Generalized symmetry in noncommutative (complex) geometry

TL;DR

This paper develops a broad framework for noncommutative complex geometry using Hopf algebroid covariance, extending previous models and including examples like transverse complex and Kähler structures.

Contribution

It introduces Hopf algebroid covariance into noncommutative differential calculus, creating a more general geometric framework that encompasses prior approaches.

Findings

Developed a general framework for noncommutative complex geometry.

Presented examples including transverse complex and Kähler structures.

Connected the new framework with existing literature.

Abstract

We introduce Hopf algebroid covariance on Woronowicz's differential calculus. Using it, we develop quite a general framework of noncommutative complex geometry that subsumes the one in [2]. We present transverse complex and K\"ahler structures as examples and discuss several other examples. Relation with past literature is described.

Equations

$$\Gamma (P)\text{-equivariant structures on }N\equiv C_c^{\infty }(\Gamma (P))\text{-covariant structures on }{C}^{\infty }(N).$$

$$d(a\cdot \omega )=d({\epsilon }_l(a))\wedge \omega +a\cdot d(\omega )$$

$$(\Omega (N{)}^{\Gamma (P)},d)=(\Omega (N{)}_{C_c^{\infty }(\Gamma (P))},d)$$

$$H{\times }_{A}H:=\{\sum _i{h}_i{\otimes }_A{h}_i^{\prime }\in H{\otimes }_{A}H\mid \sum _i{h}_{i}t(a)\otimes h_i^{\prime }=\sum _i{h}_i\otimes h_i^{\prime }s(a)\forall a\in A\}$$

$$a_1\cdot h\cdot a_2:=s(a_1)t(a_2)h,a_1,a_2\in A,h\in H.$$

$$(\Delta {\otimes }_A\mathrm{id})\Delta =(\mathrm{id}{\otimes }_A\Delta )\Delta ,(\mathrm{id}{\otimes }_A\epsilon )\Delta =(\epsilon {\otimes }_A\mathrm{id})\Delta =\mathrm{id}.$$

$${\epsilon }_l(h{h}^{\prime })={\epsilon }_l(h{s}_l({\epsilon }_l{h}^{\prime }))={\epsilon }_l(h{t}_l({\epsilon }_l{h}^{\prime }))h,h^{\prime }\in H_l.$$

$${\epsilon }_l^2(h)={\epsilon }_l^2(s_l{\epsilon }_l^1(h_1)h_2)={\epsilon }_l^1(h_1){\epsilon }_l^2(h_2)={\epsilon }_l^1(t_l{\epsilon }_l^2(h_2)h_1)={\epsilon }_l^1(h).$$

$$a_1\cdot h\cdot a_2=ht(a_1)s(a_2),a_1,a_2\in Ah\in H.$$

$$H{\times }^{A}H:=\{\sum _i{h}_i{\otimes }_A{h}_i^{\prime }\in H{\otimes }_{A}H\mid \sum _{i}s(a)h_i\otimes h_i^{\prime }=\sum _i{h}_i\otimes t(a)h_i^{\prime }\forall a\in A\}.$$

$${\epsilon }_r(h{h}^{\prime })={\epsilon }_r(s_r({\epsilon }_{r}h)h^{\prime })={\epsilon }_r(t_r({\epsilon }_{r}h)h^{\prime })h,h^{\prime }\in H_r.$$

$$s_l{\epsilon }_l{t}_r=t_r,t_l{\epsilon }_l{s}_r=s_r,s_r{\epsilon }_r{t}_l=t_l,t_r{\epsilon }_r{s}_l=s_l;$$

$$({\Delta }_l\otimes {\mathrm{id}}_H){\Delta }_r=({\mathrm{id}}_H\otimes {\Delta }_r){\Delta }_l,({\Delta }_r\otimes {\mathrm{id}}_H){\Delta }_l=({\mathrm{id}}_H\otimes {\Delta }_l){\Delta }_r;$$

$$S(t_l(a_1)h{t}_r(a_2))=s_r(a_2)S(h)s_l(a_1);$$

$${\mu }_H(S\otimes {\mathrm{id}}_H){\Delta }_l=s_r{\epsilon }_r,{\mu }_H({\mathrm{id}}_H\otimes S){\Delta }_r=s_l{\epsilon }_l.$$

$$\begin{array}{r}\varphi :={\epsilon }_r{s}_l:A_l^{op}\to A_r,{\varphi }^{-1}:={\epsilon }_l{t}_r:A_r\to A_l^{op},\\ \theta :={\epsilon }_r{t}_l:A_l\to A_r^{op},{\theta }^{-1}:={\epsilon }_l{s}_r:A_r^{op}\to A_l.\end{array}$$

$$\mathrm{flip}(S\otimes S){\Delta }_l={\Delta }_{r}S,\mathrm{flip}(S\otimes S){\Delta }_r={\Delta }_{l}S,$$

$$\begin{array}{cccc}{s}_r{\epsilon }_r{s}_l=S{s}_l,& s_l{\epsilon }_l{s}_r=S{s}_r,& s_r{\epsilon }_r{t}_l=S^{-1}{s}_l,& s_l{\epsilon }_l{t}_r=S^{-1}{s}_r,\\ t_r{\epsilon }_r{s}_l=S{t}_l,& t_l{\epsilon }_l{s}_r=S{t}_r,& t_r{\epsilon }_r{t}_l=S^{-1}{t}_l,& t_l{\epsilon }_l{t}_r=S^{-1}{t}_r,\\ {\epsilon }_r{s}_l{\epsilon }_l={\epsilon }_{r}S,& {\epsilon }_l{s}_r{\epsilon }_r={\epsilon }_{l}S,& {\epsilon }_r{t}_l{\epsilon }_l={\epsilon }_r{S}^{-1},& {\epsilon }_l{t}_r{\epsilon }_r={\epsilon }_l{S}^{-1},\end{array}$$

$$\begin{array}{cc}{\mu }_H(S\otimes s_l{\epsilon }_l){\Delta }_l=S,& {\mu }_H(s_r{\epsilon }_r\otimes S){\Delta }_r=S,\\ {\mu }_{H^{op}}({\mathrm{id}}_H\otimes S^{-1}){\Delta }_l=t_r{\epsilon }_r,& {\mu }_{H^{op}}(S^{-1}\otimes {\mathrm{id}}_H){\Delta }_r=t_l{\epsilon }_l,\\ {\mu }_{H^{op}}(t_l{\epsilon }_l\otimes S^{-1}){\Delta }_l=S^{-1},& {\mu }_{H^{op}}(S^{-1}\otimes t_r{\epsilon }_r){\Delta }_r=S^{-1}.\end{array}$$

$$S_2(h)=s_r{\epsilon }_r(h^{(1)})S_2(h^{(2)})$$

$$=S_1(h_{(1)})h_{(2)}^{(1)}{S}_2(h_{(2)}^{(2)})=S_1(h_{(1)})s_l{\epsilon }_l(h_{(2)})=S_1(h).$$

$$G_2$$

$$\Gamma (P)(x,y)=\{ger{m}_{x}g\mid g\in P,x\in dom(g),g(x)=y\}.$$

$$({\varphi }_{\ast }s)(y)=\sum _{\varphi (x)=y}\alpha (s(x)),$$

$$\begin{array}{r}{\Omega }_{s,t}:C_c^{\infty }(G){\otimes }_{C^{\infty }(G_0)}^{rl}{C}_c^{\infty }(G)\to C_c^{\infty }(G{}^s{\times }_{G_0}^{t}G)=C_c^{\infty }(G_2)\\ {\Omega }_{t,t}:C_c^{\infty }(G){\otimes }_{C^{\infty }(G_0)}^{ll}{C}_c^{\infty }(G)\to C_c^{\infty }(G{}^t{\times }_{G_0}^{t}G)=C_c^{\infty }(G_2)\\ {\Omega }_{s,s}:C_c^{\infty }(G){\otimes }_{C^{\infty }(G_0)}^{rr}{C}_c^{\infty }(G)\to C_c^{\infty }(G{}^s{\times }_{G_0}^{s}G)=C_c^{\infty }(G_2)\\ {\Omega }_{t,s}:C_c^{\infty }(G){\otimes }_{C^{\infty }(G_0)}^{lr}{C}_c^{\infty }(G)\to C_c^{\infty }(G{}^t{\times }_{G_0}^{s}G)=C_c^{\infty }(G_2)\end{array}$$

$${\Omega }_{-.-}(u{\otimes }_{C^{\infty }(G_0)}^{--}{u}^{\prime })(g,g^{\prime })=u(g)u(g^{\prime }),$$

$$\ast :C_c^{\infty }(G){\otimes }_{C^{\infty }(G_0)}^{rl}{C}_c^{\infty }(G)\stackrel{{\Omega }_{s,t}}{\to }{C}_c^{\infty }(G_2)\stackrel{m_{\ast }}{\to }{C}_c^{\infty }(G).$$

$$(u\ast v)(g):=\ast (u\otimes v)=(m_{\ast }{\Omega }^{s,t}(u\otimes v))(g)=\sum _{g=g_1{g}_2}u(g_1)u(g_2),$$

$$(f\ast u)(g)=f(t(g))u(g)\text{and}(u\ast f)(g)=u(g)f(s(g)).$$

$$s_l\equiv s_r\equiv t_l\equiv t_r\equiv 1_{\ast }:C^{\infty }(G_0)\to C_c^{\infty }(G),$$

$$s_l:f\mapsto \overline{f},\text{where}\overline{f}(g)=\{\begin{array}{ll}f(x)& \text{if }g=1_x\text{ for some }x\in G_0\\ 0& \text{otherwise.}\end{array}$$

Full text

1. Introduction

Symmetry plays an important, often decisive, role in almost all areas of mathematics; especially in geometry and topology. Classically, symmetry means group action on spaces. However, one has to pass from groups to groupoids to capture local symmetry in an efficient way. For example, it is more natural to consider the (Lie)groupoid of isometries of a Riemannian manifold which is not globally symmetric or homogeneous. The natural domain for the characteristic classes of certain geometric structures are in fact the cohomology of the classifying spaces of (Lie)groupoids. One can go further, saying that groupoids provide a concept of generalized symmetry that is essential, as exemplified above and spectacularly apparent in the theory of foliations. In the realm of noncommutative geometry, symmetry is captured by action or coaction of Hopf algebras on (co)algebras, which is the noncommutative version of a space.

The concept of Hopf algebroids generalizes that of groupoids, providing a way of considering generalized symmetry in noncommutative geometry. They can be thought of as Hopf algebras over noncommutative bases. Initially conceived by algebraic topologists, Hopf algebroids over commutative bases have been used extensively in geometry and topology. Problems appear when one considers generalizing the definition to noncommutative bases. The pre-Hopf algebroid level, i.e., the definition of a bialgebroid is usually accepted as the correct generalization of a bialgebra over a noncommutative base. The problem is with the addition of an antipode. A description of the various definitions is given in the introduction of [MR2817646]. The definition used in this article also comes from that paper which was first given in [MR2043373].

The theory of noncommutative complex geometry was initiated in \citelist[MR3428362][MR3720811], although there are precursors, see \citelist[MR3073899][MR1614993][MR2773332][MR1977884]. It attempts to provide a fresh insight into various aspects of noncommutative geometry, such as the construction of spectral triples for quantum groups, by considering “complex structures”. It also promises a fruitful interaction between noncommutative geometry and noncommutative projective algebraic geometry. Identifying “differential forms” as the basic objects of study, the framework of noncommutative complex geometry is developed in the setting of Woronowicz’s differential calculus, see [MR994499]. The classical complex geometry being the obvious example, the set-up in [MR3720811] takes as its motivating example the family of quantum flag manifolds. One can go, as shown in there, as far as proving a version of Hard Lefschetz theorem.

Singular spaces, such as the leaf space of a foliation, have been studied extensively in classical geometry as well as noncommutative geometry. These spaces provided the main impetus for the development of noncommutative geometry, see [MR679730]. Classically, “transverse geometry” attempts to study such singular spaces using symmetry, which most of the time turns out to be a pseudogroup. This was exemplified in the beautiful paper [MR614370]. It led to the systematic study of spaces with pseudogroup symmetry. It is natural to ask whether one can do complex geometry over such spaces. That one can, was done in a volume of works, \citelist[MR1151583][MR1042454][MR1042454], to name a few.

Now, pseudogroups and groupoids are very much noncommutative in their nature. This led to Connes’ construction of the highly noncommutative groupoid $C^{*}$-algebra of the holonomy groupoid of a foliation, which is successfully applied to the questions in index theory. However, the fact that groupoids consist of symmetries is lost in this construction. To take the symmetry into account, one is naturally led to the language of Hopf algebroids, as shown in \citelist[MR2853081][MR1722796][MR2311185].

Thus, the study of complex geometry over such singular spaces consists of studying regular spaces with highly noncommutative symmetry, which are also generalized, in that they are not Hopf algebras. Let us mention a few of our results to exemplify what we have said above. For the convenience of the reader, let us recall the definition of a foliation first.

Definition**.**

A foliation ${\mathcal{F}}$ on $M$ is given by a cocycle ${\mathcal{U}}=\{U_{i},f_{i},g_{ij}\}$ modeled on a manifold $N_{0}$ (${\mathbb{R}}^{n}$ or ${\mathbb{C}}^{n}$), i.e.,

1. i)

$\{U_{i}\}$* is an open covering of $M$;* 2. ii)

$f_{i}:U_{i}\rightarrow N_{0}$* are submersions with connected fibers defining ${\mathcal{F}}$;* 3. iii)

$g_{ij}$* are local diffeomorphisms of $N_{0}$ and $g_{ij}f_{j}=f_{i}$ on $U_{i}\cap U_{j}$.*

The manifold $N=\sqcup f_{i}(U_{i})$ is called the transverse manifold of ${\mathcal{F}}$ associated to the cocycle ${\mathcal{U}}$, and the pseudogroup $P$ generated by $g_{ij}$ is called the holonomy pseudogroup on the transverse manifold. To such a pseudogroup $P$, one can associate an étale (effective) groupoid $\Gamma(P)$ over $N$, called the holonomy groupoid. Then a “geometric structure” on the foliated manifold $(M,{\mathcal{F}})$ is a “structure” on the manifold $N$ equivariant under the action of the groupoid $\Gamma(P)$. Now, it is known that (see Subsection 2.2) the space $C_{c}^{\infty}(\Gamma(P))$ of compactly supported smooth functions on $\Gamma(P)$ is a Hopf algebroid in the sense of Definition 2.1.9. We need a conversion such as

[TABLE]

The first candidate of such a structure is the differential forms. The translation in this case is more or less routine but what is hard is to identify the property that relates the groupoid action and the exterior derivative. One can see the difficulty from the fact that groupoids generalize both spaces and groups. In the case of spaces, the derivative satisfies a Leibniz rule and in the group case, it commutes with the action. Our first nontrivial result is the translation of this property to the Hopf algebroid language, more precisely,

Lemma**.**

The differential $d$ on $N$ satisfies

[TABLE]

*for $a\in C_{c}^{\infty}(\Gamma(P))$ and $\omega\in\Omega(N)$. *

Denote by $\Omega(N)^{\Gamma(P)}$ the $\Gamma(P)$-invariant forms. Then forms on the “orbit or leaf space” are captured as follows.

Proposition**.**

The pair $(\Omega(N),\,d)$ is a $C_{c}^{\infty}(\Gamma(P))$-covariant differential calculus, and we have

[TABLE]

as differential graded algebras, where the right-hand side denotes the invariant subalgebra under the action of the Hopf algebroid $C_{c}^{\infty}(\Gamma(P))$.

The framework in which to talk about $C_{c}^{\infty}(\Gamma(P))$-covariant differential calculus and such statements is our principal achievement in this paper. In fact, the goal of the present article is to introduce Hopf algebroid symmetry in noncommutative geometry and build such a framework. We formulate and study a quite general framework of Hopf algebroid-covariance of noncommutative complex and Kähler structures. We have been able to accommodate all the existing examples in our framework. Another notable and novel aspect of our work is a new definition of Hopf algebroid action or covariance on differential calculus which seems to work in a very general context. We present the Connes-Moscovici Hopf algebroid as one of the most interesting examples of our setup. Let us briefly describe the plan of the paper. In Section 2, we recall the definition of a Hopf algebroid. Foliations and étale groupoids in general, are also discussed in some detail and is shown to provide examples. In the last subsection, we recall the already-existing Connes-Moscovici left bialgebroid and we prove that it is a Hopf algebroid by putting a right bialgebroid structure. Section 3 introduces modules and module-algebras over a Hopf algebroid. In the final subsection of this section, we put $\ast$-structures on Hopf algebroids which is essential in order to view them as symmetry objects in noncommutative geometry. In Section 4, our framework and applications to noncommutative Kähler structures are described. Hopf algebroid covariance is introduced. The necessary modifications of the framework in [MR3720811] are described and along the way, examples coming from foliations are provided. Finally, Section 5 discusses some future directions. We could prove a version of the Hodge decomposition theorem and the formality theorem, which generalizes the case for foliations but we decided to omit these two from the present article, owing to its length.

Before ending this introduction, we would like to mention that it is not our intention to bring in Hopf algebroids and investigate such a general situation just for generality’s sake. Even in the classical case, one cannot always avoid having symmetries encoded in a more general object than a group. One needs a framework to work with these situations, and the theory of foliations amply exemplify such a need. We are after such a framework where one can take the naturally-occurring Hopf algebroid symmetry into account. If one needs further justification of where indeed these objects occur, we refer to subfactor theory. The knowledgeable reader might immediately recall \citelist[MR1745634][MR1800792][MR1913440] where Hopf algebroids appear in the form of weak Hopf algebras as the “Galois group of automorphisms.” If one forgets the analysis, there is no way one can escape Hopf algebroids as is wonderfully explained in [MR2004729]. Further pointers are given at the end of this article (see 5.3).

2. Preliminaries

2.1. Hopf algebras over noncommutative base - Hopf algebroids

We recall the definition of Hopf algebroids from [MR2817646]. See also [MR2553659, MR2043373]. We begin by defining a generalization of bialgebras.

Definition 2.1.1**.**

Let $A$ be a ${\mathbb{C}}$-algebra. An $(s,t)$-ring over $A$ is a ${\mathbb{C}}$-algebra $H$ with homomorphisms $s:A\rightarrow H$ and $t:A^{op}\rightarrow H$ whose images commute in $H$.

The functions $s$ and $t$ are referred to as the source and target maps respectively. An $(s,t)$-ring structure is equivalent to the structure of an $A^{e}$-algebra on $H$.

Definition 2.1.2**.**

Let $H$ be an $(s,t)$-ring over $A$. The Takeuchi product is the subspace

[TABLE]

of $H\otimes_{A}H$, where the tensor product $\otimes_{A}$ is defined with respect to the following $(A,A)$-bimodule structure on $H$:

[TABLE]

This Takeuchi product becomes a unital algebra with factorwise multiplication as well as an $(s,t)$-ring. Before we go onto the definition of a bialgebroid, let us recall the definition of an $A$-coalgebra.

Definition 2.1.3**.**

Let $A$ be a ${\mathbb{C}}$-algebra. A coalgebra over $A$ is a triple $(C,\Delta,\varepsilon)$ with $C$ an $(A,A)$-bimodule, $\Delta$ an $(A,A)$-bimodule morphism called the comultiplication, $\varepsilon:C\rightarrow A$ an $(A,A)$-bimodule morphism called the counit, and such that

[TABLE]

A left bialgebroid over $A$ is then an algebra with a compatible coalgebra structure over $A$. More precisely,

Definition 2.1.4**.**

Let $A_{l}$ be a ${\mathbb{C}}$-algebra. A left bialgebroid over $A_{l}$ is an $(s_{l},t_{l})$-ring $H_{l}$ equipped with the structure of an $A_{l}$-coalgebra $(\Delta_{l},\varepsilon_{l})$ with respect to the $(A_{l},A_{l})$-bimodule structure (2.1), subject to the following conditions:

1. i)

the (left) coproduct $\Delta_{l}:H_{l}\rightarrow H_{l}\otimes_{A_{l}}H_{l}$ maps into the subset $H_{l}\times_{A_{l}}H_{l}$ and defines a morphism $\Delta_{l}:H_{l}\rightarrow H_{l}\times_{A_{l}}H_{l}$ of unital ${\mathbb{C}}$-algebras; 2. ii)

the (left) counit has the property:

[TABLE]

We denote the above left bialgebroid by $(H_{l},A_{l},s_{l},t_{l},\Delta_{l},\varepsilon_{l})$ or simply by $H_{l}$.

Remark 2.1.5**.**

From (2.3) above and the fact that $\varepsilon_{l}$ is an $(A_{l},A_{l})$-bimodule morphism, it follows that $\varepsilon_{l}(s_{l}(a)h)=a\varepsilon_{l}(h)$, $\varepsilon_{l}(t_{l}(a)h)=\varepsilon_{l}(h)a$, and it also follows that $\varepsilon_{l}(1_{H_{l}})=1_{A_{l}}$. So we have that $\varepsilon_{l}s_{l}=\varepsilon_{l}t_{l}=\operatorname{id}_{A_{l}}$.

Lemma 2.1.6**.**

In a left bialgebroid, the left counit is unique.

Proof.

Indeed, if both $\varepsilon_{l}^{1}$ and $\varepsilon_{l}^{2}$ make $(H_{l},A_{l},s_{l},t_{l},\Delta_{l},\varepsilon_{l}^{1})$ and $(H_{l},A_{l},s_{l},t_{l},\Delta_{l},\varepsilon_{l}^{2})$ left bialgebroids, then we have:

[TABLE]

∎

Given an $(s,t)$-ring $H$, there is another $(A,A)$-bimodule structure on $H$:

[TABLE]

With respect to this bimodule structure, the tensor product $\otimes_{A}$ is defined. Inside $H\otimes_{A}H$, there is the Takeuchi product:

[TABLE]

This again becomes a unital algebra with factorwise multiplication and also is an $(s,t)$-ring.

Definition 2.1.7**.**

Let $A_{r}$ be a ${\mathbb{C}}$-algebra. A right bialgebroid over $A_{r}$ is an $(s_{r},t_{r})$-ring $H_{r}$ equipped with the structure of an $A_{r}$-coalgebra $(\Delta_{r},\varepsilon_{r})$ with respect to the $(A_{r},A_{r})$-bimodule structure (2.4), subject to the following conditions:

1. i)

the (right) coproduct $\Delta_{r}:H_{r}\rightarrow H_{r}\otimes_{A_{r}}H_{r}$ maps into $H_{r}\times^{A_{r}}H_{r}$ and defines a morphism $\Delta_{r}:H_{r}\rightarrow H_{r}\times^{A_{r}}H_{r}$ of unital ${\mathbb{C}}$-algebras; 2. ii)

the (right) counit has the property:

[TABLE]

We denote a right bialgebroid by $(H_{r},A_{r},s_{r},t_{r},\Delta_{r},\varepsilon_{r})$ or simply by $H_{r}$. Note that if $(H_{l},A_{l},s_{l},t_{l},$ $\Delta_{l},\varepsilon_{l})$ is a left bialgebroid, then $(H_{l}^{op},A_{l},t_{l},s_{l},\Delta_{l},\varepsilon_{l})$ is a right bialgebroid.

Remark 2.1.8**.**

As in Remark 2.1.5, we have $\varepsilon_{r}s_{r}=\varepsilon_{r}t_{r}=\operatorname{id}_{A_{r}}$. Also as above, the right counit is unique.

Sweedler notation

We shall use Sweedler notation with subscripts $\Delta_{l}(h)=h_{(1)}\otimes h_{(2)}$ for left comultiplication while the right comultiplication are indicated by superscripts: $\Delta_{r}(h)=h^{(1)}\otimes h^{(2)}.$

We now define a Hopf algebroid as an algebra endowed with a left and a right bialgebroid structure together with an antipode “intertwining” the left bialgebroid and the right bialgebroid structures. More precisely:

Definition 2.1.9**.**

A Hopf algebroid is given by a triple $(H_{l},H_{r},S)$, where $H_{l}=(H_{l},A_{l},s_{l},t_{l},\Delta_{l},\varepsilon_{l})$ is a left $A_{l}$-bialgebroid and $H_{r}=(H_{r},A_{r},s_{r},t_{r},\Delta_{r},\varepsilon_{r})$ is a right $A_{r}$-bialgebroid on the same ${\mathbb{C}}$-algebra $H$, and $S:H\rightarrow H$ is invertible ${\mathbb{C}}$-linear. These structures are subject to the following four conditions:

1. i)

the images of $s_{l}$ and $t_{r}$ as well as those of $t_{l}$ and $s_{r}$, coincide:

[TABLE] 2. ii)

mixed coassociativity holds:

[TABLE] 3. iii)

for all $a_{1}\in A_{l}$, $a_{2}\in A_{r}$ and $h\in H$, we have

[TABLE] 4. iv)

the antipode axioms hold:

[TABLE]

We apply $\varepsilon_{r}$ to the first two and $\varepsilon_{l}$ to the second pair of identities in (2.6) and get that $A_{l}$ and $A_{r}$ are anti-isomorphic as ${\mathbb{C}}$-algebras:

[TABLE]

The antipode is anti-algebra and anti-coalgebra morphism (between different coalgebras) and satisfies the equations

[TABLE]

where $\operatorname{flip}:H\otimes_{{\mathbb{C}}}H\rightarrow H\otimes_{{\mathbb{C}}}H$ is the flip permuting two factors of the tensor product (this becomes an $(A_{l},A_{l})$-respectively $(A_{r},A_{r})$-bimodule). Similar formulas hold for the inverse $S^{-1}$. The following identities will be used:

[TABLE]

and

[TABLE]

Lemma 2.1.10**.**

In a Hopf algebroid, the antipode is unique.

Proof.

Indeed, if both $S_{1}$ and $S_{2}$ make $(H_{l},H_{r},S_{1})$ and $(H_{l},H_{r},S_{2})$ Hopf algebroids then we have

[TABLE]

∎

Finally, note that if $(H_{l},H_{r},S)$ is a Hopf algebroid, then $(H_{r}^{op},H_{l}^{op},S^{-1})$ is also a Hopf algebroid.

2.2. The main example - Étale groupoids

We now introduce our main example besides Hopf algebras. A Hopf algebra is a Hopf algebroid with $A_{l}=A_{r}={\mathbb{C}}$. We follow [MR2012261]. See also [MR1303779, MR3448330, MR2853081].

Definition 2.2.1**.**

A groupoid $G$ is a small category in which each arrow is invertible. More explicitly, a groupoid consists of a space of objects $G_{0}$, a space of arrows $G_{1}$ (often denoted by $G$ itself) and five structure maps relating the two:

1. i)

source and target maps $s,t:G_{1}\rightarrow G_{0}$, assigning to each arrow $g$ its source $s(g)$ and target $t(g)$; one says that $g$ is from $s(g)$ to $t(g)$; 2. ii)

a partially defined composition of arrows, that is, only for those arrows $g,h$ for which source and target match, that is $s(g)=t(h)$; in other words, a map $m:G_{2}:=G_{1}~{}^{s}\times^{t}_{G_{0}}G_{1}\rightarrow G_{1}$, $(g,h)\mapsto gh$ that is associative whenever defined, producing the composite arrow going from $s(gh)=s(h)$ to $t(gh)=t(g)$; 3. iii)

a unit map $1:G_{0}\rightarrow G_{1}$, $x\mapsto 1_{x}$, that has the property $1_{t(g)}g=g1_{s(g)}=g$; 4. iv)

an inversion $inv:G_{1}\rightarrow G_{1}$, $g\mapsto g^{-1}$ that produces the inverse arrow going from $s(g^{-1})=t(g)$ to $t(g^{-1})=s(g)$, fulfilling $g^{-1}g=1_{s(g)}$, $gg^{-1}=1_{t(g)}$.

These maps can be assembled into a diagram

[TABLE]

An arrow may be denoted by $x\xrightarrow{g}y$ to indicate that $y=s(g)$ and $x=t(g)$.

A topological groupoid is a groupoid in which both $G_{1}$ and $G_{0}$ are topological spaces and all the structure maps are continuous. Similarly one defines smooth groupoids, where in addition $s$ and $t$ are required to be surjective submersions in order to ensure that $G_{2}=G_{1}~{}^{s}\times^{t}_{G_{0}}G_{1}$ remains a manifold. A topological (or smooth) groupoid is called étale if the source map is a local homeomorphism (or local diffeomorphism); this condition implies that all structures maps are local homeomorphisms (or local diffeomorphisms, respectively). In the smooth case, this equivalently amounts to saying that $\dim G_{1}=\dim G_{0}$. In particular, an étale groupoid has zero-dimensional source and target fibers, and hence they are discrete. We shall only be dealing with smooth étale groupoids.

We give some examples of étale groupoids below.

Example 2.2.2**.**

**

1. i)

The unit groupoid has a single manifold $M$ as both its object and arrow space. All the maps are identity functions. 2. ii)

A (discrete) group is a one-object groupoid (called the point groupoid). 3. iii)

The translation groupoid $\Gamma\ltimes M$ of a smooth left action of a discrete group has as object space $M$ and arrow space $\Gamma\times M$. The source is $(g,m)\mapsto m$, the target is $(g,m)\mapsto gm$ and the multiplication is $(g,m)(g^{\prime},m^{\prime})=(gg^{\prime},m^{\prime})$. 4. iv)

Orbifold groupoids or proper étale groupoids. We refer to [MR2012261, MR3448330] for more details. 5. v)

Let $(M,{\mathcal{F}})$ be a foliated manifold. Then the (reduced) holonomy groupoid is étale.

As the last example is one of our main motivating examples, we shall describe it in a slightly greater details. See [MR1151583, MR2012261, MR824240, MR1813430]. A foliation ${\mathcal{F}}$ on $M$ is given by a cocycle ${\mathcal{U}}=\{U_{i},f_{i},g_{ij}\}$ modeled on a manifold $N_{0}$ (${\mathbb{R}}^{n}$ or ${\mathbb{C}}^{n}$), i.e.,

1. i)

$\{U_{i}\}$ is an open covering of $M$; 2. ii)

$f_{i}:U_{i}\rightarrow N_{0}$ are submersions with connected fibers defining ${\mathcal{F}}$; 3. iii)

$g_{ij}$ are local diffeomorphisms of $N_{0}$ and $g_{ij}f_{j}=f_{i}$ on $U_{i}\cap U_{j}$.

The manifold $N=\sqcup f_{i}(U_{i})$ is called the transverse manifold of ${\mathcal{F}}$ associated to the cocycle ${\mathcal{U}}$, and the pseudogroup $P$ generated by $g_{ij}$ is called the holonomy pseudogroup on the transverse manifold. To any pseudogroup $P$ on some manifold $X$ we can associate an étale (effective) groupoid $\Gamma(P)$ over $X$ as follows: for any $x,y\in X$ let

[TABLE]

The multiplication in $\Gamma(P)$ is given by the composition of transitions. Equipped with classical sheaf topology $\Gamma(P)_{1}$ becomes a smooth manifold and $\Gamma(P)$ becomes an étale groupoid. In our case, $\Gamma(P)$ is called the reduced holonomy groupoid of $(M,{\mathcal{F}})$ and is denoted $Hol_{N}(M,{\mathcal{F}})$ (but also we write $\Gamma(P)$ sometimes).

We now show one gets Hopf algebroids naturally from étale groupoids following [MR2817646, MR2311185]. Before that we introduce the following.

Fiber sum notation.

Let $E$ and $F$ are vector bundles over two manifolds $X$ and $Y$, respectively. Suppose $\phi:X\rightarrow Y$ is an étale map (i.e., a local homeomorphism) and $\alpha:E\cong\phi^{*}F$ an isomorphism of vector bundles. Then the push-forward (or fiber sum) of $\phi$, denoted by $\phi_{*}:\Gamma_{c}(X,E)\rightarrow\Gamma_{c}(Y,F)$, is defined by

[TABLE]

where $x\in X,y\in Y$ and $s\in\Gamma_{c}(X,E)$. Here we identify the fiber ${\phi}^{*}F_{z}$ with $F_{\phi(z)}$ using the definition of pullback.

If $G$ is an étale groupoid over a compact Hausdorff $G_{0}$, the space $C_{c}^{\infty}(G)$ of smooth functions on $G=G_{1}$ with compact support carries a Hopf algebroid structure. Although $G=G_{1}$ often happens to be non-Hausdorff in examples, we assume this condition in this paper since the reduced holonomy groupoid of a Riemannian foliation is always Hausdorff. We have two $C^{\infty}(G_{0})$-actions on $C_{c}^{\infty}(G)$ by left and right multiplication with respect to which we define the four tensor products denoted by $\otimes^{ll}_{C^{\infty}(G_{0})}$, $\otimes^{rr}_{C^{\infty}(G_{0})}$, $\otimes^{rl}_{C^{\infty}(G_{0})}$ and $\otimes^{lr}_{C^{\infty}(G_{0})}$. We need the following isomorphisms

[TABLE]

all given by the formulas

[TABLE]

for $u,u^{\prime}\in C_{c}^{\infty}(G)$ and $(g,g^{\prime})$ in the respective pullback $G~{}^{-}\times^{-}_{G_{0}}G$. The maps are isomorphism, as it was shown in [MR2311185]. We now give the Hopf algebroid structure maps for $C_{c}^{\infty}(G)$ over $C^{\infty}(G_{0})$:

Ring structure

On the base algebra $C^{\infty}(G_{0})$ one has the commutative pointwise product, whereas the total algebra $C_{c}^{\infty}(G)$ is equipped with a convolution product, defined as the composition

[TABLE]

Explicitly,

[TABLE]

which can be used in showing associativity of the product.

Source and target maps

For $f\in C^{\infty}(G_{0})$ and $u\in C_{c}^{\infty}(G)$,

[TABLE]

It can be shown that $C^{\infty}(G_{0})$, identified with those functions in $C_{c}^{\infty}(G)$ having support on $1_{G_{0}}\subset G$, is a commutative subalgebra of $C_{c}^{\infty}(G)$. We put for the (left and right bialgebroid) source and target maps

[TABLE]

i.e., the injection as subalgebra given by the fiber sum of the unit map $1:G_{0}\rightarrow G$. More explicitly,

[TABLE]

Left and right comultiplications

Using the isomorphism $\Omega_{-,-}$, the left and right comultiplications are given as follows:

[TABLE]

Alternatively, $\Delta_{l}=d^{l}_{\ast}$ and $\Delta_{r}=d^{r}_{\ast}$ for the diagonal maps $d^{l}:G\rightarrow G~{}^{t}\times^{t}_{G_{0}}G$, $g\mapsto(g,g)$ and $d^{r}:G\rightarrow G~{}^{s}\times^{s}_{G_{0}}G$, $g\mapsto(g,g)$.

Left and right counits

Both left and right counits are respectively determined by the fiber sum of the target and source maps of the groupoid. For any $x\in G_{0}$,

[TABLE]

Antipode

The antipode is given by the groupoid inversion,

[TABLE]

Theorem 2.2.3**.**

With the above structure maps, $C_{c}^{\infty}(G)$ becomes a Hopf algebroid over $C^{\infty}(G_{0})$.

The proof is in [MR2817646]. See also [MR679730, MR823176, MR2456103, MR2530918].

2.3. More examples of Hopf algebroids

So far we have focused on a single example, that of coming from étale groupoids. We have also mentioned Hopf algebras and weak Hopf algebras and built our framework using these as guiding examples. In this section we describe another example, namely, the Connes-Moscovici Hopf algebroid, which is over a noncommutative base, thus providing wider scope of our framework. Before we plunge into the Connes-Moscovici Hopf algebroid, we describe a special case, namely the following.

2.3.1. The enveloping Hopf algebroid of an algebra

Given an arbitrary ${\mathbb{C}}$-algebra $A$, let $H=A\otimes_{{\mathbb{C}}}A^{op}$. The left bialgebroid structure over $A$ is given as

[TABLE]

and the right bialgebroid structure over $A^{op}$ is given as

[TABLE]

for $a,b\in A$. Finally, the antipode

[TABLE]

makes $H$ into a Hopf algebroid. We now come to

2.3.2. The Connes-Moscovici Hopf algebroid

Let $Q$ be a Hopf algebra over ${\mathbb{C}}$ with antipode $T$ satisfying $T^{2}=\operatorname{id}$ and $A$ a $Q$-module algebra. Consider $H=A\otimes_{{\mathbb{C}}}Q\otimes_{{\mathbb{C}}}A$ with multiplication given by

[TABLE]

for $a,b,a^{\prime},b^{\prime}\in A$ and $q,q^{\prime}\in Q$. A left bialgebroid structure over $A$, known as the Connes-Moscovici bialgebroid, is given as

[TABLE]

for $a,b\in A$ and $q\in Q$. $\varepsilon$ is the counit of $Q$ and we have used Sweedler notation for the coproduct of $Q$. This much is in the literature, see for example [MR2553659]. We now put a right bialgebroid structure on $H$ over $A^{op}$ as

[TABLE]

for $a,b\in A$ and $q\in Q$. Observe that, for the above structure maps,

[TABLE]

for $a_{1},a_{2},b,b^{\prime}\in A$ and $q\in Q$. From this, it at once follows that $\Delta_{r}$ and $\varepsilon_{r}$ are bimodule morphisms. Coassociativity of $\Delta_{r}$ and counitarity of $\varepsilon_{r}$ are easy to verify. We now check the Takeuchi condition. Given $a,b,c\in A$ and $q\in Q$, we have

[TABLE]

thus proving the Takeuchi condition. The verification of the character property of $\varepsilon_{r}$ is left to the reader. So this proves that we indeed have a right bialgebroid. Now we define the antipode $S$ as

[TABLE]

Again, the antipode axioms are straightforward to check. As an example we show that $\mu(S\otimes\operatorname{id}_{H})\Delta_{l}=s_{r}\varepsilon_{r}$ holds:

[TABLE]

Thus we have:

Theorem 2.3.1**.**

With the structures described above, $H$ becomes a Hopf algebroid, which we call the Connes-Moscovici Hopf algebroid.

Remark 2.3.2**.**

Observe that taking $Q={\mathbb{C}}$ gives the enveloping Hopf algebroid back and $A={\mathbb{C}}$ reduces $H$ to a Hopf algebra. Thus it is a simultaneous generalization of the cases discussed above.

Remark 2.3.3**.**

We have used $T^{2}=\operatorname{id}$ to make $H$ into a Hopf algebroid. We think that it is possible to remove this condition by introducing a “modular pair in involution”, that in turn produces a “twisted antipode” for $Q$, hence for $H$.

3. Hopf algebroids as symmetries

In this section, we recall Hopf algebroids representation and introduce $\ast$-structures, necessary for viewing Hopf algebroids as symmetry objects.

3.1. Modules over Hopf algebroids

Let $H=(H_{l},H_{r},S)$ be a Hopf algebroid. A left module over $H$ is simply a left module over the underlying ${\mathbb{C}}$-algebra $H$. We denote the structure map by $(h,m)\mapsto h\cdot m$. The left bialgebroid structure $H_{l}$ induces an $(A_{l},A_{l})$-bimodule structure on each module and a monoidal structure on the category of modules. More explicitly, let $M$ be an $H$-module. Then the $(A_{l},A_{l})$-bimodule structure is given by

[TABLE]

for all $a_{1},a_{2}\in A_{l}$ and $m\in M$. The left coproduct defines the monoidal structure $(M,N)\mapsto M\otimes_{A}N$, where $M\otimes_{A}N$ is equipped with the $H$-module structure

[TABLE]

The monoidal unit is given by $A_{l}$ with left $H$-action $h\cdot a=\varepsilon_{l}(hs_{l}(a)).$ Note that $\varepsilon_{l}(ht_{l}(a))=\varepsilon_{l}(hs_{l}(\varepsilon_{l}(t_{l}(a))))=\varepsilon_{l}(hs_{l}(a)).$ Also $A_{l}$ being the monoidal unit it is an algebra in the category of $H$-modules, i.e., it is an $H$-module algebra. This structure will be important for us in the examples we consider.

Remark 3.1.1**.**

We state the definition of an $H$-module algebra explicitly. It is a ${\mathbb{C}}$-algebra and left $H$-module $B$ such that the multiplication in $B$ is $A_{l}$-balanced and

1. i)

$h\cdot 1_{B}=s_{l}\varepsilon_{l}(h)\cdot 1_{B}$; 2. ii)

$h\cdot(bb^{\prime})=(h_{(1)}\cdot b)(h_{(2)}\cdot b^{\prime})$.

for $b,b^{\prime}\in B$ and $h\in H$. Note that $B$ has a canonical $A_{l}$-ring structure. Its unit is the map $A_{l}\rightarrow B$, $a\mapsto s_{l}(a)\cdot 1_{B}=t_{l}(a)\cdot 1_{B}$.

Similarly, one can consider right $H$-modules as modules over the ${\mathbb{C}}$-algebra $H$. Such modules get the structure of an $(A_{r},A_{r})$-bimodule and the category becomes monoidal using the right coproduct. The monoidal unit is $A_{r}$. We now see some examples coming from the geometry of groupoids. We follow [MR2853081].

Definition 3.1.2**.**

A smooth left action of a Lie groupoid $G$ on a smooth manifold $P$ along a smooth map $\pi:P\rightarrow G_{0}$ is a smooth map $\mu:G_{1}{}^{s}\times^{\pi}_{G_{0}}P\rightarrow P$, $(g,p)\mapsto g\cdot p$, which satisfies the conditions $\pi(g\cdot p)=t(g)$, $1_{\pi(p)}\cdot p=p$ and $g^{\prime}\cdot(g\cdot p)=(g^{\prime}g)\cdot p$ for all $g^{\prime},g\in G_{1}$ and $p\in P$ with $s(g^{\prime})=t(g)$ and $s(g)=\pi(p)$.

We define right actions of étale groupoids on smooth manifolds in a similar way.

Definition 3.1.3**.**

Let $G$ be an étale groupoid, and let $E$ be a smooth complex vector bundle over $G_{0}$. A representation of the groupoid $G$ on $E$ is a smooth left action $\rho:G_{1}~{}^{s}\times^{p}_{G_{0}}E\rightarrow E$, denoted by $\rho(g,v)=g\cdot v$, of $G$ on $E$ along the bundle projection $p:E\rightarrow G_{0}$ such that for any arrow $x\xrightarrow{g}y$ the induced map $g_{*}:E_{x}\rightarrow E_{y}$, $v\mapsto g\cdot v$, is a linear isomorphism. A section $u:G_{0}\rightarrow E$ is called $G$-invariant if for any arrow $x\xrightarrow{g}y$, it holds that $g\cdot u(x)=u(y)$.

Let us see what representations mean in the examples above.

Example 3.1.4**.**

**

1. i)

Representations of the unit groupoid associated to a smooth manifold correspond precisely to complex vector bundles. 2. ii)

Representations of the point groupoid associated to a (discrete) group $\Gamma$ correspond to representations of the group on finite dimensional complex vector spaces. 3. iii)

Representations of the translation groupoid $\Gamma\ltimes M$ corresponds to $\Gamma$-equivariant complex vector bundles over $M$. 4. iv)

Representations of the orbifold groupoid are the orbibundles. 5. v)

Representations of the holonomy groupoid are the transversal vector bundles. 6. vi)

For an étale groupoid $G$ the complexified tangent bundle of $G_{0}$ becomes a representation of $G$. The cotangent bundle, exterior bundle all inherit this natural representation, so it makes sense to speak of vector fields, differential forms or Riemannian metrics etc. on étale groupoids (vector fields, differential forms or Riemannian metrics etc. on $G_{0}$, respectively, invariant under the action). Also note that the exterior derivative $d$ is invariant under the $G$-action. This follows from naturality of $d$ and a local argument.

Proposition 3.1.5**.**

Let $E$ be representation of the étale groupoid $G$. The space of smooth sections $\Gamma^{\infty}(E)$ over $G_{0}$ becomes a module over $C_{c}^{\infty}(G)$ by the formulas

[TABLE]

for $a\in C_{c}^{\infty}(G)$ and $u\in\Gamma^{\infty}(E)$.

The proof is in [MR2853081]. Moreover, each module of finite type and constant rank appears in this way, giving a version of Serre-Swan theorem. See [MR823176] for an example coming from Sobolev spaces.

3.2. $\ast$-structures and conjugate modules

We introduce $\ast$-structures on Hopf algebroids which will be needed in order to view them as symmetry objects. This is one of the main results of the present paper. We view the ensuing structures as the first step in defining a “compact”-type Hopf algebroid in analogy with $CQG$-algebras [MR1310296], though we do not go in that direction here.

Let $(H_{l},H_{r},S)$ be a Hopf algebroid such that $H$, $A_{l}$ and $A_{r}$ are $\ast$-algebras, $s_{l}$ and $s_{r}$ are $\ast$-preserving (the involutions for $H$, $A_{r}$ and $A_{l}$ are denoted by the same symbol $\ast$). Assume that

[TABLE]

hold for all $a_{1}\in A_{r}$, $a_{2}\in A_{l}.$

Lemma 3.2.1**.**

We have

[TABLE]

Proof.

We compute

[TABLE]

∎

Lemma 3.2.1 says that the map $(\ast\otimes\ast):H_{l}\otimes_{{\mathbb{C}}}H_{l}\rightarrow H_{r}\otimes_{A_{r}}H_{r}$ descends to an isomorphism $(\ast\otimes\ast):H_{l}\otimes_{A_{l}}H_{l}\rightarrow H_{r}\otimes_{A_{r}}H_{r}$. So we can make sense of

[TABLE]

In Sweedler notation,

[TABLE]

Definition 3.2.2**.**

Let $(H_{l},H_{r},S)$ be a Hopf algebroid such that $H$, $A_{l}$ and $A_{r}$ are $\ast$-algebras while $s_{l}$ and $s_{r}$ are $\ast$-preserving. Then $(H_{l},H_{r},S)$ is said to be a Hopf $\ast$-algebroid if (3.4) and (3.7) hold.

Some immediate corollaries of Definition 3.2.2 are:

1. i)

$(\ast\otimes\ast):H_{l}\otimes_{A_{l}}H_{l}\rightarrow H_{r}\otimes_{A_{r}}H_{r}$ induces an isomorphism $H_{l}\times_{A_{l}}H_{l}\rightarrow H_{r}\times^{A_{r}}H_{r}$. 2. ii)

From (2.6), $t_{l}\ast=s_{r}\varepsilon_{r}t_{l}\ast=s_{r}\ast\varepsilon_{r}s_{l}=\ast s_{r}\varepsilon_{r}s_{l}=\ast Ss_{l}$, with the last equality following from (2.12). 3. iii)

Similarly, $t_{r}\ast=\ast Ss_{r}.$

Proposition 3.2.3**.**

Let $(H_{l},H_{r},S)$ be a Hopf $\ast$-algebroid. Then the counits and the antipode satisfy

[TABLE]

and $A_{l}$ becomes an $H$-module $\ast$-algebra, i.e., the $H$-action satisfies

[TABLE]

Proof.

We have

[TABLE]

so

[TABLE]

Similarly,

[TABLE]

Now,

[TABLE]

Similarly,

[TABLE]

So we conclude that $\ast\varepsilon_{r}t_{l}\varepsilon_{l}\ast$ satisfies the right counit axioms. Hence $\varepsilon_{r}=\ast\varepsilon_{r}t_{l}\varepsilon_{l}\ast=\ast\varepsilon_{r}S^{-1}\ast$. Similarly, $\varepsilon_{l}=\ast\varepsilon_{l}S^{-1}\ast.$ From this we observe that $s_{l}\varepsilon_{l}\ast=\ast t_{r}\varepsilon_{r}$ and $s_{r}\varepsilon_{r}\ast=\ast t_{l}\varepsilon_{l}$. Using the above observation and proceeding exactly as before, it follows that $\ast S^{-1}\ast$ satisfies the antipode axioms. By uniqueness, we have $S=\ast S^{-1}\ast$, which implies $S\ast S\ast=\operatorname{id}_{H}$.

Finally,

[TABLE]

∎

Besides Hopf $\ast$-algebras, the Hopf algebroid in Theorem 2.2.3 becomes a central example of Hopf $\ast$-algebroids:

Proposition 3.2.4**.**

The space $C_{c}^{\infty}(G)$ becomes a Hopf $\ast$-algebroid over $C^{\infty}(G_{0})$ with $\ast$-structure given by

[TABLE]

Proof.

This follows from direct computations. ∎

If $A$ is a $\ast$-algebra then $A\otimes_{{\mathbb{C}}}A^{op}$ as in Subsubsection 2.3.1 is a Hopf $\ast$-algebroid. Furthermore, if $Q$ is a Hopf $\ast$-algebra and $A$ is a $Q$-module $\ast$-algebra then the Connes-Moscovici Hopf algebroid in Theorem 2.3.1 $H$ becomes a Hopf $\ast$-algebroid in our sense. Another class of examples, which we have not mentioned above, comes from weak Hopf algebras studied in [MR1726707]. Our $\ast$-structure is the same as $C^{*}$-structure mentioned in [MR1726707]. Following this and the standard theory of $CQG$-algebras, leads to opening up a new direction of study, namely, (co)representation theory of Hopf $\ast$-algebroids and the interplay of the $\ast$-structure and (co)integrals.

We shall systematically use the language of conjugate modules in order to keep track of various aspects, see [MR2501177, MR3073899]. Let $(H_{l},H_{r},S)$ be a Hopf $\ast$-algebroid and $M$ an $H$-module. We define the conjugate module $\overline{M}$ by declaring that

1. i)

$\overline{M}=M$ as abelian group; 2. ii)

we write $\overline{m}$ for an element $m\in M$ when we consider it as an element of $\overline{M}$; 3. iii)

the module operation for $\overline{M}$ is $h\cdot\overline{m}=\overline{S(h)^{*}\cdot m}$.

Again, let $B$ be a $\ast$-algebra and let $E$ be a $(B,B)$ bimodule. The conjugate bimodule $\overline{E}$ is defined by the following three conditions:

1. i)

$\overline{E}=E$ as abelian group; 2. ii)

We write $\overline{e}$ for an element $e\in E$ when we consider it as an element of $\overline{E}$; 3. iii)

The bimodule operations for $\overline{E}$ are $b\cdot\overline{e}=\overline{e\cdot b^{*}}$ and $\overline{e}\cdot b=\overline{b^{*}\cdot e}.$

If $\theta:E\rightarrow F$ is any morphism, then we define $\overline{\theta}:\overline{E}\rightarrow\overline{F}$ by $\overline{\theta}(\overline{e})=\overline{\theta(e)}$. We make $\overline{B}$ an associative algebra by defining the multiplication $\overline{b}{}\overline{b^{\prime}}:=\overline{b^{\prime}b}$. As an ${\mathbb{R}}$-algebra, $\overline{B}$ is isomorphic to $B^{op}$ via the map $b\mapsto\overline{b}$. We make $\overline{B}$ a ${\mathbb{C}}$-algebra through the algebra homomorphism ${\mathbb{C}}\rightarrow\overline{B}$, $\lambda\mapsto\overline{\lambda^{*}1_{B}}$. We now define $\#:B\rightarrow\overline{B}$, $b\mapsto\overline{b^{*}}$. Then $\#$ is an isomorphism of ${\mathbb{C}}$ algebras. If $\theta:B\rightarrow B^{\prime}$ is a morphism then we say that $\theta$ is $\ast$-preserving if $\#\theta=\overline{\theta}\#$.

So we see that the conclusion in (3.10) that $A_{l}$ is an $H$-module $\ast$ algebra, is nothing but the assertion that $\#:A_{l}\rightarrow\overline{A_{l}}$ is an $H$-module morphism. We also see that for an $H$-module $M$, the induced $(A_{l},A_{l})$-bimodule structure matches with the prescription above. Thus our $\ast$-structure naturally produces examples of “Bar categories” in the sense of [MR2501177].

Lemma 3.2.5**.**

Let $B$ be an $H$-module $\ast$-algebra, and let the invariant subalgebra $B_{H}$ be defined as $B_{H}=\{b\in B\mid h\cdot b=s_{l}\varepsilon_{l}(h)\cdot b\}$. Then $\#:B\rightarrow\overline{B}$ induces an isomorphism $\#:B_{H}\rightarrow\overline{B}_{H}$.

Proof.

This follows from the fact that $\#$ is an $H$-module morphism. ∎

In fact, we can say more:

Proposition 3.2.6**.**

Let $B$ be an $H$-module $\ast$-algebra. Then $B_{H}$ is also a $\ast$-algebra. So that, by Lemma 3.2.5 we can identify $(\overline{B})_{H}=\overline{(B_{H})}$ as algebras.

Proof.

Let $b\in B_{H}$. We compute

[TABLE]

so that

[TABLE]

Next observe that taking $h=s_{l}(a)$ for $a\in A_{l}$ in the last equality gives $s_{l}(a)\cdot b^{*}=t_{l}(a)\cdot b^{*}$. So that for all $h\in H$ we get

[TABLE]

which in turn implies that $b^{*}\in B_{H}$. ∎

4. Applications to noncommutative Kähler structures

In this section, we introduce Hopf algebroid action on differential graded algebras, a not-so-straightforward generalization of the Hopf algebra case. Then we introduce noncommutative Kähler structures admitting Hopf algebroid equivariance, generalizing the case of foliations along the way.

4.1. Noncommutative Kähler structures

We begin with a brief review of noncommutative Kähler structures in the sense of [MR3720811].

4.1.1. Complexes and Double Complexes

For $(S,+)$ a commutative semigroup, an $S$-graded algebra is an algebra of the form $A=\bigoplus_{s\in S}A^{s}$, where each $A^{s}$ is a linear subspace of $A$, and $A^{s}A^{t}\subset A^{s+t}$, for all $s,t\in S$. If $a\in A^{s}$, then we say that $a$ is a homogeneous element of degree $s$. A homogeneous mapping of degree $t$ on $A$ is a linear mapping $L:A\rightarrow A$ such that if $a\in A^{s}$, then $L(a)\in A^{s+t}$. We say that a subspace $B$ of $A$ is homogeneous if it admits a decomposition $B=\oplus_{s\in S}B^{s}$, with $B^{s}\subset A^{s}$, for all $s\in S$.

A pair $(A,d)$ is called a complex if $A$ is an ${\mathbb{N}}_{0}$-graded algebra, and $d$ is a homogeneous mapping of degree $1$ such that $d^{2}=0$. A triple $(A,\partial,\overline{\partial})$ is called a double complex if $A$ is an ${\mathbb{N}}^{2}_{0}$-graded algebra, $\partial$ is homogeneous mapping of degree $(1,0)$, $\overline{\partial}$ is homogeneous mapping of degree $(0,1)$, and

[TABLE]

4.1.2. Differential $*$-Calculi

A complex $(A,d)$ is called a differential graded algebra if $d$ is a graded derivation, which is to say, if it satisfies the graded Leibniz rule

[TABLE]

for all $\alpha\in A^{k}$, $\beta\in A$. The operator $d$ is called the differential of the differential graded algebra.

Definition 4.1.1**.**

A differential calculus over an algebra $A$ is a differential graded algebra $(\Omega,d)$ such that $\Omega^{0}=A$, and

[TABLE]

We use $\wedge$ to denote the multiplication between elements of a differential calculus when both are of order greater than [math]. We call an element of a differential calculus a form. A differential map between two differential calculi $(\Omega,\delta_{\Omega})$ and $(\Gamma,d_{\Gamma})$, defined over the same algebra $A$, is a bimodule map $\phi:\Omega\rightarrow\Gamma$ such that $\phi d_{\Omega}=d_{\Gamma}\phi$.

We call a differential calculus $(\Omega,d)$ over a $*$-algebra $A$ a differential $\ast$-calculus if the involution of $A$ extends to an involutive conjugate-linear map on $\Omega$, for which $(d\omega)^{*}=d\omega^{*}$, for all $\omega\in\Omega$, and

[TABLE]

for all $\omega\in\Omega^{k}$, $\nu\in\Omega^{l}$. We say that a form $\omega\in\Omega$ is real if $\omega^{*}=\omega$.

4.1.3. Orientability and Closed Integrals

We say that a differential calculus has total dimension $n$ if $\Omega^{k}=0$, for all $k>n$, and $\Omega^{n}\neq 0$. If in addition there exists an $(A,A)$-bimodule isomorphism $\operatorname{vol}:\Omega^{n}\simeq A$, then we say that $\Omega$ is orientable. We call a choice of such an isomorphism an orientation. If $\Omega$ is a $\ast$-calculus over a $\ast$-algebra, then a $\ast$-orientation is an orientation which is also a $*$-map. A $\ast$-orientable calculus is one which admits a $\ast$-orientation.

4.1.4. Complex structures

We now recall the definition of a noncommutative complex structure.

Definition 4.1.2**.**

An almost complex structure for a differential $\ast$-calculus $\Omega$, over a $\ast$-algebra $A$, is an ${\mathbb{N}}^{2}_{0}$-algebra grading $\bigoplus_{(a,b)\in{\mathbb{N}}^{2}_{0}}\Omega^{(a,b)}$ for $\Omega$ such that

1. i)

$\Omega^{k}=\bigoplus_{a+b=k}\Omega^{(a,b)}$, for all $k\in{\mathbb{N}}_{0}$, 2. ii)

$(\Omega^{(a,b)})^{*}=\Omega^{(b,a)}$, for all $(a,b)\in{\mathbb{N}}^{2}_{0}$.

We call an element of $\Omega^{(a,b)}$ an $(a,b)$-form. Let $\partial$ and $\overline{\partial}$ be the unique homogeneous operators of order $(1,0)$, and $(0,1)$ respectively, defined by

[TABLE]

where $\operatorname{proj}_{\Omega^{(a+1,b)}}$, and $\operatorname{proj}_{\Omega^{(a,b+1)}}$, are the projections from $\Omega^{a+b+1}$ onto $\Omega^{(a+1,b)}$, and $\Omega^{(a,b+1)}$, respectively. The proof of the following lemma carries over directly from the classical setting [MR2093043].

Lemma 4.1.3**.**

If $\bigoplus_{(a,b)\in{\mathbb{N}}^{2}_{0}}\Omega^{(a,b)}$ is an almost complex structure for a differential $\ast$-calculus $\Omega$ over an algebra $A$, then the following two conditions are equivalent:

1. i)

$d=\partial+\overline{\partial}$, 2. ii)

the triple \big{(}\bigoplus_{(a,b)\in{\mathbb{N}}^{2}}\Omega^{(a,b)},\partial,\overline{\partial}\big{)} is a double complex.

With this in hand,

Definition 4.1.4**.**

When the conditions in Lemma 4.1.3 hold for an almost complex structure, then we say that it is integrable.

We call an integrable almost complex structure a complex structure, and the double complex $(\bigoplus_{(a,b)\in{\mathbb{N}}^{2}}\Omega^{(a,b)},\partial,\overline{\partial})$ its Dolbeault double complex. An easy consequence of integrability is that

[TABLE]

for all $\omega\in\Omega$.

4.1.5. Hermitian and Kähler structures

Let $\Omega$ denote a differential $\ast$-calculus, over an algebra $A$, of total dimension $2n$. As a first step towards the definition of a hermitian form, we present a direct noncommutative generalization of the classical definition of an almost symplectic form.

Definition 4.1.5**.**

An almost symplectic form for $\Omega$ is a central real $2$-form $\sigma$ such that, with respect to the Lefschetz operator

[TABLE]

isomorphisms are given by

[TABLE]

for all $1\leq k<n$.

Note that since $\sigma$ is a central real form, $L$ is an $(A,A)$-bimodule $*$-homomorphism.

Definition 4.1.6**.**

For $L$ the Lefschetz operator of any almost symplectic form, the space of primitive $k$-forms is

[TABLE]

Now we present the following noncommutative generalization of the classical notion of a symplectic form [MR2093043].

Definition 4.1.7**.**

A symplectic form is a $d$-closed almost symplectic form.

We now introduce a hermitian structure for a differential $\ast$-calculus, which is essentially just a symplectic form interacting with a complex structure in a natural way. In the commutative case each such form is the fundamental form of a uniquely identified hermitian metric [MR2093043].

Definition 4.1.8**.**

An hermitian structure for a $\ast$-calculus $\Omega$ is a pair $(\Omega^{(\cdot,\cdot)},\sigma)$ where $\Omega^{(\cdot,\cdot)}$ is a complex structure and $\sigma$ is an almost symplectic form, called the hermitian form, such that $\sigma\in\Omega^{(1,1)}$.

Definition 4.1.9**.**

A Kähler structure for a differential $*$-calculus is a hermitian structure $(\Omega^{(\cdot,\cdot)},\kappa)$ such that the hermitian form $\kappa$ is $d$-closed. We call such a $\kappa$ a Kähler form.

Every $2$-form in a $*$-calculus with total dimension $2$ is obviously $d$-closed. Hence, just as in the classical case, with respect to any choice of complex structure, every $\kappa\in\Omega^{(1,1)}$ is a Kähler form.

4.2. Covariant noncommutative Kähler structures

In this subsection, we introduce Hopf algebroid covariance into the setup of the previous subsection.

4.2.1. Covariant differential calculi

Let $H=(H_{l},H_{r},S)$ be a Hopf $\ast$-algebroid. We start by defining a differential calculus.

Definition 4.2.1**.**

An ${\mathbb{N}}_{0}$-graded $H$-module is an ${\mathbb{N}}_{0}$-graded ${\mathbb{C}}$-vector space which is also an $H$-module such that the $H$-action preserves the ${\mathbb{N}}_{0}$-grading.

Definition 4.2.2**.**

An ${\mathbb{N}}_{0}$-graded $H$-module algebra is an ${\mathbb{N}}_{0}$-graded algebra which is also an $H$-module algebra such that the $H$-action preserves the ${\mathbb{N}}_{0}$-grading.

Definition 4.2.3**.**

A pair $(B,\,d)$ is called an $H$-covariant complex if $B$ is an ${\mathbb{N}}_{0}$-graded $H$-module algebra, and $d$ is homogeneous of degree one satisfying $d^{2}=0$, such that

[TABLE]

where

[TABLE]

Definition 4.2.4**.**

A triple $(B,\partial,\overline{\partial})$ is called an $H$-covariant double complex if $B$ is an ${\mathbb{N}}_{0}^{2}$-graded $H$-module algebra, $\partial$ is homogeneous of degree $(1,0)$, and $\overline{\partial}$ is homogeneous of degree $(0,1)$, such that $\partial^{2}=0$, $\overline{\partial}^{2}=0$, $\partial\overline{\partial}+\overline{\partial}\partial=0$ and they satisfy (4.10).

For any $H$-covariant complex $(B,\,d)$, we call an element $d$-closed if it is contained in $\ker(d)$ and $d$-exact if it is contained in $im(d)$. For an $H$-covariant double complex $(B,\partial,\overline{\partial})$, we define $\partial$-closed, $\overline{\partial}$-closed, $\partial$-exact and $\overline{\partial}$-exact elements analogously.

Definition 4.2.5**.**

An $H$-covariant complex $(B,\,d)$ is called an $H$-covariant differential graded algebra if $d$ satisfies the graded Leibniz rule

[TABLE]

Definition 4.2.6**.**

An $H$-covariant differential calculus over an $H$-module algebra $B$ (with unit map $\iota_{B}$) is an $H$-covariant differential graded algebra $(\Omega,\,d)$ (with unit map $\iota_{\Omega}$) such that $\Omega^{0}=B$, the two $H$-action on $B$ coming from $B$ itself and $\Omega^{0}$ coincide, and

[TABLE]

Observe that the coincidence of the two $H$-actions on $B$ implies that the two unit maps also coincide. Observe also that the induced $(A_{l},A_{l})$-bimodule structure on $\Omega$ coincide with the one coming from the unit map.

Definition 4.2.7**.**

An $H$-covariant differential calculus $(\Omega,\,d)$ over an $H$-module $\ast$-algebra $B$ is a $\ast$-differential calculus if the involution of $B$ extends to a degree zero involutive conjugate linear map on $\Omega$, for which $(d\omega)^{*}=d(\omega^{*})$ for all $\omega\in\Omega$, and

[TABLE]

making $\Omega$ an $H$-module $\ast$-algebra.

Recall the Connes-Moscovici Hopf algebroid from Theorem 2.3.1. We then have

Proposition 4.2.8**.**

Let $(\Omega,\,d)$ be a $Q$-covariant differential calculus on $A$, $Q$ being a Hopf algebra. Then $(\Omega,\,d)$ can be made into an $H$-covariant differential calculus on $A$ in the sense of Definition 4.2.6. Furthermore, if $Q$ is a Hopf $\ast$-algebra, $A$ is a $Q$-module $\ast$-algebra and $(\Omega,\,d)$ is a $Q$-covariant $\ast$-differential calculus then it can be made into an $H$-covariant $\ast$-differential calculus in the sense of Definition 4.2.7.

Proof.

We define the $H$-action on $\Omega$ as follows:

[TABLE]

The only non-trivial part to check is that (4.10) holds. This is easy because $H_{0}$ contains $1\otimes_{{\mathbb{C}}}Q\otimes_{{\mathbb{C}}}1$. ∎

Lemma 4.2.9**.**

For an $H$-covariant $\ast$-differential calculus $(\Omega,\,d)$, we have

1. i)

$[h-s_{l}\varepsilon_{l}(h),d]=0$* $\implies$ $[S^{-1}(h^{*})-t_{l}\varepsilon_{l}(S^{-1}(h^{*})),d]=0$;* 2. ii)

$[h-t_{l}\varepsilon_{l}(h),d]=0$* $\implies$ $[S^{-1}(h^{*})-s_{l}\varepsilon_{l}(S^{-1}(h^{*})),d]=0$;*

for $h\in H$. Thus combining the two, we get that $h\in H_{0}$ if and only if $S^{-1}(h^{*})\in H_{0}$.

Proof.

For $\omega\in\Omega$, we compute

[TABLE]

And similarly, $0=([h-t_{l}\varepsilon_{l}(h),d](\omega^{*}))^{*}=[(S(h-t_{l}\varepsilon_{l}(h)))^{*},d](\omega)$. Now

[TABLE]

and

[TABLE]

Thus we get $S^{-1}(h^{*})\in H_{0}$ if $h\in H_{0}$. The other direction follows from $(S\ast)^{2}=\operatorname{id}$ ∎

Lemma 4.2.10**.**

On $\overline{\Omega}$, defining the product as $\overline{\omega}\wedge\overline{\eta}=(-1)^{kl}\overline{\eta\wedge\omega}$ for $\omega\in\Omega^{k}$, $\eta\in\Omega^{l}$ makes $(\overline{\Omega},\overline{d})$ an $H$-covariant differential graded algebra. Then an $H$-covariant $\ast$-differential calculus is an $H$-covariant differential calculus such that $\#:(\Omega,\,d)\rightarrow(\overline{\Omega},\overline{d})$ is $H$-linear and a differential graded algebra homomorphism.

Proof.

The second part follows from the discussion prior to Lemma 3.2.5. For the first part, we observe that given $\omega\in\Omega$ and $h\in H$,

[TABLE]

and similarly, $[h-t_{l}\varepsilon_{l}(h),\overline{d}](\overline{\omega})=\overline{[S(h-t_{l}\varepsilon_{l}(h))^{*},d](\omega)}$. Now the lemma follows from Lemma 4.2.9. ∎

Definition 4.2.11**.**

We define the space of invariant forms $\Omega_{0}$ of $\Omega$ as

[TABLE]

Observe that we recover the usual definition of invariant subalgebra as in Lemma 3.2.5 if the differential $d$ is identically [math].

Proposition 4.2.12**.**

For the space of invariant forms we have,

1. i)

$(\Omega_{0},d|_{\Omega_{0}})$* is a differential graded algebra;* 2. ii)

$\Omega_{0}$* is a $\ast$-algebra;* 3. iii)

$d|_{\Omega_{0}}$* satisfies $d|_{\Omega_{0}}(\omega^{*})=(d|_{\Omega_{0}}\omega)^{*}$ for all $\omega\in\Omega_{0}$;* 4. iv)

$\#:(\Omega_{0},d|_{\Omega_{0}})\rightarrow(\overline{\Omega}_{0},\overline{d}|_{\overline{\Omega}_{0}})$* is a differential graded algebra homomorphism.*

Proof.

i) That $\Omega_{0}$ is an algebra follows from the same proof as in $d$ identically [math] case. Moreover, that $d$ preserves $\Omega_{0}$ follows from the definition of $H_{0}$.

ii) Observe that for $h\in H_{0}$ and $\omega\in\Omega_{0}$

[TABLE]

so that

[TABLE]

Again

[TABLE]

so that

[TABLE]

iii) holds because $d$ satisfies the property.

iv) Follows from ii). ∎

We shall denote the differential on $\Omega_{0}$ only by $d$, assuming that it really means $d$ is restricted to $\Omega_{0}$.

Definition 4.2.13**.**

**

1. i)

We say that an $H$-covariant differential calculus $(\Omega,\,d)$ over an $H$-module algebra $B$ has total dimension $n$ if $\Omega^{k}=0$, for all $k>n$, and $\Omega^{n}\neq 0$. 2. ii)

If in addition, there exists a $(B,B)$-bimodule and an $H$-module isomorphism $\operatorname{vol}:\Omega^{n}\rightarrow B$, then we say that $\Omega$ is orientable. 3. iii)

If $\Omega$ is a $\ast$-calculus over a $\ast$-algebra, then a $\ast$-orientation is an orientation which is also $\ast$-preserving, meaning $\overline{\operatorname{vol}}\#=\#\operatorname{vol}$. 4. iv)

A $\ast$-orientable calculus is one which admits a $\ast$-orientation. 5. v)

Let $\tau$ be a state on $B$, i.e., a unital linear functional $\tau:B\rightarrow{\mathbb{C}}$ such that $\tau(b^{*}b)\geq 0$. We call the functional $\tau\operatorname{vol}$ the integral associated to $\tau$ and denote it by $\int_{\tau}$. 6. vi)

We say that the integral is closed if $\int_{\tau}(d\omega)=0$ for all $\omega\in\Omega^{n-1}$.

Lemma 4.2.14**.**

Assume that $(\Omega,\,d)$ is $\ast$-oriented with orientation $\operatorname{vol}$ and of total dimension $2n$. Then $(\Omega_{0},\,d)$ is $\ast$-oriented.

Proof.

Since $\operatorname{vol}$ is assumed to be $H$-linear, it restricts to $\Omega_{0}$, which in turn shows that $\Omega^{2n}_{H}\neq 0$ so that it also has total dimension $2n$. The lemma now follows from Lemma 3.2.5 and Proposition 4.2.12. ∎

4.2.2. Covariant complex structures

As mentioned before, the setup below is due to [MR3720811] and we follow it closely. We shall omit the proofs of some of the results here as they are essentially given in [MR3720811].

Definition 4.2.15**.**

An $H$-covariant almost complex structure for an $H$-covariant $\ast$-differential calculus $(\Omega,\,d)$ over an $H$-module $\ast$-algebra $B$ is an ${\mathbb{N}}_{0}^{2}$-algebra grading $\oplus_{(k,l)\in{\mathbb{N}}_{0}^{2}}\Omega^{(k,l)}$ for $\Omega$ such that

1. i)

the $H$-action preserves the ${\mathbb{N}}_{0}^{2}$-grading; 2. ii)

$\Omega^{n}=\oplus_{k+l=n}\Omega^{(k,l)}$, for all $n\in{\mathbb{N}}_{0}$; 3. iii)

$\#:\Omega\rightarrow\overline{\Omega}$* preserves the ${\mathbb{N}}_{0}^{2}$-grading, where the ${\mathbb{N}}_{0}^{2}$-grading on $\overline{\Omega}$ is given by $\overline{\Omega}^{(k,l)}=\overline{\Omega^{(l,k)}}$.*

Let $\partial$ and $\overline{\partial}$ be the unique homogeneous operators of order $(1,0)$ and $(0,1)$ respectively, defined by

[TABLE]

where $\operatorname{proj}_{\Omega^{(k,l+1)}}$ and $\operatorname{proj}_{\Omega^{(k,l+1)}}$ are the projections from $\Omega^{(k+l+1)}$ onto $\Omega^{(k+1,l)}$ and $\Omega^{(k,l+1)}$, respectively.

As in [MR3720811], we have:

Lemma 4.2.16**.**

If $\oplus_{(k,l)\in{\mathbb{N}}_{0}^{2}}\Omega^{(k,l)}$ is an $H$-covariant almost complex structure for an $H$-covariant $\ast$-differential calculus $(\Omega,\,d)$ over an $H$-module $\ast$-algebra $B$, then the following two conditions are equivalent:

1. i)

$d=\partial+\overline{\partial}$; 2. ii)

the triple $(\oplus_{(k,l)\in{\mathbb{N}}_{0}^{2}}\Omega^{(k,l)},\partial,\overline{\partial})$ is an H-covariant double complex.

Proof.

The proof of the equivalence is in [MR3720811]. All we have to show is the $H$-covariant part in ii). Observe that $\operatorname{proj}_{\Omega^{(k+1,l)}}$ and $\operatorname{proj}_{\Omega^{(k,l+1)}}$ are $H$-linear. Then for $h\in H_{0}$,

[TABLE]

Thus we get (4.10) for $\partial\mid_{\Omega^{(k,l)}}$, and similarly for $\overline{\partial}\mid_{\Omega^{(k,l)}}$, hence the covariance. ∎

Definition 4.2.17**.**

When the conditions in Lemma 4.2.16 hold for an almost complex structure, then we say that the almost complex structure is integrable.

We also call an integrable almost complex structure a complex structure and the double complex $(\oplus_{(k,l)\in{\mathbb{N}}_{0}^{2}}\Omega^{(k,l)},\partial,\overline{\partial})$ its Dolbeault double complex. Note that

[TABLE]

as they follow from the integrability condition.

Lemma 4.2.18**.**

Suppose that $(\Omega,\,d)$ admits an $H$-covariant complex structure. Then $(\Omega_{0},\,d)$ admits a complex structure. We call this a transverse complex structure on $B_{0}$.

Proof.

Condition i) in Definition 4.2.15 implies that $(\Omega_{0},\,d)$ admits an ${\mathbb{N}}^{2}_{0}$-algebra grading by $(\Omega_{0})^{(k,l)}=\Omega^{(k,l)}_{0}$, $(k,l)\in{\mathbb{N}}^{2}_{0}$. Condition ii) follows automatically, while Condition iii) follows from that fact that $\#$ is $H$-linear. $\partial$ and $\overline{\partial}$ restrict to the space of invariant forms as in Proposition 4.2.12. Finally, $d=\partial+\overline{\partial}$ then follows automatically. ∎

4.2.3. Covariant hermitian and Kähler structures

We fix an $H$-covariant $\ast$-differential calculus $(\Omega,\,d)$ over an $H$-module $\ast$-algebra $B$ of total dimension $2n$. As in [MR3720811], the following is a non-commutative generalization of an almost symplectic form.

Definition 4.2.19**.**

An almost symplectic form for $\Omega$ is a central real $H$-invariant 2-form $\sigma$ ($h\cdot\sigma=s_{l}\varepsilon_{l}(h)\cdot\sigma$ for all $h\in H$) such that, the Lefschetz operator

[TABLE]

satisfies the following condition: the maps

[TABLE]

are isomorphisms for all $0\leq k<n$.

Since $\sigma$ is a central real form, $L$ is a $(B,B)$-bimodule morphism and $\ast$-preserving $(\overline{L}\#=\#L)$. Moreover, the $H$-invariance condition implies that $L$ is also an $H$-module morphism. Indeed, we have

[TABLE]

[TABLE]

Definition 4.2.20**.**

A symplectic form is a $d$-closed almost symplectic form.

Ó Buachalla, [MR3720811], introduced hermitian structure which is an almost symplectic form compatible with a complex structure.

Definition 4.2.21**.**

A hermitian structure for $\Omega$ is a pair $(\Omega^{(\cdot,\cdot)},\sigma)$, where $\Omega^{(\cdot,\cdot)}$ is an $H$-covariant complex structure, and $\sigma$ is an almost symplectic form, called the hermitian form, such that $\sigma\in\Omega^{(1,1)}$.

We have:

Lemma 4.2.22**.**

Suppose that $(\Omega^{(\cdot,\cdot)},\sigma)$ is a hermitian structure for $(\Omega,\,d)$. Then $\sigma$ induces a hermitian structure on $(\Omega_{0},\,d)$.

Proof.

By definition, $\sigma\in\Omega_{0}$. The $H$-linearity of $L$ shows that $\sigma$ is an almost symplectic form for $(\Omega_{0},\,d)$. Finally, $\sigma\in(\Omega^{(1,1)})_{0}=(\Omega_{0})^{(1,1)}$, by Lemma 4.2.18. ∎

We say that an almost complex structure is of diamond type if $\Omega^{(a,b)}=0$ whenever $a>n$ or $b>n$. Supposing $a>n$ and observing that the isomorphism $L^{a+b-n}$ maps $\Omega^{(n-b,n-a)}$ onto $\Omega^{(a,b)}$, we see that the existence of a hermitian structure implies that the complex structure has to be of diamond type.

Definition 4.2.23**.**

The Hodge map associated to a hermitian structure is the morphism uniquely defined by

[TABLE]

Recall the notion of primitive forms from Definition 4.1.6. Observe that $\star$ is an $H$-module morphism. Hence it descends to $\Omega_{0}$.

Lemma 4.2.24**.**

We have

1. i)

$\star^{2}(\omega)=(-1)^{k}\omega$* for all $\omega\in\Omega^{k}$,* 2. ii)

$\star$* is an isomorphism,* 3. iii)

$\star(\Omega^{(a,b)})=\Omega^{(n-b,n-a)}$, 4. iv)

$\star$* is a $\ast$-preserving.*

Given a hermitian structure $(\Omega^{(\cdot,\cdot)},\kappa)$, we first recover the hermitian metric associated to it:

Definition 4.2.25**.**

Define $g:\Omega\otimes_{B}\overline{\Omega}\rightarrow B$ by $g(\omega\otimes\overline{\eta})=0$ for $\omega\in\Omega^{k}$, $\eta\in\Omega^{l}$, $k\neq l$, and

[TABLE]

for $\omega,\eta\in\Omega^{k}$.

A metric on the orbit space should be an invariant one as is showed in the following lemma.

Lemma 4.2.26**.**

For $\omega,\eta\in\Omega^{k}$ and $h\in H$, it holds that

[TABLE]

so that $g$ is $H$-covariant.

Proof.

We compute

[TABLE]

∎

Proposition 4.2.27**.**

The following decompositions are orthogonal with respect to $\langle,\rangle$:

1. i)

The degree decomposition $\Omega=\oplus_{k}\Omega^{k}$; 2. ii)

The bidegree decomposition $\Omega^{k}=\oplus_{(a,b)}\Omega^{(a,b)}$; 3. iii)

The Lefschetz decomposition $\Omega^{k}=\oplus_{j\geq 0}L^{j}(P^{k-2j})$.

The above proposition implies the following.

Corollary 4.2.28**.**

We have $g(\omega\otimes\overline{\eta})=g(\eta\otimes\overline{\omega})^{*}$ for $\omega,\eta\in\Omega$.

The hermitian structure is said to be positive definite if $g(\omega\otimes\overline{\omega})>0$ for all nonzero $\omega\in\Omega$. In that case, we define an inner product (positive definite, hermitian) on $\Omega$ by setting

[TABLE]

for $\omega,\eta\in\Omega$ and a fixed faithful state $\tau$ on $B$. We denote the corresponding norm of $\omega$ by $\|\omega\|$. Moreover, Lemma 4.2.26 shows that $g$ induces a metric on $\Omega_{0}$ that takes values in $B_{0}$. Applying $\tau$, we get an inner product on $\Omega_{0}$ which is really the restriction of $\langle\cdot,\cdot\rangle$ to $\Omega_{0}$. From now on, we assume that the hermitian structure to be positive definite.

Proposition 4.2.29**.**

The Hodge map $\star$ is unitary.

Proof.

See the proof of Lemma 5.10 in [MR3720811]. ∎

We now define the Laplacians.

Definition 4.2.30**.**

**

1. i)

The codifferential is defined as $d^{*}:=-\star d\star$; 2. ii)

the holomorphic codifferential is defined as $\partial^{*}:=-\star\overline{\partial}\star$; 3. iii)

the anti-holomorphic codifferential is defined as $\overline{\partial}^{*}=-\star\partial\star$.

Observe that for $\omega\in\Omega$,

[TABLE]

Now, it is natural to define the $d$-, $\partial$- and $\overline{\partial}$- Laplacians, respectively as

[TABLE]

Proposition 4.2.31**.**

The operator adjoints of $d$, $\partial$ and $\overline{\partial}$ are $d^{*}$, $\partial^{*}$ and $\overline{\partial}^{*}$, respectively.

The following will be used later.

Corollary 4.2.32**.**

The Laplacians $\Delta_{d}$, $\Delta_{\partial}$ and $\Delta_{\overline{\partial}}$ are symmetric.

We have:

Lemma 4.2.33**.**

The operator $d^{*}$ (respectively $\partial$, $\overline{\partial}$) and hence $\Delta_{d}$ (respectively $\Delta_{\partial}$, $\Delta_{\overline{\partial}}$) descends to $\Omega_{0}$.

Proof.

Since $\star$ is $H$-linear, we have for $h\in H_{0}$,

[TABLE]

Hence $d^{*}$ descends to $\Omega_{0}$. ∎

According to [MR3720811], Kähler structures are defined as follows.

Definition 4.2.34**.**

A Kähler structure is a hermitian structure $(\Omega^{(\cdot,\cdot)},\kappa)$ such that the hermitian form $\kappa$ is $d$-closed. Such a form is called a Kähler form.

Lemma 4.2.35**.**

A Kähler structure $(\Omega^{(\cdot,\cdot)},\kappa)$ on $(\Omega,\,d)$ induces via $\kappa$ a Kähler structure on $(\Omega_{0},\,d)$.

Proof.

Since $\kappa$ is automatically $d|_{\Omega_{0}}$-closed, the lemma follows from Lemma 4.2.22. ∎

4.3. The main example - Transversely Kähler foliations

In this subsection, we show how transversely Kähler foliations fit into our framework.

According to Haefliger [MR2456103]:

Definition 4.3.1**.**

A transverse structure on a foliated manifold $(M,{\mathcal{F}})$ is a structure on the transversal manifold $N$, invariant under the action of the holonomy pseudogroup $P$.

Since the groupoid $\Gamma(P)$ is constructed out of $P$, it follows that $P$ invariant structures are $\Gamma(P)$ invariant. The normal bundle $N(M,{\mathcal{F}})$ of the foliation ${\mathcal{F}}$ is isomorphic to the tangent bundle $TN$ of $N$. Thus, basic forms on the foliated manifold $(M,{\mathcal{F}})$ are in bijective correspondence with $\Gamma(P)$-invariant forms on the transverse manifold $N$ (see [MR2456103]). To see what does $\Gamma(P)$ invariant forms correspond to, we introduce the following.

Definition 4.3.2**.**

A local bisection of a Lie groupoid $G$ is a local section $\sigma:U\rightarrow G$ of $s:G\rightarrow G_{0}$ defined on an open subset $U\subset G_{0}$ such that $t\sigma$ is an open embedding.

If $G$ is étale, any arrow $g$ induces a germ of a homeomorphism $\sigma_{g}:(U,s(g))\rightarrow(V,t(g))$ from a neighborhood $U$ of $s(g)$ to a neighborhood $V$ of $t(g)$ as follows: choosing $U$ small enough such that a bisection $\sigma$ exists and $t|_{\sigma U}$ is a homeomorphism into $V:=t(\sigma U)$, we set $\sigma_{g}:=t\sigma$. We do not distinguish between $\sigma_{g}$ and the actual germ of this map at the point $s(g)$.

Lemma 4.3.3**.**

Let $G$ be an étale groupoid, and let $E$ be a smooth complex vector bundle over $G_{0}$ with a $G$-representation. Then a section $u:G_{0}\rightarrow E$ is $G$-invariant if and only if it is $C_{c}^{\infty}(G)$-invariant.

Proof.

Recall that a section $u$ of the bundle $E$ is $G$ invariant, if $g\cdot u(x)=u(y)$ for all arrow $x\xrightarrow{g}y$, while $u$ is $C_{c}^{\infty}(G)$ invariant if $a\cdot u=\varepsilon_{l}(a)u$ for all $a\in C_{c}^{\infty}(G)$. That $G$-invariance implies $C_{c}^{\infty}(G)$-invariance is clear. For the converse, pick an arrow $x\xrightarrow{g}y$ and a bisection $(U,\sigma)$ such that $g\in\sigma(U)$ [MR2012261]. Then choose any function $a\in C_{c}^{\infty}(G)$ with support in $\sigma(U)$ and $a(g)=1$. Note that on a bisection $\sigma(U)$, we have $a(t|_{\sigma(U)})^{-1}=\varepsilon_{l}(a)$ and $a\cdot u=a(t|_{\sigma(U)})^{-1}u=\varepsilon_{l}(a)u$. Hence the lemma follows. ∎

Now take $B=C^{\infty}(G_{0})$ and $\Omega=\Omega(G_{0})$, the ${\mathbb{C}}$-valued smooth functions and forms on $G_{0}$, respectively.

Lemma 4.3.4**.**

The differential $d$ on $G_{0}$ satisfies

[TABLE]

for $a\in C_{c}^{\infty}(G)$ and $\omega\in\Omega(G_{0})$. Hence $[a-\varepsilon_{l}(a),d]=0$ for all $a\in C_{c}^{\infty}(G)$, thus implying $H_{0}=C_{c}^{\infty}(G)$ (see (4.11) for $H_{0}$).

Proof.

As observed above in the proof of Lemma 4.3.3, on a bisection $\sigma(U)$, we have $a(t|_{\sigma(U)})^{-1}=\varepsilon_{l}(a)$ and $a\cdot u=a(t|_{\sigma(U)})^{-1}u=\varepsilon_{l}(a)u$. Now (4.24) follows from Leibniz rule and locality of $d$. The last statement follows from (4.24) and the fact that $s_{l}\equiv t_{l}$. ∎

Denote by $\Omega(G_{0})^{G}$ the $G$-invariant forms. Then forms on the “orbit or leaf space” are captured as follows.

Proposition 4.3.5**.**

The pair $(\Omega(G_{0}),\,d)$ is a $C_{c}^{\infty}(G)$-covariant differential calculus, and we have $(\Omega(G_{0})^{G},\,d)=(\Omega(G_{0})_{C_{c}^{\infty}(G)},\,d)$ as differential graded algebras.

Proof.

Since $G$ acts by local diffeomorphisms, it follows that $d$ is $G$-invariant. So $d$ descends to $\Omega(G_{0})^{G}$. The proposition now follows from Lemma 4.3.3 and Lemma 4.3.4. ∎

Definition 4.3.6**.**

An étale groupoid $G$ is oriented if $G_{0}$ is oriented in the ordinary sense and $G$ acts by orientation-preserving local diffeomorphisms.

Proposition 4.3.7**.**

With $B=C^{\infty}(G_{0})$ and $\Omega=\Omega(G_{0})$, orientation in the sense of Definition 4.2.13 coincide with groupoid orientation on $G$.

Proof.

This follows from Proposition 4.3.5. ∎

As in [MR1151583], we define:

Definition 4.3.8**.**

The foliation ${\mathcal{F}}$ on a foliated manifold $(M,{\mathcal{F}})$ is transversely holomorphic if it carries a transverse complex structure in the sense of Definition 4.3.1.

If the foliation ${\mathcal{F}}$ is transversely holomorphic, the normal bundle $N(M,{\mathcal{F}})$ of ${\mathcal{F}}$ has a complex structure corresponding to the complex structure on $N$. Therefore any complex-valued basic $k$-form can be represented as a sum of the $k$-forms of pure type $(r,s)$ corresponding to the decomposition of $k$-forms on the complex manifold $N$. Let $\Omega^{k}_{{\mathbb{C}}}(M,{\mathcal{F}})$ denote the space of complex-valued basic $k$-forms on the foliated manifold $(M,{\mathcal{F}})$, and denote by $\Omega^{(r,s)}_{{\mathbb{C}}}(M,{\mathcal{F}})$ the space of complex-valued basic forms of pure type $(r,s)$. Then we have $\Omega^{k}_{{\mathbb{C}}}(M,{\mathcal{F}})=\oplus_{r+s=k}\Omega^{(r,s)}_{{\mathbb{C}}}(M,{\mathcal{F}})$. The exterior derivative $d:\Omega^{k}_{{\mathbb{C}}}(M,{\mathcal{F}})\rightarrow\Omega^{k+1}_{{\mathbb{C}}}(M,{\mathcal{F}})$ decomposes into two components $d=\partial+\overline{\partial}$, where $\partial$ is of bidegree $(1,0)$ and $\overline{\partial}$ is of bidegree $(0,1)$, i.e., $\partial:\Omega^{(r,s)}\rightarrow\Omega^{(r+1,s)}$ and $\overline{\partial}:\Omega^{(r,s)}\rightarrow\Omega^{(r,s+1)}$.

Keeping in mind Definition 4.3.8 and the case for orbifolds (see [MR3751961]), we make

Definition 4.3.9**.**

An étale groupoid $G$ is holomorphic if $G_{0}$ is a complex manifold and $G$ acts by local biholomorphic transformations.

This fits into our framework as follows:

Proposition 4.3.10**.**

An étale groupoid $G$ is holomorphic if and only if $(\Omega(G_{0}),\,d)$ admits a $C_{c}^{\infty}(G)$-covariant complex structure.

Proof.

First observe that an almost complex structure on $G_{0}$ is also given by a bundle map $J:T^{*}(G_{0})\rightarrow T^{*}(G_{0})$ (and its extension to the exterior algebra bundle) such that $J^{2}=-\operatorname{id}_{T^{*}(G_{0})}$. The bidegree decomposition is a consequence of this fact. Since bundle maps are sections of the $\operatorname{HOM}$-bundle, $G$ is almost complex if and only if $(\Omega(G_{0}),\,d)$ admits a $C_{c}^{\infty}(G)$-covariant almost complex structure, by Lemma 4.3.3. Since integrability is same in both sense, we have the proposition proved. ∎

The orbit space inherits a complex structure:

Corollary 4.3.11**.**

If $G$ is holomorphic, then $(\Omega(G_{0})^{G},\,d)$ admits a complex structure.

Proof.

This follows from Proposition 4.3.10 and Lemma 4.2.18. ∎

We recall from [MR2012261]:

Definition 4.3.12**.**

The foliation ${\mathcal{F}}$ on a foliated manifold $(M,{\mathcal{F}})$ is transversely Riemannian if it carries a transverse Riemannian structure in the sense of Definition 4.3.1.

The metric on $N(M,{\mathcal{F}})$ is induced from a bundle-like metric on $M$. Recall from [MR1151583]:

Definition 4.3.13**.**

The foliation ${\mathcal{F}}$ on a foliated manifold $(M,{\mathcal{F}})$ is transversely hermitian if it carries a transverse hermitian structure in the sense of Definition 4.3.1.

The operator $\star:\Lambda^{k}(M,{\mathcal{F}})\rightarrow\Lambda^{2q-k}(M,{\mathcal{F}})$ defined via the transverse part of the bundle-like metric of ${\mathcal{F}}$ extends to $\Lambda^{k}_{{\mathbb{C}}}(M,{\mathcal{F}})\rightarrow\Lambda^{2q-k}_{{\mathbb{C}}}(M,{\mathcal{F}})$, where $q$ is the complex codimension of ${\mathcal{F}}$.

Being motivated by this, we make the following definition.

Definition 4.3.14**.**

An étale groupoid $G$ is hermitian if $G_{0}$ admits a $G$-invariant hermitian structure.

Again, algebraically we have the following proposition.

Proposition 4.3.15**.**

An étale groupoid $G$ is hermitian if and only if $(\Omega(G_{0}),\,d)$ admits a $C_{c}^{\infty}(G)$-covariant hermitian structure.

Proof.

The proof of the statement that $G$ is hermitian implies that $(\Omega(G_{0}),\,d)$ admits a $C_{c}^{\infty}(G)$-invariant hermitian structure is straightforward. For the converse, we recover the hermitian metric as in Definition 4.2.25, and Lemma 4.2.26 shows that it is $G$-invariant. Compatibility follows from Proposition 4.2.27. ∎

Corollary 4.3.16**.**

If $G$ is hermitian, then $(\Omega(G_{0})^{G},\,d)$ admits a hermitian structure.

Proof.

This follows from Proposition 4.3.15 and Lemma 4.2.22. ∎

Following [MR1151583], we have:

Definition 4.3.17**.**

The foliation ${\mathcal{F}}$ on a foliated manifold $(M,{\mathcal{F}})$ is transversely Kähler if it carries a transverse Kähler structure in the sense of Definition 4.3.1.

The Kähler form of $N$ defines a basic $(1,1)$-form on $(M,{\mathcal{F}})$ which is called the transverse Kähler form of the foliation ${\mathcal{F}}$. Motivated by this and the case for orbifolds, we define:

Definition 4.3.18**.**

An étale groupoid $G$ is Kähler if $G_{0}$ admits a $G$-invariant Kähler structure.

The following is routine:

Proposition 4.3.19**.**

An étale groupoid $G$ is Kähler if and only if $(\Omega(G_{0}),\,d)$ admits a $C_{c}^{\infty}(G)$-covariant Kähler structure.

Proof.

This follows from Proposition 4.3.15 and Proposition 4.3.5. ∎

Corollary 4.3.20**.**

If $G$ is Kähler, then $(\Omega(G_{0})^{G},\,d)$ admits a Kähler structure.

Proof.

This follows from Proposition 4.3.19 and Lemma 4.2.35. ∎

4.4. Left bialgebroid covariance of universal $1$-forms

Let $X$ be the finite set $\{1,\dots,n\}$. In this subsection, we sketch the constrution of a left bialgebroid over $C(X)$ whose action on $C(X)$ lifts to the space of universal one forms on $C(X)$, in the sense of Definition 4.2.3. For that, let us consider the left bialgebroid $C(X\times X\times X\times X)$ over $C(X\times X)$, constructed in the same way as above. We identify $C(X)$ as the first copy of the product $C(X)\otimes C(X)=C(X\times X)$. Let $f\in C(X)$ and $h\in C(X\times X\times X\times X)$. Then

[TABLE]

So a sufficient condition that $h$ takes $f\otimes 1$ to an element of the same form is that $\sum_{w}h(z,w,x,y)$ does not depend on $y$. Therefore we can write

[TABLE]

Now the space of universal one forms can be identified with functions on $X\times X$ vanishing on the diagonal. If $\sum_{i}f_{i}\otimes g_{i}$ is such an element then

[TABLE]

Thus a sufficient condition that $h$ preserves this space is that

[TABLE]

for $z\neq w$. Observe that, (4.28) together with (4.26) imply

[TABLE]

A sufficient condition on $h$ such that $[h-s\varepsilon(h),d]=0$ holds is

[TABLE]

Let $H_{0}$ be the set of all $h\in C(X\times X\times X\times X)$ such that

1. i)

$\sum_{w}h(z,w,x,y)=h(z,z,x,x);$ 2. ii)

$h(z,w,x,x)=0$ for $z\neq w;$ 3. iii)

$\sum_{z}h(z,w,x,y)=h(w,w,y,y);$

And let $H$ be the smallest subalgebra of $C(X\times X\times X\times X)$ containing $C(X)$ and $H_{0}$.

Proposition 4.4.1**.**

$H$* is a left bialgebroid over $C(X)$ such that the action on $C(X)$ lifts to an action on the space of universal one forms in the sense of Definition 4.2.3. *

Conjecture 4.4.2**.**

$H$* is not of the form $C(X)\#Q$ for any Hopf algebra $Q$ acting on $C(X)$.*

Remark 4.4.3**.**

We remark that the space $H_{0}$ is at least $n^{2}$-dimensional and hence $H$ is at least $n^{3}$-dimensional, sufficiently large for our purposes.

Question 4.4.4**.**

Give a nice characterization/description of $H$.

Finally, we ask a question which is not directly related to this work but interesting in its own right. In [MR3777415], it is shown that a coaction of a compact quantum group on an algebra can be lifted to a differential calculus (at least in the classical situation) under some suitable (unitarity of the coaction, technically, see also (4.19)) conditions, like one expects from a group action. So we ask

Question 4.4.5**.**

Is the above true for unitary action (i.e., (4.19) is satisfied) of a Hopf algebroid?

Remark 4.4.6**.**

We have computed the complex structures on a three-point space in [three], none of which are covariant with respect to the natural $S_{3}$-action. It would be interesting to investigate covariance of the Hopf algebroid constructed in this subsection.

4.5. Covariant complex structures on toral deformed manifolds

In this subsection, we sketch the construction of a natural Hopf algebroid and a covariant complex structure on toral deformed manifolds a la Connes, Dubois-Violette, Landi, and Rieffel. This provides a rather large class of genuine (non-classical) examples of Hopf algebroid covriant complex structures.

Let $(M,T)$ be a toral manifold, by which we mean a compact manifold together with a smooth action of a torus $T$. We shall consider toral deformations of $M$ in the sense of [MR1846904, MR1937657], see also [MR1184061] for the analytic aspects. Making repeated use of Lemma 7.18 and Lemma 7.19 of [bgm], we obtain

Proposition 4.5.1**.**

Let $H$ be a Hopf algebroid over $C^{\infty}(M)$ and let $T$ act on $H$ in such a way so that all the structure maps are $T$-equivariant. Then $H_{\theta}$ is a Hopf algebroid over $C^{\infty}(M)_{\theta}$.

As a concrete example, let $\mathcal{D}(M)$ be the algebra of differential operators. It is well-known (see [MR1815717]) that $\mathcal{D}(M)$ is a Hopf algebroid. By the Proposition above, we obtain

Proposition 4.5.2**.**

$\mathcal{D}(M)_{\theta}$* is a Hopf algebroid over $C^{\infty}(M)_{\theta}$.*

Since a differential operator is given by vector fields locally, and since Lie derivative with respect to a vector field commutes with the exterior derivative $d$, we see that the canonical action of $\mathcal{D}(M)$ on $\Omega^{*}(M)$ satisfies our definition, where we take $H_{0}$ (as in Definition 4.2.3, Equation 4.11) to be the set of $L_{X}$, $X$ being a vector field.

Proposition 4.5.3**.**

$\Omega^{*}(M)_{\theta}$* is a $\mathcal{D}(M)_{\theta}$-covariant differential calculus in the sense of Definition 4.2.6.*

Proof.

It follows from the general construction that $\Omega^{*}(M)_{\theta}$ is indeed a differential calculus over $C^{\infty}(M_{\theta})$. Now since the $T$-action keeps the space $H_{0}$ invariant, the result follows from a totality argument similar to Lemma 7.28 of [bgm]. ∎

By considering complex conjugation, we see that $\Omega^{*}(M)_{\theta}$ is even a $\mathcal{D}(M)_{\theta}$-covariant $\ast$-differential calculus. Before introducing complex structure in this example, let us summarize the above facts as follows:

Theorem 4.5.4**.**

Let $(M,T)$ be a toral manifold and let $\mathcal{D}(M)$ be the algebra of differential operators on $M$. Then $\Omega^{*}(M)_{\theta}$ is a $\mathcal{D}(M)_{\theta}$-covariant $\ast$-differential calculus over $C^{\infty}(M)_{\theta}$.

Now let $(M,T)$ be a complex toral manifold, by which we mean that $M$ is a complex manifold (with equivariant complex structure $J$), $T$ is a complex torus and the action $T\times M\rightarrow M$ is holomorphic. Let $\mathcal{D}^{\mathrm{hol}}(M)$ be the subalgebra of $\mathcal{D}(M)$ generated by $C^{\infty}(M)$ and differential operators $D$ such that $D(J)=0$. For example, if $X$ is a holomorphic vector field, then $L_{X}$ is such an operator. It is well-known that such a $D$ preserves the $(p,q)$-decomposition (the ${\mathbb{N}}_{0}^{2}$-grading) of $\Omega^{\ast}(M)$.

Proposition 4.5.5**.**

$\mathcal{D}^{\mathrm{hol}}(M)$* is a Hopf algebroid over $C^{\infty}(M)$ and the complex structure on $M$ is a $\mathcal{D}^{\mathrm{hol}}(M)$-covariant complex structure on $\Omega^{*}(M)$.*

Remark 4.5.6**.**

The ${\mathbb{N}}_{0}^{2}$-grading comes from the eigenspaces of $J$ as in [MR3073899].

The holomorphicity of the toral action yields

Proposition 4.5.7**.**

$\mathcal{D}^{\mathrm{hol}}(M)_{\theta}$* is a Hopf algebroid over $C^{\infty}(M)_{\theta}$ and $J_{\theta}$ is a $\mathcal{D}^{\mathrm{hol}}(M)_{\theta}$-covariant complex structure on $\Omega^{*}(M)_{\theta}$.*

Summarizing all these,

Theorem 4.5.8**.**

Let $(M,T)$ be a complex toral manifold and let $\mathcal{D}^{\mathrm{hol}}(M)$ be the algebra of differential operators on $M$ defined as above. Then $\Omega^{*}(M)_{\theta}$ is a $\mathcal{D}^{\mathrm{hol}}(M)_{\theta}$-covariant $\ast$-differential calculus over $C^{\infty}(M)_{\theta}$ and $J_{\theta}$ is a $\mathcal{D}^{\mathrm{hol}}(M)_{\theta}$-covariant complex structure.

As a concrete example, let us consider $\mathbb{T}^{2n}$ acting on itself by translation. Then

Theorem 4.5.9**.**

$\mathcal{D}^{\mathrm{hol}}(\mathbb{T}^{2n})_{\theta}$* is a Hopf algebroid over $\mathbb{T}^{2n}_{\theta}$ and $\Omega^{*}(\mathbb{T}^{2n})_{\theta}$ is a $\mathcal{D}^{\mathrm{hol}}(\mathbb{T}^{2n})_{\theta}$-covariant differential calculus over $\mathbb{T}^{2n}_{\theta}$ together with a $\mathcal{D}^{\mathrm{hol}}(\mathbb{T}^{2n})_{\theta}$-covariant complex structure.*

5. Further directions and comments

We end this paper by discussing some directions that we have not touched upon.

5.1. Comparison with Connes’ approach

In [MR679730, MR823176, MR866491], the approach taken to study singular spaces, in particular, the leaf space of a foliation is as follows. One models the singular space by a groupoid $G$ and then considers the convolution algebra $C_{c}^{\infty}(G)$ as the function algebra of the space in question. We have considered the groupoid here also, but as symmetries. To consider noncommutative complex geometry on the singular space, we need a differential calculus on the algebra $C_{c}^{\infty}(G)$. Here there are many choices and it is a priori not clear what is the correct choice to make. In fact, if one takes a discrete group and view it as a groupoid then the convolution algebra is the group algebra and we don’t know what a choice of differential calculus would be (neither the universal one nor a bicovariant one), let alone the study of noncommutative complex structure and the meaning of it. So before moving onto arbitrary groupoids, one needs to answer the following question.

Question 5.1.1**.**

Construct (or even classify) differential calculi on the group algebra ${\mathbb{C}}\Gamma$ of a discrete group $\Gamma$. Are there any complex structures on it? If so, what does it mean to have a complex structure on ${\mathbb{C}}\Gamma$?

5.2. Comparison with Fröhlich et al.’s approach

In [MR1614993], they study spectral data associated to hermitian, Kähler structure. [MR3720811] already mentions this and it is being taken up by him and collaborators [2019arXiv190307599D]. We sketch this in our set up. Note that $H$ is represented on $L^{2}(\Omega)$ by unbounded operators with common domain $\Omega$. We first show that these operators are closable, by exhibiting densely defined adjoint operators. Taking ideas from [MR2817646], we exploit the $(A_{r},A_{r})$-bimodule structure on $\Omega\otimes_{B}\overline{\Omega}$ which is given by (3.1) via $\theta^{-1}:A_{r}\rightarrow A_{l}^{op}$; explicitly,

[TABLE]

for $a_{1},a_{2}\in A_{r}$ and $\omega,\eta\in\Omega$. We assume that the faithful state $\tau$ used to define the inner product (4.21) is right invariant, i.e.,

[TABLE]

for $h\in H$ and $b\in B$. We have the following lemma.

Lemma 5.2.1**.**

For $\omega,\eta\in\Omega$ and $h\in H$,

[TABLE]

holds, where $g$ is as in Definition 4.2.25. Thus $\langle h\cdot\omega,\eta\rangle=\langle\omega,(S^{2}(h))^{*}\cdot\eta\rangle.$

Proof.

The proof is essentially contained in [MR2817646]. We compute

[TABLE]

The last statement follows from the definition of $H$-action on $\overline{\Omega}$. ∎

Thus $H$ is represented by closable operators having a common dense domain. We denote the adjoint of $h\in H$ by $h^{\dagger}$ so that $h^{\dagger}=(S^{2}(h))^{*}$ on $\Omega$. From now on, let us allow a notational abuse of denoting by $h$ both the operator on $\Omega$ and its closure in $L^{2}(\Omega)$. At this point, we make an additional regularity assumption:

Assumption

Given $h\in H$, ${\mathcal{D}}_{h}=\{\omega\in\Omega\mid\sum_{0}^{\infty}\frac{\|h^{n}\omega\|}{n!}<\infty\}$ is dense in $L^{2}(\Omega)$.

Lemma 5.2.2**.**

For $h\in H$ with $h=h^{\dagger}$ and $\omega\in{\mathcal{D}}_{h}$, define $U_{h}$ by

[TABLE]

which is well-defined by the above $\mathbf{Assumption}$. Then $U_{h}$ extends to a unitary operator on $L^{2}(\Omega)$ denoted by $e^{ih}$.

Proof.

The result follows from the observations that for such an $h$, ${\mathcal{D}}_{h}={\mathcal{D}}_{-h}$ and that $U_{h}U_{-h}=U_{-h}U_{h}=\operatorname{id}$. ∎

Lemma 5.2.3**.**

If the commutator $[h,d+d^{*}]$ extends to a bounded operator on $L^{2}(\Omega)$, then so does $[e^{ih},d+d^{*}]$.

Proof.

Observe that

[TABLE]

As $e^{ith}$ is unitary, the integrand is bounded and the result follows. ∎

Combining Lemma 5.2.2 and Lemma 5.2.3, we get the following proposition.

Proposition 5.2.4**.**

Let ${\mathcal{A}}$ be the $\ast$-algebra generated by operators of the form $ae^{i(h+h^{\dagger})}b$ with $a,b\in A_{l}$ and $h\in H_{0}$ in $B(L^{2}(\Omega))$. Then $({\mathcal{A}},L^{2}(\Omega),d+d^{*})$ forms a spectral triple.

Proof.

We first observe that the representation of $A_{l}$ on $L^{2}(\Omega)$ is induced by restricting through $A_{l}\rightarrow B$, $a\mapsto s_{l}(a)\cdot 1_{B}$. For $b\in B$ and $\omega\in\Omega$,

[TABLE]

implying that left multiplication by $b$ extends to a bounded operator. In the above estimate, the inequality comes from the fact that $g(\omega,\omega)$ is positive in $B$.

Next, observing that $[s_{l}(a),d]$ is left multiplication by $d(s_{l}(a)\cdot 1_{B})$, we prove that left multiplication by $d(s_{l}(a)\cdot 1_{B})$ is bounded on $L^{2}(\Omega)$. Again we do this for $b\in B$. Note that

[TABLE]

The first term in the above expression is estimated as follows:

[TABLE]

where we used Cauchy-Schwarz inequality and that left multiplication by $b$ is bounded. The second term is estimated as follows:

[TABLE]

where we again used the boundedness of left multiplication by $b\in B$. Combining the two, we conclude that left multiplication by $db$ is bounded on $L^{2}(\Omega)$. Now observe that $s_{l}(a)\cdot 1_{B}$ is adjointable (see the discussion after Lemma 5.2.1) with adjoint again an element of $B$ (use Eq. (2.12) and compute: $(S^{2}(s_{l}(a)))^{*}=(Ss_{r}\varepsilon_{r}s_{l}(a))^{*}=(s_{l}\varepsilon_{l}s_{r}\varepsilon_{r}s_{l}(a))^{*}=s_{l}((\varepsilon_{l}s_{r}\varepsilon_{r}s_{l}(a))^{*})$) and for such an element the adjoint of $[d,b]$ is precisely $-[d^{*},b^{\dagger}]$, $b^{\dagger}$ is the adjoint of $b$. Since adjoint of $[d,b]$ is bounded, we can conclude that $[b,d+d^{*}]$ extends to a bounded operator on $L^{2}(\Omega)$.

Finally, $[s_{l}(a),d+d^{*}]$ extends to a bounded operator yields that $[h,d+d^{*}]$ too extends to a bounded operator for $h\in H_{0}$. The result follows from 5.2.3. ∎

If we assume that $\Delta_{d}$ has purely discrete spectrum then we get a spectral triple of compact type. We also note that [MR3720811] computes the spectrum for the concrete examples. In our abstract setup, we propose a way of doing it generally. It would be interesting to know the answer of the following:

Question 5.2.5**.**

If we assume an analogue of Relich’s lemma $H^{k}\hookrightarrow H^{k+2}$ is compact then does it follow that $\Delta_{d}$ has purely discrete spectrum? See [MR2277669] for the setup and more on abstract pseudo-differential calculi which has motivated this question.

This would give a uniform way of proving that the Laplacian $\Delta_{d}$ has purely discrete spectrum in the setting of noncommutative differential calculi.

5.3. Further examples

As examples for our framework, we have mentioned étale groupoids, Hopf algebras, weak Hopf algebras and the Connes-Moscovici Hopf algebroid. There is another class of examples coming from Lie-Rinehart algebras and associated jet spaces; see [MR2817646]. It would be interesting to know the answer of the following

Question 5.3.1**.**

Investigate if these examples fit into our framework. If so, what is the meaning of having a complex structure on a Lie-Rinehart algebra?

As alluded to in the Introduction, Hopf algebroids appear naturally as symmetries of subfactors. A well-explored rich source of subfactors is the theory of conformal nets (\citelist[MR3966864][MR1491122][MR3796433]). Moreover, there has been some work ([MR2585992]) on construction of spectral triples, i.e., noncommutative manifold structures, on ($C^{*}$-) algebras associated with conformal nets. Under certain conditions, the corresponding von Neumann algebras give rise to subfactors and as Hopf algebroids occur naturally as symmetries (\citelist[MR1913440][galois]) of such subfactors, it is plausible that one may be able to construct noncommutative complex/Hermitian/Kähler structures on the module of one-forms of such spectral triples which will have equivariance with respect to some nontrivial and interesting Hopf algebroids. We have plans to explore this idea in our future work.

Furthermore, Hopf algebroid equivariance is not unexpected from another related viewpoint of [MR1815717], in which the author argues as follows. At the space level, one knows quantization of a Poisson manifold yields a noncommutative space and at the group level, one obtains quantum groups. If one agrees to view a groupoid as the joint generalization of a space and a group, then what should a quantization of a groupoid yield? The answer is a quantum groupoid or dually a Hopf algebroid, as shown in [MR1815717]. In fact, it is shown therein that the algebra of differential operators on a Poisson manifold quantize to a canonical Hopf algebroid over the deformed function algebra. We also have plans to pursue this harmony between Hopf algebroid equivariance and deformation quantization.

Let us end this article by noting that we only consider a fixed pseudogroup of local biholomorphisms of a given complex structure. It is indeed interesting and difficult to classify the complex structures, given a fixed set of local biholomorphisms. In the noncommutative setting, this amounts to the same thing as fixing a Hopf algebroid and investigating the moduli of all covariant complex structures. For quantum groups and spectral triples, these investigations have already been taken up, see [MR2036597, MR1995873, MR2174423]. Our modest approach does not answer these questions and it would be very interesting to answer them.

Acknowledgments

The first author is grateful to Aritra Bhowmick for several discussions on foliations, to Yuri Kordyukov for answering many questions, to Edwin Beggs and Shahn Majid for helpful comments, and finally to Réamonn Ó Buachalla for countless discussions and his interest in this work. He also thanks the second author for answering many questions on complex geometry and the third author for introducing him to the theory of foliations. The second author is partially supported by J.C. Bose National Fellowship. The third author is partially supported by J.C. Bose National Fellowship and Research Grant awarded by D.S.T. (Govt. of India).