathcal {T}V\) specified by
and the unit \(\eta _{\mathcal {T}V}^{}{:}\,\mathbb {K}\rightarrow \mathcal {T}V\) given by
The algebra object \(\mathcal {T}V\) does not satisfy the commutativity constraint (3.2); hence, it is not an object of the category \({^{H}}\mathscr {A}\). We may enforce the commutativity constraint by taking the quotient of \(\mathcal {T}V\) by the two-sided ideal \(I\subseteq \mathcal {T}V\) generated by
for all \(v,v^\prime \in V\). The ideal I is stable under the left H-coaction, i.e., \(\rho ^{\mathcal {T}V}{:}\,I \rightarrow H\otimes I\). Hence, the quotient
is an object in \({^{H}}\mathscr {A}\) when equipped with the induced left H-coaction, product and unit. Given now any \({^{H}}\mathscr {M}\)-morphism \(L{:}\,V\rightarrow W\), we define an \({^{H}}\mathscr {A}\)-morphism \(\mathrm {Free}(L){:}\,\mathrm {Free}(V)\rightarrow \mathrm {Free}(W)\) by setting
This is compatible with the quotients because of (2.8). Finally, let us confirm that \(\mathrm {Free}{:}\,{^{H}}\mathscr {M}\rightarrow {^{H}}\mathscr {A}\) is the left adjoint of the forgetful functor \(\mathrm {Forget}{:}\,{^{H}}\mathscr {A}\rightarrow {^{H}}\mathscr {M}\), i.e., that there exists a (natural) bijection
between the morphism sets, for any object V in \({^{H}}\mathscr {M}\) and any object A in \({^{H}}\mathscr {A}\). This is easy to see from the fact that any \({^{H}}\mathscr {A}\)-morphism \(\kappa {:}\,\mathrm {Free}(V)\rightarrow A\) is uniquely specified by its restriction to the vector space \(V = V^{\otimes 1}\subseteq \mathrm {Free}(V)\) of generators and hence by an \({^{H}}\mathscr {M}\)-morphism \(V\rightarrow \mathrm {Forget}(A)\).
From a geometric perspective, the free \({^{H}}\mathscr {A}\)-algebras \(\mathrm {Free}(V)\) 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 \({^{H}}\mathscr {A}\)-algebras.
Definition 3.2
Let A be an object in \({^{H}}\mathscr {A}\). An \({^{H}}\mathscr {A}\)-ideal I of A is a two-sided ideal \(I\subseteq A\) of the algebra underlying A which is stable under the left H-coaction, i.e., the coaction \(\rho ^A\) induces a linear map \(\rho ^A{:}\,I \rightarrow H\otimes I\).
This definition immediately implies
Lemma 3.3
If A is an object in \({^{H}}\mathscr {A}\) and I is an \({^{H}}\mathscr {A}\)-ideal of A, the quotient A / I is an object in \({^{H}}\mathscr {A}\) when equipped with the induced coaction, product and unit.
This lemma allows us to construct a variety of \({^{H}}\mathscr {A}\)-algebras by taking quotients of free \({^{H}}\mathscr {A}\)-algebras by suitable \({^{H}}\mathscr {A}\)-ideals. We are particularly interested in the case where the object V in \({^{H}}\mathscr {M}\) that underlies the free \({^{H}}\mathscr {A}\)-algebra \(\mathrm {Free}(V)\) 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 \(H = \mathcal {O}(\mathbb {T}^n)\) can be characterized by a label \({\varvec{m}}\in \mathbb {Z}^n\). The corresponding left H-coactions are given by
We shall use the notation \(\mathbb {K}_{{\varvec{m}}} := (\mathbb {K},\rho ^{{\varvec{m}}})\) for these objects in \({^{H}}\mathscr {M}\). The coproduct \(\mathbb {K}_{{\varvec{m}}}\sqcup \mathbb {K}_{{\varvec{m^\prime }}}\) of two such objects is given by the vector space \(\mathbb {K}\oplus \mathbb {K}\simeq \mathbb {K}^2\) together with the component-wise coaction, i.e.,
The free \({^{H}}\mathscr {A}\)-algebra corresponding to a finite coproduct of objects \(\mathbb {K}_{{\varvec{m}}_i}\), for \(i=1,\ldots ,N\), in \({^{H}}\mathscr {M}\) will be used frequently in this paper. Hence, we introduce the compact notation
By construction, the \({^{H}}\mathscr {A}\)-algebras \(F_{{\varvec{m}}_1,\ldots , {\varvec{m}}_N}\) are generated by N elements \(x_i\in F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}\) whose transformation property under the left H-coaction is given by \(\rho ^{F_{{\varvec{m}}_1,\ldots , {\varvec{m}}_N}}(x_i) = t_{{\varvec{m}}_i}\otimes x_i\) and whose commutation relations read as
for all \(i,j=1,\ldots ,N\).
We can now introduce the category of finitely presented \({^{H}}\mathscr {A}\)-algebras.
Definition 3.4
An object A in \({^{H}}\mathscr {A}\) is finitely presented if it is isomorphic to the quotient \(F_{{\varvec{m}}_1,\ldots , {\varvec{m}}_N}/I\) of a free \({^{H}}\mathscr {A}\)-algebra \(F_{{\varvec{m}}_1,\ldots , {\varvec{m}}_N}\) by an \({^{H}}\mathscr {A}\)-ideal \(I= (f_k)\) that is generated by a finite number of elements \(f_k\in F_{{\varvec{m}}_1,\ldots , {\varvec{m}}_N}\), for \(k=1,\ldots , M\), with \(\rho ^{F_{{\varvec{m}}_1,\ldots , {\varvec{m}}_N}}(f_k) = t_{{\varvec{n}}_k}\otimes f_k\), for some \({\varvec{n}}_k\in \mathbb {Z}^n\). We denote by \({^{H}}\mathscr {A}_{\mathrm {fp}}\) the full subcategory of \({^{H}}\mathscr {A}\) whose objects are all finitely presented \({^{H}}\mathscr {A}\)-algebras.
Example 3.5
Let us consider the case \(\mathbb {K}=\mathbb {C}\) and R given by (2.5). Take the free \({^{H}}\mathscr {A}\)-algebra generated by \(x_i\) and \(x_i^*\), for \(i=1,\ldots ,N\), with left H-coaction specified by \(x_i\mapsto t_{{\varvec{m}}_i}\otimes x_i\) and \(x_i^*\mapsto t_{-{\varvec{m}}_i}\otimes x_i^*\), for some \({\varvec{m}}_i\in \mathbb {Z}^n\); in the notation above, we consider the free \({^{H}}\mathscr {A}\)-algebra \(F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N,-{\varvec{m}}_1,\ldots ,-{\varvec{m}}_N}\) and denote the last N generators by \(x_i^*:= x_{N+i}\), for \(i=1,\ldots ,N\). The algebra of (the algebraic version of) the \(2N{-}1\)-dimensional Connes–Landi sphere is obtained by taking the quotient with respect to the \({^{H}}\mathscr {A}\)-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 \({^{H}}\mathscr {A}\)-ideal
To obtain also the even dimensional Connes–Landi spheres, we consider the free \({^{H}}\mathscr {A}\)-algebra \(F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N,-{\varvec{m}}_1,\ldots ,-{\varvec{m}}_N,{\varvec{0}}}\), where the additional generator \(x_{2N+1}\) has trivial H-coaction \(x_{2N+1}\mapsto \mathbbm {1}_H\otimes x_{2N+1}\), and take the quotient with respect to the \({^{H}}\mathscr {A}\)-ideal
All these examples are \(*\)-algebras with involution defined by \(x_i \mapsto x_i^*\) and \(x_{2N+1}^*= x_{2N+1}\). An example which is not a \(*\)-algebra is the free \({^{H}}\mathscr {A}\)-algebra \(F_{{\varvec{m}}}\), for some \({\varvec{m}}\ne {\varvec{0}}\) in \(\mathbb {Z}^n\), which we may interpret as the algebra of (anti)holomorphic polynomials on \(\mathbb {C}\).
We will now study some properties of the categories \({^{H}}\mathscr {A}\) and \({^{H}}\mathscr {A}_{\mathrm {fp}}\) that will be used in the following. First, let us notice that the category \({^{H}}\mathscr {A}\) has (finite) coproducts: Given two objects A and B in \({^{H}}\mathscr {A}\), their coproduct \(A\sqcup B\) is the object in \({^{H}}\mathscr {A}\) whose underlying left H-comodule is \(A\otimes B\) [with coaction \(\rho ^{A\sqcup B} := \rho ^{A\otimes B}\) given in (2.9)] and whose product \(\mu _{A\sqcup B}\) and unit \(\eta _{A\sqcup B}\) are characterized by
The canonical inclusion \({^{H}}\mathscr {A}\)-morphisms \(\iota _1^{}{:}\,A\rightarrow A\sqcup B\) and \(\iota _2^{}{:}\,B\rightarrow A\sqcup B\) are given by
for all \(a\in A\) and \(b\in B\). The coproduct \(A\sqcup B\) of two finitely presented \({^{H}}\mathscr {A}\)-algebras A and B is finitely presented: If \(A = F_{{\varvec{m}}_1,\ldots , {\varvec{m}}_N}/(f_k)\) and \(B = F_{{\varvec{m^\prime }}_1,\ldots , {\varvec{m^\prime }}_{N^\prime }}/(f^\prime _{k^\prime })\), then
where we have identified \(F_{{\varvec{m}}_1,\ldots , {\varvec{m}}_N,{\varvec{m^\prime }}_1,\ldots , {\varvec{m^\prime }}_{N^\prime }}\simeq F_{{\varvec{m}}_1,\ldots , {\varvec{m}}_N} \sqcup F_{{\varvec{m^\prime }}_1,\ldots , {\varvec{m^\prime }}_{N^\prime }}\). Consequently, the category \({^{H}}\mathscr {A}_{\mathrm {fp}}\) has finite coproducts.
In addition to coproducts, we also need pushouts in \({^{H}}\mathscr {A}\) and \({^{H}}\mathscr {A}_{\mathrm {fp}}\), which are given by colimits of the form
in \({^{H}}\mathscr {A}\) or \({^{H}}\mathscr {A}_{\mathrm {fp}}\). Such pushouts exist and can be constructed as follows: Consider first the case where we work in the category \({^{H}}\mathscr {A}\). We define
where I is the \({^{H}}\mathscr {A}\)-ideal generated by \(\kappa (c)\otimes \mathbbm {1}_B^{} - \mathbbm {1}_{A}^{}\otimes \zeta (c)\), for all \(c\in C\). The dashed \({^{H}}\mathscr {A}\)-morphisms in (3.22) are given by
It is easy to confirm that \(A\sqcup _C^{}B\) defined above is a pushout of the diagram (3.22). Moreover, the pushout of finitely presented \({^{H}}\mathscr {A}\)-algebras is finitely presented: If \(A = F_{{\varvec{m}}_1,\ldots , {\varvec{m}}_N}/(f_k)\), \(B = F_{{\varvec{m^\prime }}_1,\ldots , {\varvec{m^\prime }}_{N^\prime }}/(f^\prime _{k^\prime })\) and \(C = F_{{\varvec{m^{\prime \prime }}}_1,\ldots , {\varvec{m^{\prime \prime }}}_{N^{\prime \prime }}}/ (f^{\prime \prime }_{k^{\prime \prime }})\), then
where \(x^{\prime \prime }_i\), for \(i=1,\ldots , N^{\prime \prime }\), are the generators of C. The isomorphism in (3.25) follows from the fact that the quotient by the finite number of extra relations \(\kappa (x^{\prime \prime }_{i})\otimes \mathbbm {1} - \mathbbm {1}\otimes \zeta (x^{\prime \prime }_i)\), for all generators \(x^{\prime \prime }_i\) of C, is sufficient to describe the \({^{H}}\mathscr {A}\)-ideal I in (3.23) for finitely presented \({^{H}}\mathscr {A}\)-algebras C: We can recursively use the identities [valid on the right-hand side of (3.25)]
for all generators \(x^{\prime \prime }_i\) and elements \(c\in C\), in order to show that the \({^{H}}\mathscr {A}\)-ideal I is equivalently generated by \(\kappa (x^{\prime \prime }_{i})\otimes \mathbbm {1} - \mathbbm {1} \otimes \zeta (x^{\prime \prime }_i)\). Consequently, the category \({^{H}}\mathscr {A}_{\mathrm {fp}}\) has pushouts.
We also need the localization of \({^{H}}\mathscr {A}\)-algebras A with respect to a single H-coinvariant element \(s \in A\), i.e., \(\rho ^A(s) = \mathbbm {1}_H^{}\otimes s\). Localization amounts to constructing an \({^{H}}\mathscr {A}\)-algebra \(A[s^{-1}]\) together with an \({^{H}}\mathscr {A}\)-morphism \(\ell _s{:}\,A\rightarrow A[s^{-1}]\) that maps the element \(s\in A\) to an invertible element \(\ell _s(s)\in A[s^{-1}]\) and that satisfies the following universal property: If \(\kappa {:}\,A\rightarrow B\) is another \({^{H}}\mathscr {A}\)-morphism such that \(\kappa (s)\in B\) is invertible, then \(\kappa \) factors though \(\ell _s{:}\,A\rightarrow A[s^{-1}]\), i.e., there exists a unique \({^{H}}\mathscr {A}\)-morphism \(A[s^{-1}]\rightarrow B\) making the diagram
commute. We now show that the \({^{H}}\mathscr {A}\)-algebra
together with the \({^{H}}\mathscr {A}\)-morphism
is a localization of A with respect to the H-coinvariant element \(s\in A\). The inverse of \(\ell _s(s) = [s\otimes \mathbbm {1}_{F_{{\varvec{0}}}}^{}]\) exists, and it is given by the new generator \([\mathbbm {1}_A^{}\otimes x]\in A[s^{-1}]\); then, the inverse of \(\ell _s(s^n)\) is \([\mathbbm {1}_A^{}\otimes x^n]\), because \([s\otimes \mathbbm {1}_{F_{{\varvec{0}}}}^{}]\) and \([\mathbbm {1}_A^{}\otimes x]\) commute in \(A[s^{-1}]\), cf. (3.3), (3.19) and use the fact that x and s are coinvariants. Given now any \({^{H}}\mathscr {A}\)-morphism \(\kappa {:}\,A\rightarrow B\) such that \(\kappa (s)\in B\) is invertible, say by \(t\in B\), there is a unique \({^{H}}\mathscr {A}\)-morphism \(A[s^{-1}]\rightarrow B\) specified by \([a\otimes x^{n}]\mapsto \kappa (a)\, t^n\) that factors \(\kappa \) through \(\ell _s{:}\,A\rightarrow A[s^{-1}]\). Finally, the localization of finitely presented \({^{H}}\mathscr {A}\)-algebras is finitely presented: If \(A= F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}/(f_k)\), then
4 Toric noncommutative spaces
From a geometric perspective, it is useful to interpret an object A in \({^{H}}\mathscr {A}_{\mathrm {fp}}\) as the ‘algebra of functions’ on a toric noncommutative space \(X_A\). Similarly, a morphism \(\kappa {:}\,A\rightarrow B\) in \({^{H}}\mathscr {A}_{\mathrm {fp}}\) is interpreted as the ‘pullback’ of a map \(f{:}\,X_B\rightarrow X_A\) 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 \(\kappa = f^*{:}\,A\rightarrow B\) for the \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism corresponding to \(f{:}\,X_B\rightarrow X_A\). This can be made precise with
Definition 4.1
The category of toric noncommutative spaces
is the opposite of the category \({^{H}}\mathscr {A}_{\mathrm {fp}}\). Objects in \({^{H}}\mathscr {S}\) will be denoted by symbols like \(X_A\), where A is an object in \({^{H}}\mathscr {A}_{\mathrm {fp}}\). Morphisms in \({^{H}}\mathscr {S}\) will be denoted by symbols like \(f{:}\,X_{B}\rightarrow X_{A}\), and they are (by definition) in bijection with \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphisms \(f^*{:}\,A\rightarrow B\).
As the category \({^{H}}\mathscr {A}_{\mathrm {fp}}\) has (finite) coproducts and pushouts, which we have denoted by \(A\sqcup B\) and \(A\sqcup _C^{} B\), its opposite category \({^{H}}\mathscr {S}\) has (finite) products and pullbacks. Given two objects \(X_A\) and \(X_B\) in \({^{H}}\mathscr {S}\), their product is given by
together with the canonical projection \({^{H}}\mathscr {S}\)-morphisms
specified by \(\pi _{1}^*= \iota _{1}^{}{:}\,A\rightarrow A\sqcup B\) and \(\pi _{2}^*= \iota _{2}^{}{:}\,B\rightarrow A\sqcup B\), where \(\iota _{1}^{}\) and \(\iota _{2}^{}\) are the canonical inclusion \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphisms for the coproduct in \({^{H}}\mathscr {A}_{\mathrm {fp}}\) [cf. (3.20)]. Pullbacks
in \({^{H}}\mathscr {S}\) are given by
for \(\kappa = f^*{:}\,C\rightarrow A \) and \(\zeta = g^*{:}\,C\rightarrow B \) [cf. (3.22)]. The dashed arrows in (4.3) are specified by their corresponding \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-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.
Definition 4.2
An \({^{H}}\mathscr {S}\)-Zariski covering family is a finite family of \({^{H}}\mathscr {S}\)-morphisms
where
-
(i)
\(s_i\in A\) is an H-coinvariant element, i.e., \(\rho ^A(s_i) =\mathbbm {1}_H^{}\otimes s_i\), for all i;
-
(ii)
\(f_i\) is specified by the canonical \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(f_i^*= \ell _{s_i}{:}\,A\rightarrow A[s_i^{-1}]\), for all i;
-
(iii)
there exists a family of elements \(a_i\in A\) such that \(\sum _i\, a_i\,s_i =\mathbbm {1}_A^{}\).
Example 4.3
Recall from Example 3.5 that the algebra of functions on the 2N-dimensional Connes–Landi sphere is given by
As the last generator \(x_{2N+1}\) is H-coinvariant, we can define the two H-coinvariant elements \(s_1 := \frac{1}{2}\, (\mathbbm {1}- x_{2N+1})\) and \(s_{2} := \frac{1}{2}\, (\mathbbm {1}+ x_{2N+1})\). Then, \(s_1 + s_2 =\mathbbm {1}\), and hence, we obtain an \({^{H}}\mathscr {S}\)-Zariski covering family
for the 2N-dimensional Connes–Landi sphere. Geometrically, \(X_{A_{\mathbb {S}^{2N}_\Theta }[s_1^{-1}]}\) is the sphere with the north pole removed and similarly \(X_{A_{\mathbb {S}^{2N}_\Theta }[s_2^{-1}]}\) the sphere with the south pole removed.
We now show that \({^{H}}\mathscr {S}\)-Zariski covering families are stable under pullbacks.
Proposition 4.4
The pullback of an \({^{H}}\mathscr {S}\)-Zariski covering family \(\{f_{i}{:}\,X_{A[s_i^{-1}]} \rightarrow X_{A}^{}\}\) along an \({^{H}}\mathscr {S}\)-morphism \(g{:}\,X_{B} \rightarrow X_A\) is an \({^{H}}\mathscr {S}\)-Zariski covering family, i.e., the left vertical arrows of the pullback diagrams
define an \({^{H}}\mathscr {S}\)-Zariski covering family.
Proof
By definition, \(X_{B}\times _{X_A}^{} X_{A[s_i^{-1}]} = X_{B\sqcup _A^{} A[s_i^{-1}]}\). By universality of the pushout and localization, the pushout diagram for \(B\sqcup _A^{} A[s_i^{-1}]\) extends to the commutative diagram
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, \(X_{B}\times _{X_A}^{} X_{A[s_i^{-1}]} \simeq X_{B[g^*(s_i)^{-1}]}\) and the left vertical arrow in (4.8) is of the form as required in Definition 4.2 (i) and (ii). To show also item (iii) of Definition 4.2, if \(a_i\in A\) is a family of elements such that \(\sum _i \, a_i\,s_i=\mathbbm {1}_A^{}\), then \(b_i := g^*(a_i)\in B\) is a family of elements such that \(\sum _{i}\, b_i \, g^{*}(s_i) = \mathbbm {1}_B^{}\). \(\square \)
Corollary 4.5
Let \(\{f_{i}{:}\,X_{A[s_i^{-1}]} \rightarrow X_{A}^{}\}\) be an \({^{H}}\mathscr {S}\)-Zariski covering family. Then, the pullback
is isomorphic to \(X_{A[s_i^{-1},\, s_j^{-1}]}\), where
is the localization with respect to the two H-coinvariant elements \(s_i,s_j\in A\). The dashed arrows in (4.10) are specified by the canonical \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphisms \(\ell _{s_j}{:}\,A[s_i^{-1}] \rightarrow A[s_i^{-1},\,s_j^{-1}]\) and \(\ell _{s_i}{:}\,A[s_j^{-1}] \rightarrow A[s_i^{-1},\,s_j^{-1}]\).
Proof
This follows immediately from the proof of Proposition 4.4. \(\square \)
Remark 4.6
For later convenience, we shall introduce the notation
for the morphisms of this pullback diagram.
5 Generalized toric noncommutative spaces
The category \({^{H}}\mathscr {S}\) 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 \({^{H}}\mathscr {S}\) to a bigger category that admits exponential objects. We review this procedure in our case of interest.
The desired extension of \({^{H}}\mathscr {S}\) is given by the category
of sheaves on \({^{H}}\mathscr {S}\) with covering families given in Definition 4.2. Recall that a sheaf on \({^{H}}\mathscr {S}\) is a functor \(Y{:}\,{^{H}}\mathscr {S}^\mathrm {op}\rightarrow \mathsf {Set}\) to the category of sets (called a presheaf) that satisfies the sheaf condition: For any \({^{H}}\mathscr {S}\)-Zariski covering family \(\{f_i{:}\,X_{A[s_i^{-1}]} \rightarrow X_A^{}\}\), the canonical diagram
is an equalizer in \(\mathsf {Set}\). We have used Corollary 4.5 to express the pullback of two covering morphisms by \(X_{A[s_i^{-1},\,s_j^{-1}]}\). Because \( {^{H}}\mathscr {S}= ({^{H}}\mathscr {A}_{\mathrm {fp}})^\mathrm {op}\) was defined as the opposite category of \({^{H}}\mathscr {A}_{\mathrm {fp}}\), it is sometimes convenient to regard a sheaf on \({^{H}}\mathscr {S}\) as a covariant functor \(Y{:}\,{^{H}}\mathscr {A}_{\mathrm {fp}} \rightarrow \mathsf {Set}\). In this notation, the sheaf condition (5.2) looks like
We will interchangeably use these equivalent points of view. The morphisms in \({^{H}}\mathscr {G}\) are natural transformations between functors, i.e., presheaf morphisms.
We shall interpret \({^{H}}\mathscr {G}\) as a category of generalized toric noncommutative spaces. To justify this interpretation, we will show that there is a fully faithful embedding \({^{H}}\mathscr {S}\rightarrow {^{H}}\mathscr {G}\) of the category of toric noncommutative spaces into the new category. As a first step, we use the (fully faithful) Yoneda embedding \({^{H}}\mathscr {S}\rightarrow \mathrm {PSh}({^{H}}\mathscr {S})\) in order to embed \({^{H}}\mathscr {S}\) into the category of presheaves on \({^{H}}\mathscr {S}\). The Yoneda embedding is given by the functor which assigns to any object \(X_A\) in \({^{H}}\mathscr {S}\) the presheaf given by the functor
and to any \({^{H}}\mathscr {S}\)-morphism \(f{:}\,X_A\rightarrow X_B\) the natural transformation \(\underline{f}{:}\,\underline{X_A}\rightarrow \underline{X_B}\) with components
for all objects \(X_C\) in \({^{H}}\mathscr {S}\).
Proposition 5.1
For any object \(X_{A}\) in \({^{H}}\mathscr {S}\), the presheaf \(\underline{X_A}\) is a sheaf on \({^{H}}\mathscr {S}\). As a consequence, the Yoneda embedding induces a fully faithful embedding \({^{H}}\mathscr {S}\rightarrow {^{H}}\mathscr {G}\) into the category of sheaves on \({^{H}}\mathscr {S}\).
Proof
We have to show that the functor \(\underline{X_A}{:}\,{^{H}}\mathscr {S}^\mathrm {op}\rightarrow \mathsf {Set}\) satisfies the sheaf condition (5.2), or equivalently (5.3). Given any \({^{H}}\mathscr {S}\)-Zariski covering family \(\{f_i{:}\,X_{B[s_i^{-1}]}\rightarrow X_B^{}\}\), we therefore have to confirm that
is an equalizer in \(\mathsf {Set}\), where we used \(\underline{X_A}(X_B) = \mathrm {Hom}_{{^{H}}\mathscr {S}}^{}(X_B,X_A) = \mathrm {Hom}_{{^{H}}\mathscr {A}_{\mathrm {fp}}}^{}(A,B)\). Because the \(\mathrm {Hom}\)-functor \(\mathrm {Hom}_{{^{H}}\mathscr {A}_{\mathrm {fp}}} (A,-){:}\,{^{H}}\mathscr {A}_{\mathrm {fp}} \rightarrow \mathsf {Set}\) preserves limits, it is sufficient to prove that
is an equalizer in \({^{H}}\mathscr {A}_{\mathrm {fp}}\).
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 \([b_i\otimes c_i] = \sum _{\alpha }\, [{(b_i)}_{\alpha } \otimes {(c_{i})}_\alpha ]\) which we suppress. This is an element in the desired equalizer if and only if
for all i, j, as equalities in \(B[s_{i}^{-1},\,s_j^{-1}]\). Recalling that the relations in \(B[s_{i}^{-1},\,s_j^{-1}]\) are given by \(s_i\otimes x_i \otimes \mathbbm {1} =\mathbbm {1}\) and \(s_j\otimes \mathbbm {1}\otimes x_j=\mathbbm {1}\), the equalities (5.8) hold if and only if \([b_i\otimes c_i] = [b\otimes \mathbbm {1}]\) with the same \(b\in B\), for all i. Hence, (5.6) is an equalizer. \(\square \)
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 \({^{H}}\mathscr {G}\). The sheaf \(\underline{X_A}\) 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 \({^{H}}\mathscr {G}\) (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 \({^{H}}\mathscr {S}\) and any object Y in \({^{H}}\mathscr {G}\).
The advantage of the sheaf category \({^{H}}\mathscr {G}\) of generalized toric noncommutative spaces over the original category \({^{H}}\mathscr {S}\) 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 \({^{H}}\mathscr {G}\) has all (small) limits and all exponential objects. Limits in \({^{H}}\mathscr {G}\) are computed object-wise, i.e., as in presheaf categories. In particular, the product of two objects Y, Z in \({^{H}}\mathscr {G}\) is the sheaf specified by the functor \(Y\times Z{:}\,{^{H}}\mathscr {S}^{\mathrm {op}} \rightarrow \mathsf {Set}\) that acts on objects as
where on the right-hand side \(\times \) is the Cartesian product in \(\mathsf {Set}\), and on morphisms \( f{:}\,X_A\rightarrow X_B\) as
The terminal object in \({^{H}}\mathscr {G}\) is the sheaf specified by the functor \(\{*\}{:}\,{^{H}}\mathscr {S}^{\mathrm {op}} \rightarrow \mathsf {Set}\) that acts on objects as
where on the right-hand side \(\{*\}\) is the terminal object in \(\mathsf {Set}\), i.e., a singleton set, and in the obvious way on morphisms. The fully faithful embedding \({^{H}}\mathscr {S}\rightarrow {^{H}}\mathscr {G}\) of Proposition 5.1 is limit preserving. In particular, we have
for all objects \(X_A,X_B\) in \({^{H}}\mathscr {S}\) and the terminal object \(X_{\mathbb {K}}\) in \({^{H}}\mathscr {S}\). Here \(\mathbb {K}\) is the \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-algebra with trivial left H-coaction \(c\mapsto \mathbbm {1}_H\otimes c\), i.e., the initial object in \({^{H}}\mathscr {A}_{\mathrm {fp}}\).
The exponential objects in \({^{H}}\mathscr {G}\) are constructed as follows: Given two objects Y, Z in \({^{H}}\mathscr {G}\), the exponential object \(Z^Y\) is the sheaf specified by the functor \(Z^Y{:}\,{^{H}}\mathscr {S}^{\mathrm {op}} \rightarrow \mathsf {Set}\) that acts on objects as
and on morphisms \(f{:}\,X_A\rightarrow X_B\) as
where \(\underline{f}{:}\,\underline{X_A}\rightarrow \underline{X_B}\) is the \({^{H}}\mathscr {G}\)-morphism specified by (5.4). The formation of exponential objects is functorial, i.e., there are obvious functors \((-)^Y{:}\,{^{H}}\mathscr {G}\rightarrow {^{H}}\mathscr {G}\) and \(Z^{(-)}{:}\,{^{H}}\mathscr {G}^\mathrm {op}\rightarrow {^{H}}\mathscr {G}\), for all objects Y, Z in \({^{H}}\mathscr {G}\). Moreover, there are natural isomorphisms
for all objects \(Y,Y^\prime ,Z,Z^\prime \) in \({^{H}}\mathscr {G}\).
Given two ordinary toric noncommutative spaces \(X_A\) and \(X_B\), i.e., objects in \({^{H}}\mathscr {S}\), we can form the exponential object \(\underline{X_B}^{\underline{X_A}}\) in the category of generalized toric noncommutative spaces \({^{H}}\mathscr {G}\). The interpretation of \(\underline{X_B}^{\underline{X_A}}\) 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 \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphisms.
Example 5.3
To illustrate the differences between the exponential objects \(\underline{X_B}^{\underline{X_A}}\) in \({^{H}}\mathscr {G}\) and the \(\mathrm {Hom}\)-sets \(\mathrm {Hom}_{{^{H}}\mathscr {S}}(X_A,X_B)\), let us consider the simplest example where \(A=B=F_{{\varvec{m}}}\), for some \( {\varvec{m}}\ne {\varvec{0}}\) in \(\mathbb {Z}^n\). In this case, the \(\mathrm {Hom}\)-set is given by
because by H-equivariance any \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(\kappa {:}\,F_{{\varvec{m}}}\rightarrow F_{{\varvec{m}}}\) is of the form \(\kappa (x) = c\,x\), for some \(c\in \mathbb {K}\); here x denotes the generator of \(F_{{\varvec{m}}}\), whose left H-coaction is by definition \(\rho ^{F_{{\varvec{m}}}}(x) = t_{{\varvec{m}}}\otimes x\). On the other hand, the exponential object \(\underline{X_{F_{{\varvec{m}}}}}^{\underline{X_{F_{{\varvec{m}}}}}}\) is a functor from \({^{H}}\mathscr {S}^{\mathrm {op}} = {^{H}}\mathscr {A}_{\mathrm {fp}}\) to \(\mathsf {Set}\), and hence, it gives us a set for any test \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-algebra A. Using (5.16), we obtain
For the initial object \(A = \mathbb {K}\) in \({^{H}}\mathscr {A}_{\mathrm {fp}}\), we recover the \(\mathrm {Hom}\)-set
Let us now take \(A_{\mathbb {T}_{\Theta }} = F_{-{\varvec{m}},{\varvec{m}}}/(y^*\, y -\mathbbm {1})\) to be the toric noncommutative circle, see Example 3.5. We write \(y^{-1}:=y^*\) in \(A_{\mathbb {T}_{\Theta }}\) and recall that the left H-coaction is given by \(\rho ^{F_{-{\varvec{m}},{\varvec{m}}} }(y) = t_{-{\varvec{m}}}\otimes y\) and \(\rho ^{F_{-{\varvec{m}},{\varvec{m}}} }(y^{-1}) = t_{{\varvec{m}}}\otimes y^{-1}\). We then obtain an isomorphism of sets
because elements \(a\in F_{{\varvec{m}}}\), i.e., polynomials \(a= \sum _{j}\, c_j\,x^j\), for \(c_j\in \mathbb {K}\), are in bijection with \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphisms \(\kappa {:}\,F_{{\varvec{m}}}\rightarrow A_{\mathbb {T}_{\Theta }} \sqcup F_{{\varvec{m}}}\) via
By construction, each summand on the right-hand side has left H-coaction \(\rho ^{F_{-{\varvec{m}},{\varvec{m}}} }(y^{j-1}\otimes x^j) = t_{-(j-1)\,{\varvec{m}}}\,t_{j\,{\varvec{m}}} \otimes y^{j-1}\otimes x^j = t_{{\varvec{m}}}\otimes y^{j-1}\otimes x^j\). Heuristically, this means that the exponential object \(\underline{X_{F_{{\varvec{m}}}}}^{\underline{X_{F_{{\varvec{m}}}}}}\) captures all polynomial maps \(F_{{\varvec{m}}}\rightarrow F_{{\varvec{m}}}\), while the \(\mathrm {Hom}\)-set \(\mathrm {Hom}_{{^{H}}\mathscr {S}}(X_{F_{{\varvec{m}}}},X_{F_{{\varvec{m}}}})\) captures only those that are H-equivariant which in the present case are the linear maps \(x\mapsto c\,x\). Similar results hold for generic exponential objects \(\underline{X_B}^{\underline{X_A}}\) in \({^{H}}\mathscr {G}\); in particular, their global points \(\underline{X_B}^{\underline{X_A}}(X_\mathbb {K})\) coincide with the \(\mathrm {Hom}\)-sets \(\mathrm {Hom}_{{^{H}}\mathscr {S}}(X_A,X_B)\) while their generalized points \(\underline{X_B}^{\underline{X_A}}(X_C)\), for \(X_C\) an object in \({^{H}}\mathscr {S}\), capture additional maps.
6 Automorphism groups
Associated with any object, \(X_A\) in \({^{H}}\mathscr {S}\) is its exponential object \(\underline{X_A}^{\underline{X_A}}\) in \({^{H}}\mathscr {G}\) which describes the ‘space of maps’ from \(X_A\) to itself. The object \(\underline{X_A}^{\underline{X_A}}\) has a distinguished point (called the identity) given by the \({^{H}}\mathscr {G}\)-morphism
that is specified by the natural transformation with components
given by the canonical projection \({^{H}}\mathscr {S}\)-morphisms of the product. Under the Yoneda bijections \(\mathrm {Hom}_{{^{H}}\mathscr {G}}(\{*\}, \underline{X_A}^{\underline{X_A}}) \simeq \underline{X_A}^{\underline{X_A}}(\{*\})\simeq \mathrm {Hom}_{{^{H}}\mathscr {S}}(X_A,X_A)\), e is mapped to the identity \({^{H}}\mathscr {S}\)-morphism \(\mathrm {id}_{X_A}\). Moreover, there is a composition \({^{H}}\mathscr {G}\)-morphism
that is specified by the natural transformation with components
defined by
for all \({^{H}}\mathscr {S}\)-morphisms \(g,h{:}\,X_B\times X_A\rightarrow X_A\). The diagonal \({^{H}}\mathscr {S}\)-morphism \(\mathrm {diag}_{X_B}{:}\,X_B\rightarrow X_B\times X_B\) is defined as usual via universality of products by
The object \( \underline{X_A}^{\underline{X_A}}\) together with the identity (6.1) and composition \({^{H}}\mathscr {G}\)-morphisms (6.2) is a monoid object in \({^{H}}\mathscr {G}\): It is straightforward to verify that the \({^{H}}\mathscr {G}\)-diagrams
and
commute. Notice that \(\underline{X_A}^{\underline{X_A}}\) is not a group object in \({^{H}}\mathscr {G}\) 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 \(\underline{X_A}^{\underline{X_A}}\,\), which then becomes a group object in \({^{H}}\mathscr {G}\) called the automorphism group \(\mathrm {Aut}(X_A)\) of \(X_A\).
Let us apply the fully faithful functor \(\mathrm {Sh}({^{H}}\mathscr {S}) \rightarrow \mathrm {PSh}({^{H}}\mathscr {S})\) (which assigns to sheaves their underlying presheaves) on the monoid object \(( \underline{X_A}^{\underline{X_A}},\bullet , e)\) in \({^{H}}\mathscr {G}= \mathrm {Sh}({^{H}}\mathscr {S})\) to obtain the monoid object \(( \underline{X_A}^{\underline{X_A}},\bullet , e)\) in \(\mathrm {PSh}({^{H}}\mathscr {S})\), denoted with abuse of notation by the same symbol. This monoid object in \(\mathrm {PSh}({^{H}}\mathscr {S})\) may be equivalently regarded as a functor \({^{H}}\mathscr {S}^{\mathrm {op}} \rightarrow \mathsf {Monoid}\) with values in the category of ordinary \(\mathsf {Set}\)-valued monoids (i.e., monoid objects in the category \(\mathsf {Set}\)). The functor assigns to any object \(X_B\) in \({^{H}}\mathscr {S}\) the monoid
and to any \({^{H}}\mathscr {S}\)-morphism \(f{:}\,X_B\rightarrow X_C\) the monoid morphism
For any object \(X_B\) in \({^{H}}\mathscr {S}\), we define \(\mathrm {Aut}(X_A)(X_B)\) to be the subset of elements \(g\in \underline{X_A}^{\underline{X_A}}(X_B)\) for which there exists \(g^{-1}\in \underline{X_A}^{\underline{X_A}}(X_B)\) such that
Because the inverse of an element in a monoid (if it exists) is always unique, it follows that any element \(g\in \mathrm {Aut}(X_A)(X_B)\) has a unique inverse \(g^{-1}\in \mathrm {Aut}(X_A)(X_B)\), and that the inverse of \(g^{-1}\) is g. The monoid structure on \(\underline{X_A}^{\underline{X_A}}(X_B)\) induces to \(\mathrm {Aut}(X_A)(X_B)\), because the inverse of \(e_{X_B}\) is \(e_{X_B}\) itself and the inverse of \(g\bullet _{X_B} h\) is \((g\bullet _{X_B} h)^{-1} = h^{-1}\bullet _{X_B} g^{-1}\). Denoting by
the map that assigns the inverse, we obtain for any object \(X_B\) in \({^{H}}\mathscr {S}\) a group
The monoid morphism \(\underline{X_A}^{\underline{X_A}}(f)\) in (6.5) induces a group morphism which we denote by
Hence, we have constructed a functor \({^{H}}\mathscr {S}^{\mathrm {op}} \rightarrow \mathsf {Group}\) with values in the category of ordinary \(\mathsf {Set}\)-valued groups (i.e., group objects in \(\mathsf {Set}\)), which we can equivalently regard as a group object \((\mathrm {Aut}(X_A),\bullet ,e,\mathrm {inv})\) in the category \(\mathrm {PSh}({^{H}}\mathscr {S})\). Notice further that \(\mathrm {Aut}(X_A)\) is a subobject of \(\underline{X_A}^{\underline{X_A}}\) in the category \(\mathrm {PSh}({^{H}}\mathscr {S})\).
Proposition 6.1
For any object \(X_A\) in \({^{H}}\mathscr {S}\), the presheaf \(\mathrm {Aut}(X_A)\) satisfies the sheaf condition (5.2). In particular, \((\mathrm {Aut}(X_A),\bullet ,e,\mathrm {inv})\) is the subobject of invertibles of the monoid object \((\underline{X_A}^{\underline{X_A}},\bullet ,e)\) in \({^{H}}\mathscr {G}\) and hence a group object in \({^{H}}\mathscr {G}\).
Proof
Given any \({^{H}}\mathscr {S}\)-Zariski covering family \(\{f_i{:}\,X_{B[s_i^{-1}]} \rightarrow X_{B}\}\), we have to show that
is an equalizer in \(\mathsf {Set}\). Recalling that \(\mathrm {Aut}(X_A)\) is the subpresheaf of the sheaf \(\underline{X_A}^{\underline{X_A}}\) specified by the invertibility conditions (6.6), an element in \(\prod _{i}\, \mathrm {Aut}(X_A)(X_{B[s_i^{-1}]})\) can be represented by an element
such that each \(g_i\) has an inverse \(g_i^{-1}\in \underline{X_A}^{\underline{X_A}}\big (X_{B[s_i^{-1}]} \big )\) in the sense that
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 \(\underline{X_A}^{\underline{X_A}}\) is a sheaf, we can represent \(\prod _{i} \, g_i\) by the element \(g\in \underline{X_A}^{\underline{X_A}}(X_B)\) that is uniquely specified by \(\underline{X_A}^{\underline{X_A}}(f_i)(g) = g_i\), for all i.
We have to show that \(g\in \mathrm {Aut}(X_A)(X_B)\subseteq \underline{X_A}^{\underline{X_A}}(X_B)\), i.e., that there exists \(g^{-1}\in \underline{X_A}^{\underline{X_A}}(X_B)\) such that \(g\bullet _{X_B}g^{-1}= g^{-1}\bullet _{X_B} g = e_{X_B}\). Since \(\underline{X_A}^{\underline{X_A}}(f_{i;j})\) and \(\underline{X_A}^{\underline{X_A}}(f_{j;i})\) are monoid morphisms, both sides of the equality (6.12) are invertible and the inverse is given by
Using again the property that \(\underline{X_A}^{\underline{X_A}}\) is a sheaf, we can represent \(\prod _{i} \, g^{-1}_i\) by the element \(\tilde{g} \in \underline{X_A}^{\underline{X_A}}(X_B) \) that is uniquely specified by \(\underline{X_A}^{\underline{X_A}}(f_i)(\tilde{g}) = g^{-1}_i\), for all i. It is now easy to check that \(\tilde{g}\) is the inverse of g: Using once more the property that \(\underline{X_A}^{\underline{X_A}}(f_i)\) are monoid morphisms, we obtain \(\underline{X_A}^{\underline{X_A}}(f_i)(\tilde{g}\bullet _{X_B} g)= g_i^{-1}\bullet _{X_{B[s_i^{-1}]}} g_i = e_{X_{B[s_i^{-1}]}}\) and similarly \(\underline{X_A}^{\underline{X_A}}(f_i)(g\bullet _{X_B} \tilde{g})= e_{X_{B[s_i^{-1}]}}\), for all i. Because \(\underline{X_A}^{\underline{X_A}}\) is a sheaf, this implies \(\tilde{g}\bullet _{X_B} g = g\bullet _{X_B} \tilde{g} = e_{X_B}\) and hence that \(\tilde{g} = g^{-1}\). \(\square \)
7 Lie algebras of automorphism groups
The category \({^{H}}\mathscr {G}\) of generalized toric noncommutative spaces has a distinguished object \(\underline{K}:=\underline{X_{F_{{\varvec{0}}}}}\), where \(F_{{\varvec{0}}}\) is the free \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-algebra with one coinvariant generator x, i.e., \(x\mapsto \mathbbm {1}_H\otimes x\). We call \(\underline{K}\) the line object as it describes the toric noncommutative line. The line object \(\underline{K}\) is a ring object in \({^{H}}\mathscr {G}\): The sum \({^{H}}\mathscr {G}\)-morphism \(+{:}\,\underline{K}\times \underline{K}\rightarrow \underline{K}\) is induced (via going opposite and the Yoneda embedding) by the \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(F_{{\varvec{0}}} \rightarrow F_{{\varvec{0}}}\sqcup F_{{\varvec{0}}},~x\mapsto x\otimes \mathbbm {1} + \mathbbm {1}\otimes x\). The multiplication \({^{H}}\mathscr {G}\)-morphism \(\cdot {:}\,\underline{K}\times \underline{K}\rightarrow \underline{K}\) is induced by the \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(F_{{\varvec{0}}} \rightarrow F_{{\varvec{0}}}\sqcup F_{{\varvec{0}}},~x\mapsto x\otimes x\). The (additive) zero element is the \({^{H}}\mathscr {G}\)-morphism \(0{:}\,\{*\}\rightarrow \underline{K}\) induced by the \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(F_{{\varvec{0}}}\rightarrow \mathbb {K},~x\mapsto 0\in \mathbb {K}\), and the (multiplicative) unit element is the \({^{H}}\mathscr {G}\)-morphism \(1{:}\,\{*\}\rightarrow \underline{K}\) induced by the \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(F_{{\varvec{0}}}\rightarrow \mathbb {K},~x\mapsto 1\in \mathbb {K}\). Finally, the additive inverse \({^{H}}\mathscr {G}\)-morphism \( \mathrm {inv}_{+}{:}\,\underline{K}\rightarrow \underline{K}\) is induced by the \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(F_{{\varvec{0}}}\rightarrow F_{{\varvec{0}}},~x\mapsto -x\). It is straightforward, but slightly tedious, to confirm that these structures make \(\underline{K}\) into a ring object in \({^{H}}\mathscr {G}\).
Remark 7.1
Regarding the line object as a functor \(\underline{K}{:}\,{^{H}}\mathscr {S}^\mathrm {op}\rightarrow \mathsf {Set}\), it assigns to an object \(X_B\) in \({^{H}}\mathscr {S}\) the set
where \(B^{{\varvec{0}}} := \{b\in B{:}\,\rho ^B(b)=\mathbbm {1}_H\otimes b \}\) is the set of coinvariants; in the last step, we have used the fact that \(F_{{\varvec{0}}}\) is the free \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-algebra with one coinvariant generator; hence, \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphisms \(F_{{\varvec{0}}} \rightarrow B\) are in bijection with \(B^{{\varvec{0}}}\). To an \({^{H}}\mathscr {S}\)-morphism \(f{:}\,X_B\rightarrow X_{B^\prime }\), it assigns the restriction of \(f^*{:}\,B^\prime \rightarrow B\) to coinvariants, i.e.,
The \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-algebra structure on \(B,B^\prime \) induces a (commutative) ring structure on \(B^{{\varvec{0}}},{B^\prime }\,^{{\varvec{0}}}\) and \(f^*\) preserves this ring structure. Hence, we have obtained a functor \({^{H}}\mathscr {S}^{\mathrm {op}}\rightarrow \mathsf {CRing}\) with values in the category of commutative rings (in \(\mathsf {Set}\)), which is an equivalent way to describe the ring object structure on \(\underline{K}\) introduced above.
The \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(F_{{\varvec{0}}} \rightarrow F_{{\varvec{0}}}/(x^2)\) given by the quotient map induces a monomorphism \(\underline{D} := \underline{X_{F_{{\varvec{0}}}/(x^2)}} \rightarrow \underline{X_{F_{{\varvec{0}}}}}= \underline{K}\) in \({^{H}}\mathscr {G}\). The zero element, sum and additive inverse of \(\underline{K}\) induce to \(\underline{D}\), i.e., we obtain \({^{H}}\mathscr {G}\)-morphisms \(0{:}\,\{*\}\rightarrow \underline{D}\), \(+{:}\,\underline{D}\times \underline{D}\rightarrow \underline{D}\) and \(\mathrm {inv}_{+}{:}\,\underline{D}\rightarrow \underline{D}\) which give \(\underline{D}\) the structure of an Abelian group object in \({^{H}}\mathscr {G}\). Moreover, \(\underline{D}\) is a \(\underline{K}\)-module object in \({^{H}}\mathscr {G}\) with left \(\underline{K}\)-action \({^{H}}\mathscr {G}\)-morphism \(\cdot {:}\,\underline{K}\times \underline{D}\rightarrow \underline{D}\) induced by the \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(F_{{\varvec{0}}}/(x^2) \rightarrow F_{{\varvec{0}}}\sqcup F_{{\varvec{0}}}/(x^2),~ x\mapsto x\otimes x\). Heuristically, \(\underline{D}\) describes the infinitesimal neighborhood of 0 in \(\underline{K}\), i.e., \(\underline{D}\) is an infinitesimally short line, so short that functions on \(\underline{D}\) (which are described by \(F_{{\varvec{0}}}/(x^2)\)) are polynomials of degree 1.
Following the ideas of synthetic (differential) geometry [19, 21, 26], we may use \(\underline{D}\) to define the tangent bundle of a generalized toric noncommutative space.
Definition 7.2
Let Y be any object in \({^{H}}\mathscr {G}\). The (total space of) tangent bundle of Y is the exponential object
in \({^{H}}\mathscr {G}\). The projection \({^{H}}\mathscr {G}\)-morphism is
where we use the property that \(Y^{(-)}{:}\,{^{H}}\mathscr {G}^\mathrm {op}\rightarrow {^{H}}\mathscr {G}\) is a functor.
Remark 7.3
Focusing on the underlying functors \(Y{:}\,{^{H}}\mathscr {S}^\mathrm {op}\rightarrow \mathsf {Set}\) of objects Y in \({^{H}}\mathscr {G}\), there is an equivalent, and practically useful, characterization of the tangent bundle TY. By Yoneda’s Lemma and the fact that the exponential \((-)^{\underline{D}}{:}\,{^{H}}\mathscr {G}\rightarrow {^{H}}\mathscr {G}\) is the right adjoint of the product \(-\times \underline{D}{:}\,{^{H}}\mathscr {G}\rightarrow {^{H}}\mathscr {G}\), there are natural isomorphisms
Hence, the set \(TY(X_B)\) at stage \(X_B\) is simply given by \(Y(X_B\times D)\) at stage \(X_B\times D\). The components of the projection then read as
where \(\mathrm {id}_{X_B}\times 0{:}\,X_B\simeq X_B\times \{*\} \rightarrow X_B\times D\) is the product of the \({^{H}}\mathscr {S}\)-morphisms \(\mathrm {id}_{X_B}{:}\,X_B\rightarrow X_B\) and \(0:\{*\} \rightarrow D\).
We shall now study in more detail the tangent bundle
of the automorphism group of some object \(X_A\) in \({^{H}}\mathscr {S}\). We are particularly interested in the fiber \(T_e\mathrm {Aut}(X_A)\) of this bundle over the identity \(e{:}\,\{*\} \rightarrow \mathrm {Aut}(X_A) \), because it defines the Lie algebra of \(\mathrm {Aut}(X_A) \). The fiber \(T_e\mathrm {Aut}(X_A)\) is defined as the pullback
in \({^{H}}\mathscr {G}\). In particular, \(T_e\mathrm {Aut}(X_A)\) is an object in \({^{H}}\mathscr {G}\).
Using the perspective explained in Remark 7.3, we obtain
for all objects \(X_B\) in \({^{H}}\mathscr {S}\). The pullback (7.6) then introduces a further condition
for all objects \(X_B\) in \({^{H}}\mathscr {S}\). Using (6.6) and (5.16), it follows that any \(g\in \mathrm {Aut}(X_A)(X_B\times D)\) is an \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(A\rightarrow B\sqcup F_{{\varvec{0}}}/(x^2) \sqcup A\) satisfying the invertibility condition imposed in (6.6). For our purposes, it is more convenient to equivalently regard g as an \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(g{:}\,A\rightarrow F_{{\varvec{0}}}/(x^2)\sqcup B \sqcup A\) with target \( F_{{\varvec{0}}}/(x^2)\sqcup B \sqcup A\) instead of \(B\sqcup F_{{\varvec{0}}}/(x^2) \sqcup A\) (flipping \(F_{{\varvec{0}}}/(x^2)\) and B is the usual flip map because the left H-coaction on \(F_{{\varvec{0}}}/(x^2)\) is trivial). Because any element in \(F_{{\varvec{0}}}/(x^2)\) is of the form \(c_0 + c_1\,x\), for some \(c_0,c_1\in \mathbb {K}\), we obtain two \({^{H}}\mathscr {M}\)-morphisms \(g_0,g_1{:}\,A\rightarrow B\sqcup A\) which are characterized uniquely by
for all \(a\in A\). Since \(g{:}\,A\rightarrow F_{{\varvec{0}}}/(x^2)\sqcup B \sqcup A\) is an \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism, it follows that \(g_0{:}\,A\rightarrow B\sqcup A\) is an \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism and that \(g_1{:}\,A\rightarrow B\sqcup A\) satisfies the condition
for all \(a,a^\prime \in A\). From (7.8), it follows that \(g\in T_e\mathrm {Aut}(X_A)(X_B)\) if and only if \(g_0 = \iota _2{:}\,A\rightarrow B\sqcup A,~a\mapsto \mathbbm {1}_B\otimes a\) is the canonical inclusion for the coproduct. Then, (7.10) simplifies to
for all \(a,a^\prime \in A\). Notice that (7.11) is the Leibniz rule for the A-bimodule structure on \(B\sqcup A\) that is induced by the \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-algebra structure on \(B\sqcup A\) and the inclusion \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(\iota _2{:}\,A\rightarrow B\sqcup A\).
Lemma 7.4
Let \(g{:}\,A\rightarrow F_{{\varvec{0}}}/(x^2) \sqcup B \sqcup A\) be any \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism such that \(g_0 = \iota _2{:}\,A\rightarrow B\sqcup A\) in the notation of (7.9). Then, the \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(\tilde{g}{:}\, A\rightarrow F_{{\varvec{0}}}/(x^2)\sqcup B \sqcup A\) defined by \(\tilde{g}_0 = g_0 = \iota _2\) and \(\tilde{g}_1 = -g_1\) is the inverse of g in the sense that \(g\bullet _{X_B\times D} \tilde{g} = \tilde{g}\bullet _{X_B\times D} g = e_{X_B\times D}\).
Proof
From the hypothesis, (6.2) and \(x^2=0\), it follows that
for all \(a\in A\) and similarly that \(\tilde{g}\bullet _{X_B\times D} g = e_{X_B\times D}\). \(\square \)
This result allows us to give a very explicit characterization of the functor underlying the object \(T_e\mathrm {Aut}(X_A)\) in \({^{H}}\mathscr {G}\). Let us define the functor \({^{H}}\mathrm {Der}(A,-\sqcup A){:}\,{^{H}}\mathscr {S}^\mathrm {op}\rightarrow \mathsf {Set}\) on objects \(X_B\) by
which is the subset of \(v\in \mathrm {Hom}_{{^{H}}\mathscr {M}}(A,B\sqcup A)\) satisfying (7.11), and on morphisms \(f{:}\,X_B\rightarrow X_{B^\prime }\) by
Corollary 7.5
The presheaf underlying \(T_e\mathrm {Aut}(X_A)\) is isomorphic to \({^{H}}\mathrm {Der}(A,-\sqcup A)\) via the natural isomorphism with components
where \(g_1\) is defined according to (7.9). Hence, \({^{H}}\mathrm {Der}(A,-\sqcup A)\) is a sheaf, i.e., an object in \({^{H}}\mathscr {G}\), and \(T_e\mathrm {Aut}(X_A)\) is also isomorphic to \({^{H}}\mathrm {Der}(A,-\sqcup A)\) in \({^{H}}\mathscr {G}\).
Proof
Since g is uniquely specified by \(g_0,g_1\) [via (7.9)] and \(g_0 = \iota _2\) for all \(g\in T_e\mathrm {Aut}(X_A)(X_B)\), it follows that (7.14) is injective. Surjectivity of (7.14) follows from Lemma 7.4. Naturality of (7.14) is obvious. Because \(T_e\mathrm {Aut}(X_A)\) and \({^{H}}\mathrm {Der}(A,-\sqcup A)\) are isomorphic as presheaves and \(T_e\mathrm {Aut}(X_A)\) is a sheaf, it follows from the fully faithful embedding \(\mathrm {Sh}({^{H}}\mathscr {S})\rightarrow \mathrm {PSh}({^{H}}\mathscr {S})\) that \({^{H}}\mathrm {Der}(A,-\sqcup A)\) is a sheaf and that the isomorphism is in \({^{H}}\mathscr {G}\). \(\square \)
We conclude by showing that \({^{H}}\mathrm {Der}(A,-\sqcup A)\) [and hence by Corollary 7.5 also \(T_e\mathrm {Aut}(X_A)\)] is a \(\underline{K}\)-module object in \({^{H}}\mathscr {G}\) that can be equipped with a Lie bracket. From the perspective used in Remark 7.1, this is equivalent to equipping \({^{H}}\mathrm {Der}(A,B\sqcup A)\) with a \(B^{{\varvec{0}}}\)-module structure and a Lie bracket on this \(B^{{\varvec{0}}}\)-module, such that both structures are natural transformations for \({^{H}}\mathscr {S}\)-morphisms \(f{:}\,X_{B}\rightarrow X_{B^\prime }\). Recall that \({^{H}}\mathrm {Der}(A,B\sqcup A)\) is the subset of \(\mathrm {Hom}_{{^{H}}\mathscr {M}}(A,B\sqcup A)\) specified by the Leibniz rule (7.11). Because the Leibniz rule is a linear condition, it follows that \({^{H}}\mathrm {Der}(A,B\sqcup A)\) 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 \({^{H}}\mathscr {S}\)-morphisms \(f{:}\,X_{B}\rightarrow X_{B^\prime }\); hence, \({^{H}}\mathrm {Der}(A,-\sqcup A)\) is an Abelian group object in \({^{H}}\mathscr {G}\). The \(B^{{\varvec{0}}}\)-module structure
is defined by setting
for all \(a\in A\). In order to verify that \((b\cdot v)\in {^{H}}\mathrm {Der}(A,B\sqcup A)\), i.e., that it is H-equivariant and satisfies the Leibniz rule (7.11), it is essential to use the fact that b is coinvariant, \(\rho ^B{:}\,b\mapsto \mathbbm {1}_H\otimes b\). From (7.13), one immediately sees that this \(B^{{\varvec{0}}}\)-module structure is natural with respect to \({^{H}}\mathscr {S}\)-morphisms \(f{:}\,X_{B}\rightarrow X_{B^\prime }\), i.e.,
for all \(b^\prime \in {B^\prime }\,^{{\varvec{0}}}\) and \(v^\prime \in {^{H}}\mathrm {Der}(A,B^\prime \sqcup A)\). This endows \({^{H}}\mathrm {Der}(A,-\sqcup A)\) with the structure of a \(\underline{K}\)-module object in \({^{H}}\mathscr {G}\).
It remains to define a Lie bracket
on each \(B^{{\varvec{0}}}\)-module \({^{H}}\mathrm {Der}(A,B\sqcup A)\). Let us set
for all \(v,w\in {^{H}}\mathrm {Der}(A,B\sqcup A)\), where \(\mu _B{:}\,B\otimes B\rightarrow B\) is the product on B. Notice that \([v,w]_{X_B}{:}\,A\rightarrow B\sqcup A\) is an \({^{H}}\mathscr {M}\)-morphism. 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 \({^{H}}\mathrm {Der}(A,B\sqcup A)\). Antisymmetry of \([\,-,\,-\,]_{X_B}\) follows immediately from the definition, and the Jacobi identity is shown by direct computation. Moreover, \(B^{{\varvec{0}}}\)-linearity of the Lie bracket, i.e.,
for all \(b\in B^{{\varvec{0}}}\) and \(v,w\in {^{H}}\mathrm {Der}(A,B\sqcup A)\), can be easily verified by using the fact that b is coinvariant, and hence, it commutes with any other element in B [cf. (3.3)]. Naturality of the Lie bracket with respect to \({^{H}}\mathscr {S}\)-morphisms \(f{:}\,X_{B}\rightarrow X_{B^\prime }\) is a simple consequence of the fact that \(f^*{:}\,B^\prime \rightarrow B\) preserves the products entering the definition in (7.17). We have therefore obtained an explicit description of the Lie algebra of the automorphism group \(\mathrm {Aut}(X_A)\).
Proposition 7.6
The functor \({^{H}}\mathrm {Der}(A,-\sqcup A)\) equipped with the structure morphisms introduced above is a Lie algebra object in the category \(\mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\) of \(\underline{K}\)-module objects in \({^{H}}\mathscr {G}\).
8 Braided derivations
The Lie algebra object \({^{H}}\mathrm {Der}(A,-\sqcup A)\) constructed in Proposition 7.6 is (isomorphic to) the Lie algebra of the automorphism group \(\mathrm {Aut}(X_A)\). Hence, we may interpret it as the Lie algebra of infinitesimal automorphisms of the toric noncommutative space \(X_A\) with function \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-algebra A. Another (a priori unrelated) way to think about the infinitesimal automorphisms of \(X_A\) is to consider the Lie algebra \(\mathrm {der}(A)\) 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 \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-algebras A. Let us first consider the case where \(A= F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_{N}}\) is the free \({^{H}}\mathscr {A}\)-algebra with N generators \(x_i\) with left H-coaction \(x_i\mapsto t_{{\varvec{m}}_i}\otimes x_i\), for \(i=1,\ldots ,N\). Let \(\partial _j{:}\,F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_{N}} \rightarrow F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_{N}}\), for \(j=1,\ldots ,N\), be the linear map defined by
for all \(i=1,\ldots ,N\) and \(a,a^\prime \in F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_{N}} \). The map \(\partial _j\) should be interpreted as the ‘partial derivative’ along the generator \(x_j\); hence, it is natural to assign to it the left H-coaction \(\partial _j \mapsto t_{-{\varvec{m}}_j}\otimes \partial _j\). It satisfies a braided generalization of the Leibniz rule that is controlled by the cotriangular structure R. Let us define the left H-comodule
with the coproduct taken in \({^{H}}\mathscr {M}\), where for an object V in \({^{H}}\mathscr {M}\) we denote by \(V[{\varvec{m}}]\) the object in \({^{H}}\mathscr {M}\) which has the same underlying vector space as V but which is equipped with the shifted left H-coaction
We denote elements \(L \in \mathrm {der}(F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N})\) by \(L=\sum _j\, L_j\,\partial _j\), where \(L_j\in F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}\), because the H-coaction then takes the convenient form \(\rho ^{\mathrm {der}(F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N})}(L) =\sum _j\, {L_j}_{(-1)}\,t_{-{\varvec{m}}_j} \otimes {L_{j}}_{(0)} \,\partial _j\). The evaluation of \(\mathrm {der}(F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N})\) on \(F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}\) is given by the \({^{H}}\mathscr {M}\)-morphism
It is then easy to confirm the braided Leibniz rule
for all \(L\in \mathrm {der}(F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N})\) and \(a,a^\prime \in F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}\), which allows us to interpret elements of \(\mathrm {der}(F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N})\) as braided derivations.
For a finitely presented \({^{H}}\mathscr {A}\)-algebra \(A = F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}{/}(f_k)\), the left H-comodule of braided derivations is defined by
The evaluation \({^{H}}\mathscr {M}\)-morphism is similar to that in the case of free \({^{H}}\mathscr {A}\)-algebras and is given by
Notice that \(\mathrm {ev}\) 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 \({^{H}}\mathscr {A}\)-algebras.
Proposition 8.1
Let A be any object in \({^{H}}\mathscr {A}_{\mathrm {fp}}\). Then, \(\mathrm {der}(A)\) is a Lie algebra object in \({^{H}}\mathscr {M}\) with Lie bracket \({^{H}}\mathscr {M}\)-morphism \([-,\,-]{:}\,\mathrm {der}(A)\otimes \mathrm {der}(A) \rightarrow \mathrm {der}(A)\) uniquely defined by
for all \(L,L^\prime \in \mathrm {der}(A)\) and \(a\in A\).
Proof
Using the braided Leibniz rule (8.5), we can compute the right-hand side of (8.8) and obtain
Hence, \([L,L^\prime \,]\) is uniquely defined and it is a braided derivation. The braided antisymmetry and Jacobi identity on \([-,\,-]\) can be verified by a straightforward computation. \(\square \)
We would now like to compare the braided derivations \(\mathrm {der}(A)\) with the automorphism Lie algebra \({^{H}}\mathrm {Der}(A,-\sqcup A)\) constructed in Sect. 7. There is, however, a problem: While \(\mathrm {der}(A)\) is a Lie algebra object in the category \({^{H}}\mathscr {M}\) of left H-comodules, \({^{H}}\mathrm {Der}(A,-\sqcup A)\) is a Lie algebra object in the category \(\mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\) of \(\underline{K}\)-module objects in the category \({^{H}}\mathscr {G}\) of generalized toric noncommutative spaces. We show in “Appendix” that there exists a functor \(j{:}\,{^{H}}\mathscr {M}\rightarrow \mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\), which becomes a fully faithful embedding when restricted to the full subcategory \({^{H}}\mathscr {M}_{\mathrm {dec}}\) of decomposable left H-comodules (cf. Definition 9.1). Because \(\mathrm {der}(A)\) is a decomposable left H-comodule (cf. Corollary 9.4), we may use the fully faithful embedding \(j{:}\,{^{H}}\mathscr {M}_{\mathrm {dec}} \rightarrow \mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\) to relate \(\mathrm {der}(A)\) to \({^{H}}\mathrm {Der}(A,-\sqcup A)\).
Let us characterize more explicitly the object \(j(\mathrm {der}(A))\) in \(\mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\). Its underlying functor [cf. (9.7)] assigns to an object \(X_B\) in \({^{H}}\mathscr {S}\) the \(B^{\varvec{0}}\)-module
and to an \({^{H}}\mathscr {S}\)-morphism \(f{:}\,X_B\rightarrow X_C\) the module morphism
For any object \(X_B\) in \({^{H}}\mathscr {S}\), we define a map
by setting
for all \(a\in A\). It is easy to check that \(\xi _{X_B}(b\otimes L){:}\,A\rightarrow B\sqcup A\) is an \({^{H}}\mathscr {M}\)-morphism by using the property that \(b\otimes L\) is H-coinvariant. Moreover, \(\xi _{X_B}(b\otimes L)\) satisfies the Leibniz rule (7.11) because L satisfies the braided Leibniz rule (8.5) and \(b\otimes L\) is H-coinvariant. Explicitly we have
where the last step follows from (3.19) and (2.4). This shows that the image of \(\xi _{X_B}\) lies in \({^{H}}\mathrm {Der}(A,B\sqcup A)\), as we have asserted in (8.11).
The maps \(\xi _{X_B}\) are clearly \(B^{{\varvec{0}}}\)-module morphisms with respect to the \(B^{\varvec{0}}\)-module structure on \({^{H}}\mathrm {Der}(A,B\sqcup A)\) introduced in (7.15) and that on \(j(\mathrm {der}(A))(X_B)\) introduced in (9.10), and they are natural with respect to \({^{H}}\mathscr {S}\)-morphisms \(f{:}\,X_B\rightarrow X_C\). Hence, we have defined a morphism
in the category \(\mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\) of \(\underline{K}\)-module objects in \({^{H}}\mathscr {G}\). The main result of this section is
Theorem 8.2
The \(\mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\)-morphism (8.13) is an isomorphism. Hence, \(\mathrm {der}(A)\) and \({^{H}}\mathrm {Der}(A,-\sqcup A)\) are equivalent descriptions of the infinitesimal automorphisms of a toric noncommutative space \(X_A\).
Proof
Fix a presentation \(A = F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}/(f_k)\) of A and any object \(X_B\) in \({^{H}}\mathscr {S}\). We define nonequivariant linear maps \(\widehat{\partial }_j{:}\,F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N} \rightarrow B\otimes F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}\) by setting
for all generators \(x_i\) and all \(a,a^\prime \in F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}\). There is an isomorphism
given by the assignment \(v \mapsto \sum _j\, v_j\,\widehat{\partial }_j\). Because A and B are decomposable, we obtain a chain of isomorphisms
The resulting isomorphism preserves the conditions imposed in (8.15) and (8.6); hence, it induces an isomorphism between \({^{H}}\mathrm {Der}(A,B\sqcup A)\) and \(j(\mathrm {der}(A))(X_B)\). \(\square \)
Remark 8.3
Even though the functor \(j{:}\,{^{H}}\mathscr {M}\rightarrow \mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\) is not monoidal (cf. Remark 9.7), there exists a \(\mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\)-morphism
which is described explicitly in (9.13). We now confirm that the isomorphism \(\xi {:}\,j(\mathrm {der}(A)) \rightarrow {^{H}}\mathrm {Der}(A,-\sqcup A)\) established in Theorem 8.2 preserves the Lie brackets on \(\mathrm {der}(A)\) and \({^{H}}\mathrm {Der}(A,-\sqcup A)\) in the sense that the diagram
in \(\mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\) commutes. Fixing an arbitrary object \(X_B\in {^{H}}\mathscr {S}\) and going along the upper path of this diagram, we obtain
for all \(a\in A\), where in the last two steps we used the properties (2.4) of the cotriangular structure R and the fact that \(b\otimes L \in (B\otimes \mathrm {der}(A))^{{\varvec{0}}}\) is coinvariant. Going now along the lower path of the diagram, we obtain
for all \(a\in A\), 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 \(b\otimes L\in B^{{\varvec{m}}}\otimes \mathrm {der}(A)^{-{\varvec{m}}}\) and \(b^\prime \otimes L^\prime \in B^{{\varvec{m}}^\prime }\otimes \mathrm {der}(A)^{-{\varvec{m}}^\prime }\), the second term in (8.19b) can be rearranged as
for all \(a\in A\), where in the third step we used the property \(R(h\otimes g) = R(S(h)\otimes S(g))\), for all \(h,g\in H\), see, e.g., [23, Lemma 2.2.2].
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Acknowledgements
We thank Marco Benini, Giovanni Landi and Ryszard Nest for helpful comments on the material presented in this paper. This work was completed while R.J.S. was visiting the Centro de Matemática, Computação e Cognição of the Universidade de Federal do ABC in São Paulo, Brazil during June–July 2016, whom he warmly thanks for support and hospitality during his stay there. This work was supported by the Science and Technology Facilities Council [Grant number ST/L000334/1], the work of R.J.S. is supported in part by a Consolidated Grant from the UK STFC. This work was also supported in part by the Action MP1405 QSPACE from the European Cooperation in Science and Technology (COST). G.E.B. is a Commonwealth Scholar, funded by the UK government. The work of A.S. was supported by a Research Fellowship of the Deutsche Forschungsgemeinschaft (DFG, Germany). The work of R.J.S. was supported in part by the Visiting Researcher Program Grant 2016/04341-5 from the Fundação de Amparo á Pesquisa do Estado de São Paulo (FAPESP, Brazil).
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Appendix: Technical details for Sect. 8
Appendix: Technical details for Sect. 8
1.1 Decomposable objects in \({^{H}}\mathscr {M}\)
Given any object V in \({^{H}}\mathscr {M}\), we define
for all \({\varvec{m}}\in \mathbb {Z}^n\). Notice that \(V^{{\varvec{0}}}\) is the vector space of coinvariants and that \(V^{{\varvec{m}}} \subseteq V\) are \({^{H}}\mathscr {M}\)-subobjects, for all \({\varvec{m}}\).
Definition 9.1
An object V in \({^{H}}\mathscr {M}\) is decomposable if the canonical \({^{H}}\mathscr {M}\)-morphism
is an isomorphism. We denote by \({^{H}}\mathscr {M}_{\mathrm {dec}}\) the full subcategory of decomposables.
Lemma 9.2
(Properties of decomposables)
-
(a)
Tensor products of decomposables are decomposable, i.e., \({^{H}}\mathscr {M}_{\mathrm {dec}}\) is a monoidal subcategory of \({^{H}}\mathscr {M}\).
-
(b)
Coproducts of decomposables are decomposable.
-
(c)
\({^{H}}\mathscr {M}\)-subobjects of decomposables are decomposable.
Proof
To prove item (a), note that for V, W decomposable we have
hence, \(V\otimes W\) is decomposable with
The monoidal unit object \(\mathbb {K}_{{\varvec{0}}}\) is clearly decomposable. Items (b) and (c) are obvious. \(\square \)
Lemma 9.3
Let A be an object in \({^{H}}\mathscr {A}_{\mathrm {fp}}\). Then, the left H-comodule underlying A is decomposable.
Proof
Let us start with the case where \(A = F_{{\varvec{m_1}},\ldots ,{\varvec{m}}_N}\) is a free \({^{H}}\mathscr {A}\)-algebra. As \(F_{{\varvec{m_1}},\ldots ,{\varvec{m}}_N} \simeq F_{{\varvec{m}}_1} \sqcup \cdots \sqcup F_{{\varvec{m}}_M} \), where \(\sqcup \) denotes the coproduct in \({^{H}}\mathscr {A}_{\mathrm {fp}}\) (given explicitly by \(\otimes \)), we can use Lemma 9.2(a) and reduce the problem to show that \(F_{{\varvec{m}}}\) is decomposable. Notice that
where x denotes the generator of \(F_{{\varvec{m}}}\) with H-coaction \(x\mapsto t_{{\varvec{m}}}\otimes x\). The canonical \({^{H}}\mathscr {M}\)-morphism reads as
and it is easy to see that it is an isomorphism.
For the case where \(A = F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}{/}I\) is finitely presented, we use the property that \(F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}\) is decomposable and hence so is the \({^{H}}\mathscr {A}\)-ideal \(I\subseteq F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}\). Consequently, the quotient \(A = F_{{\varvec{m}}_1,\ldots ,{\varvec{m}}_N}{/}I\) is decomposable as well. \(\square \)
Corollary 9.4
Let A be an object in \({^{H}}\mathscr {A}_{\mathrm {fp}}\). Then, \(\mathrm {der}(A)\) is decomposable.
Proof
Recalling the definition of \(\mathrm {der}(A)\) in (8.6), the claim follows from the fact that A is decomposable (cf. Lemma 9.3), and Lemma 9.2(b) and (c). \(\square \)
1.2 Embedding of \({^{H}}\mathscr {M}\) into \({^{H}}\mathscr {G}\)
We first define a functor
To an object V in \({^{H}}\mathscr {M}\), the functor j assigns the presheaf \(j(V){:}\,{^{H}}\mathscr {S}^\mathrm {op}\rightarrow \mathsf {Set}\) that acts on objects \(X_B\) as
and on morphisms \(f{:}\,X_B\rightarrow X_C\) as
To a morphism \(L{:}\,V\rightarrow W\) in \({^{H}}\mathscr {M}\), the functor j assigns the presheaf morphism \(j(L){:}\,j(V) \rightarrow j(W)\) given by the natural transformation with components
Proposition 9.5
For any object V in \({^{H}}\mathscr {M}\), the presheaf j(V) is a sheaf. Hence, (9.7) induces a functor \(j{:}\,{^{H}}\mathscr {M}\rightarrow {^{H}}\mathscr {G}\).
Proof
Given any \({^{H}}\mathscr {S}\)-Zariski covering family \(\{f_i{:}\,X_{B[s_i^{-1}]} \rightarrow X_B^{}\}\), we have to verify the sheaf condition (5.2), i.e., that the diagram
is an equalizer in \(\mathsf {Set}\). This follows from the same argument that we have used in the second paragraph of the proof of Proposition 5.1. \(\square \)
For any object \(X_B\) in \({^{H}}\mathscr {S}\), the set \(j(V)(X_B) = (B\otimes V)^{{\varvec{0}}}\) is a \(B^{{\varvec{0}}}\)-module with Abelian group structure induced by the vector space structure of \(B\otimes V\) and \(B^{{\varvec{0}}}\)-action given by
These structures are natural with respect to \({^{H}}\mathscr {S}\)-morphisms \(f{:}\,X_{B}\rightarrow X_{C}\), i.e.,
for all \(c\in C^{{\varvec{0}}}\), \(c^\prime \in C\) and \(v\in V\); hence, they endow j(V) with the structure of a \(\underline{K}\)-module object in \({^{H}}\mathscr {G}\). For any \({^{H}}\mathscr {M}\)-morphism \(L{:}\,V\rightarrow W\) the \({^{H}}\mathscr {G}\)-morphism \(j(L){:}\,j(V) \rightarrow j(W)\) is compatible with this \(\underline{K}\)-module object structure, i.e., (9.8) is a \(B^{{\varvec{0}}}\)-module morphism, for all objects \(X_B\) in \({^{H}}\mathscr {S}\). We have therefore obtained
Proposition 9.6
With respect to the \(\underline{K}\)-module object structures on j(V) introduced above, \(j{:}\,{^{H}}\mathscr {M}\rightarrow \mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\) is a functor with values in the category \(\mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\) of \(\underline{K}\)-module objects in \({^{H}}\mathscr {G}\).
Remark 9.7
The functor j is not a monoidal functor, i.e., the object \(j(V\otimes W)\) is in general not isomorphic to \(j(V)\otimes j(W)\), where the tensor product in \(\mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\) is given by
for all objects \(X_B\) in \({^{H}}\mathscr {S}\). For example, take \(B=\mathbb {K}\) then \(j(V\otimes W)(X_{\mathbb {K}}) = (V\otimes W)^{{\varvec{0}}}\) but \((j(V)\otimes j(W))(X_{\mathbb {K}}) = V^{\varvec{0}}\otimes W^{{\varvec{0}}}\). However, there exists a \(\mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\)-morphism
for all objects V, W in \({^{H}}\mathscr {M}\). The components of \(\psi \) are given by
for all objects \(X_B\) in \({^{H}}\mathscr {S}\).
Let now V be decomposable, i.e., an object in \({^{H}}\mathscr {M}_{\mathrm {dec}}\). Because any object B in \({^{H}}\mathscr {A}_{\mathrm {fp}}\) is decomposable as well (cf. Lemma 9.3), we obtain
For the special case where \(B = F_{{\varvec{m}}}\) is the free \({^{H}}\mathscr {A}\)-algebra with one generator with coaction \(x\mapsto t_{{\varvec{m}}}\otimes x\), we use \(B\simeq \coprod _{k\in \mathbb {Z}_{\ge 0}} \, \mathbb {K}_{k\,{\varvec{m}}}\) to simplify this expression further to
where the coproducts here are in the category of vector spaces. Using this explicit characterization, we can establish the main result of “Appendix.”
Theorem 9.8
For any two objects V, W in \({^{H}}\mathscr {M}_{\mathrm {dec}}\) there is a bijection of \(\mathrm {Hom}\)-sets
Thus, the restricted functor \(j{:}\,{^{H}}\mathscr {M}_{\mathrm {dec}} \rightarrow \mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\) to the full subcategory of decomposables \({^{H}}\mathscr {M}_{\mathrm {dec}}\) is fully faithful.
Proof
Let \(\eta {:}\,j(V)\rightarrow j(W)\) be any morphism in \(\mathrm {Mod}_{\underline{K}}({^{H}}\mathscr {G})\). The components
are \(B^{{\varvec{0}}}\)-module morphisms, for all objects \(X_B\) in \({^{H}}\mathscr {S}\), such that for any \({^{H}}\mathscr {S}\)-morphism \(f{:}\,X_B\rightarrow X_C\) the diagram
commutes.
We first show that \(\eta \) is uniquely determined by the components \(\eta _{X_{F_{{\varvec{m}}}}}\), for all free \({^{H}}\mathscr {A}\)-algebras \(F_{{\varvec{m}}}\) with one generator. Using (9.14), we find that \(\eta _{X_B}\) is specified by its action on elements of the form \(b\otimes v \in B^{{\varvec{n}}}\otimes V^{-{\varvec{n}}}\), for all \({\varvec{n}}\). Given any such element, we define an \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(f^*{:}\,F_{{\varvec{n}}} \rightarrow B\) by sending \(x\mapsto b\). (Notice that the morphism \(f^*\) depends on the chosen element \(b\otimes v\).) Then, the commutative diagram (9.18) implies that \(\eta _{X_B}(b\otimes v) = (f^*\otimes \mathrm {id}_W)(\eta _{X_{F_{{\varvec{n}}}}}(x\otimes v))\); hence, the value of \(\eta _{X_B}\) at \(b\otimes v\) is fixed by \(\eta _{X_{F_{{\varvec{n}}}}}\). As \(b\otimes v\) was arbitrary, we find that \(\eta \) is uniquely determined by the components \(\{\eta _{X_{F_{{\varvec{m}}}}}{:}\,{\varvec{m}}\in \mathbb {Z}^n\}\).
In the next step, we show that the components \(\{\eta _{X_{F_{{\varvec{m}}}}}{:}\,{\varvec{m}}\in \mathbb {Z}^n\}\) are uniquely determined by an \({^{H}}\mathscr {M}\)-morphism \(L{:}\,V\rightarrow W\). Consider the \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(f^*{:}\,F_{{\varvec{m}}}\rightarrow F_{{\varvec{m}}}\) defined by \(x\mapsto c\,x\), where \(c\in \mathbb {K}\) 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 \(v\in V^{-k\,{\varvec{m}}}\) to \(c^k\,v \in \coprod _{k\in \mathbb {Z}_{\ge 0}}\, V^{-k\,{\varvec{m}}}\) (and similarly for \(w\in W^{-k\,{\varvec{m}}}\)), where the power in \(c^k\) depends on the term in the coproduct. Hence, by \({F_{{\varvec{m}}}}^{{\varvec{0}}}\)-linearity of \(\eta _{X_{F_{{\varvec{m}}}}}\) (which in particular implies \(\mathbb {K}\)-linearity), we find that \(\eta _{X_{F_{{\varvec{m}}}}}\) decomposes into \(\mathbb {K}\)-linear maps
It remains to show that \(L_{{\varvec{m}}, k} = L_{k\,{\varvec{m}},1}\), for all \({\varvec{m}}\in \mathbb {Z}^n\) and all \(k\in \mathbb {Z}_{\ge 0}\). Consider the \({^{H}}\mathscr {A}_{\mathrm {fp}}\)-morphism \(f^*{:}\,F_{k\,{\varvec{m}}}\rightarrow F_{{\varvec{m}}}\) defined by \(x\mapsto x^k\). The corresponding commutative diagram (9.18) then relates \(\eta _{X_{F_{k\,{\varvec{m}}}}} \) to \(\eta _{X_{F_{{\varvec{m}}}}} \), and we obtain the desired result \(L_{{\varvec{m}}, k} = L_{k\,{\varvec{m}},1}\). This defines a unique \({^{H}}\mathscr {M}\)-morphism
and hence by the assumption that V and W are decomposable also a unique \({^{H}}\mathscr {M}\)-morphism \(L{:}\,V\rightarrow W\). \(\square \)
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Barnes, G.E., Schenkel, A. & Szabo, R.J. Mapping spaces and automorphism groups of toric noncommutative spaces. Lett Math Phys 107, 1591–1628 (2017). https://doi.org/10.1007/s11005-017-0957-8
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DOI: https://doi.org/10.1007/s11005-017-0957-8