# On Lie algebroid over algebraic spaces

Ashis Mandal and Abhishek Sarkar

###### Abstract.

We consider Lie algebroids over algebraic spaces (in short we call it as
$a$-spaces) by considering the sheaf of Lie-Rinehart algebras. We discuss
about properties of universal enveloping algebroid
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ of a Lie algebroid $\mathcal{L}$
over an $a$-space $(X,\mathcal{O}_{X})$. This is done by sheafification of the
presheaf of universal enveloping algebras for Lie-Rinehart algebras. We review
the extent to which structure of the universal enveloping algebroid of Lie
algebroids (over special $a$-spaces) resembles a sheaf of bialgebras. In the
sequel we present a version of Poincaré-Birkhoff-Witt theorem and Cartier-
Milnor-Moore theorem for the Lie algebroid.

###### Key words and phrases:

Lie algebroids, Lie-Rinehart algebras, tangent sheaf, universal enveloping
algebra

111AMS Mathematics Subject Classification : $17$B$35$, $16$T$10$.

##  1\. Introduction

The notion of Lie algebroid plays a prominent role in geometry as generalized
infinitesimal symmetries of spaces related to the corresponding global
symmetries of spaces, called Lie groupoid (see [14]). Lie algebroid over a
smooth manifold is a joint generalization of tangent vector bundle over the
manifold and Lie algebras, whose algebraic analogue is known as Lie-Rinehart
algebra [2, 9, 17].

There are many instances for consideration of spaces obtained as the solution
space of a system of first order partial differential equations (generalized
involutive distribution), the vanishing set of a system of holomorphic
functions in complex geometry and the vanishing set of a system of polynomials
in algebraic geometry. At places, the solution space may not be a smooth
object and it is considered as basic analytic space or affine variety. In the
presence of singularity, an analogue of tangent bundle is the tangent sheaf.
It is the sheaf of derivations over its structure sheaf. Another important
sheaf associated to these spaces is the sheaf of logarithmic vector fields
(vector fields tangent to these subspaces). In both cases we find the common
underlying structure of coherent sheaf of Lie-Rinehart algebras, which we
consider as a Lie algebroid. The Lie algebroid over analytic spaces and over
algebraic varieties are joint generalization of the tangent sheaf [2, 18, 19]
and the sheaf of Lie algebras. We consider Lie algebroids over algebraic
spaces $(X,\mathcal{O}_{X})$ (in short $a$-spaces) where $\mathcal{O}_{X}$ is
a sub-sheaf of algebras of the sheaf of continuous functions $C^{0}$ on $X$,
it combines the three types of base spaces as special cases [16, 19].

Unlike to the case of (smooth) vector bundles over a smooth manifold, the
space of global sections does not capture whole information for a holomorphic
(algebraic) vector bundle over a complex manifold (algebraic variety). In the
later case, we need to consider sheaf of sections [9, 22] of the corresponding
vector bundles. The necessity is mostly due to lack of the notion of bump
functions and partition of unity, which is present automatically for the case
of real vector bundles over real smooth manifold. One requires a sheaf-
theoretic approach for analytic and algebraic geometry, even for not
necessarily singular cases. In doing global calculus over real smooth
manifolds, the sheaf-theoretic approach opens up a new way of doing classical
differential geometry [19]. But in the other cases it becomes mandatory except
special situations like Stein manifolds and affine varieties.

The universal enveloping algebra of a Lie-Rinehart algebra is an associative,
non-commutative algebra (except the trivial case when it become symmetric
algebra). In [15], I. Moerdijk and J. Mrčun showed that the universal
enveloping algebra of a $(\mathbb{K},R)$-Lie-Rinehart algebra has a
$R/{\mathbb{K}}$-bialgebra structure. As a special case they deduced
$C^{\infty}(M)/{\mathbb{R}}$-bialgebra structure for Lie algebroid over a
smooth manifold $M$. As in the universal enveloping algebra of a Lie algebra,
the bialgebra is equipped with a canonical ascending filtration and by using
the co-algebra structure we get an ascending primitive filtration which
coincides for the case of projective Lie-Rinehart algebras. They extended the
Cartier-Milnor-Moore theorem from Lie algebras to Lie algebroids over smooth
manifolds and more generally for projective Lie-Rinehart algebras.

In this paper we discuss properties of universal enveloping algebroid denoted
by $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ of a Lie algebroid $\mathcal{L}$
over an $a$-space $(X,\mathcal{O}_{X})$. Here
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ is obtained by sheafification of
the presheaf of universal enveloping algebras of Lie-Rinehart algebras
associated with the space of sections of $\mathcal{L}$. In section $2$, we
define the notion of Lie algebroids over algebraic spaces. Then in section
$3$, we present a version of Poincare-Birkhoff-Witt theorem for universal
enveloping algebroid $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ of a Lie
algebroid $\mathcal{L}$ over $(X,\mathcal{O}_{X})$. In section $4$, we
consider the sheaf of logarithmic differential operators and Weyl algebra
associated with some principal divisors, namely nilpotent cone and normal
crossing divisor. Next, in section $5$, we present a sheaf of generalized
bialgebra structure on $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ which we
call $\mathcal{O}_{X}/{\mathbb{K}_{X}}$\- bialgebra. In section $6$, we
present a version of Cartier-Milnor-Moore theorem for locally free Lie
algebroids over some special type of $a$-spaces. This yields a categorical
equivalence between special subcategories of the Lie algebroids and of the
generalized sheaf of bialgebras over special types of $a$-spaces.

##  2\. Lie algebroids over special $a$-spaces

###  2.1. Algebraic spaces or $a$-spaces

A class of objects in geometry, for example real $C^{k}$-manifolds, analytic
spaces, algebraic varieties, schemes all are described as locally ringed
spaces [22]. Here, we consider a special type of it which is called algebraic
spaces and in short we like to refer as $a$-spaces (see [16]). It facilitates
to study all the three kinds of geometry (smooth, analytic and algebraic) in a
common framework. The notation $\mathbb{K}$ is used for $\mathbb{R,C}$ (the
real or complex number fields respectively) or a general algebraically closed
field of characteristic zero. We denote by $\mathbb{K}_{X}$, the constant
sheaf on a base space $X$ with stalks being isomorphic to $\mathbb{K}$. The
sheaf of $\mathbb{K}$-valued continuous functions on $X$ is denoted by
$C^{0}$.

######  Definition 2.1.

Algebraic space or $a$-space: Let $X$ be a topological space and
$\mathcal{O}_{X}$ be a $\mathbb{K}_{X}$-subalgebra of the sheaf of
$\mathbb{K}$-algebras $C^{0}$. In other words, let $\mathcal{O}_{X}$ be a
sheaf of $\mathbb{K}$-algebras where for an open set U $\subset$ X we assign
subalgebra $\mathcal{O}_{X}(U)$ of the algebra $C^{0}(U)$ and the assignment
is compatible with restriction morphisms.

The pair $(X,\mathcal{O}_{X})$ is said to be an algebraic space over the
topological space $X$ or, simply $(X,\mathcal{O}_{X})$ is an $a$-space.

In most of the cases, we consider an $a$-space $(X,\mathcal{O}_{X})$ where
$\mathcal{O}_{X}$ is the structure sheaf on $X$.

A morphism
$(\mathcal{\phi,\phi^{*}}):(X,\mathcal{O}_{X})\Longrightarrow(Y,\mathcal{O}_{Y})$
between two $a$\- spaces is made up by a continuous map $\phi:X\longrightarrow
Y$ and a family of ring homomorphisms

$\phi_{V}^{*}:\mathcal{O}_{Y}(V)\longrightarrow\mathcal{O}_{X}(\phi^{-1}(V))$

$g\longmapsto g\circ{\phi}|_{\phi^{-1}(V)}$,

where $V$ is an open subset of $Y$ and $g\in\mathcal{O}_{Y}(V)$. All maps
commute with inherent restriction maps for the sheaves.

######  Remark 2.2.

An appropriate choice for the structure sheaf $\mathcal{O}_{X}$ provides
various examples of $a$-spaces $(X,\mathcal{O}_{X})$ coming from smooth
manifolds, complex manifolds, analytic spaces and algebraic varieties.

Tangent Sheaf : The tangent sheaf $\mathcal{T}_{X}$ on a smooth manifold
$(X,C^{\infty})$ is defined by
$\mathcal{T}_{X}(U):=Der_{\mathbb{R}}(C^{\infty}(U))$ for any open subset $U$
of $X$. Elements of $\mathcal{T}_{X}(U)$ are linear operators on
$C^{\infty}(U)$ satisfying Leibniz rule, which are equivalent to smooth vector
fields on $U$.

Analogous consideration holds for Stein spaces and affine varieties [22], in
view of the Cartan-Serre theorem [8]. But, in a more general case for any
analytic space or algebraic variety $(X,\mathcal{O}_{X})$, firstly we need to
assign $U\mapsto Der_{\mathbb{K}}(\mathcal{O}_{X}(U))$ for each Stein open set
or affine open set $U$ and it induces a presheaf. By considering
$X=\cup_{i}U_{i}$ with an open Stein or affine cover, we get a canonical
surjective homomorphisms $Der_{\mathbb{K}}(\mathcal{O}_{X}(U_{i}))\rightarrow
Der_{\mathbb{K}}(\mathcal{O}_{X,x})$ of $\mathbb{K}$-vector spaces for each
$x\in U_{i}$. For an open set $U(=\cap_{i}U\cap U_{i})$, we define restriction
morphism from $U_{i}$ to $U\cap U_{i}$. A section $\tilde{D}$ on $U$ is given
by a collection of sections $D_{i}$ associated with good open sets $U\cap
U_{i}$ and using this we can define restriction for the above presheaf.
Consequently, the sheaf $\mathcal{D}er_{\mathbb{K}_{X}}(\mathcal{O}_{X})$ is
obtained by sheafification of this presheaf.

In view of this, the tangent sheaf over an $a$-space $(X,\mathcal{O}_{X})$ is
given by $\mathcal{D}er_{\mathbb{K}_{X}}(\mathcal{O}_{X})$\- the sheaf of
derivations over $\mathcal{O}_{X}$. It has a canonical structures of a
$\mathbb{K}_{X}$-Lie algebra and an $\mathcal{O}_{X}$-module with the
compatibility condition via Leibniz rule.

The Cartan-Serre Theorem states that for any coherent analytic (or algebraic)
sheaf $\mathcal{F}$ on a Stein space (or affine variety) $X$ the stalk
$\mathcal{F}_{x}$ of $\mathcal{F}$ at a point $x\in X$ is generated as an
$\mathcal{O}_{X,x}$-module by global sections of the sheaf $\mathcal{F}$.
Similar results holds for a coherent $\mathcal{O}_{X}$-module on a smooth
manifold $X$.

Next, we consider Lie algebroids over algebraic spaces as a sheaf of Lie-
Rinehart algebras equipped with a morphism to the tangent sheaf on $a$-spaces.
Let ${R}$ be a commutative associative algebra over a field $\mathbb{K}$, then
the set $Der_{\mathbb{K}}({R})$ of all $\mathbb{K}$-derivations of ${R}$, is a
$\mathbb{K}$-Lie algebra as well as an ${R}$-module.

A $(\mathbb{K},R)$-Lie-Rinehart algebra $L$ is a Lie algebra $L$ (over
$\mathbb{K}$) with an $R$-module structure and an $R$-module homomorphism
$\rho:L\rightarrow Der_{\mathbb{K}}(R)$ such that

  * •

the map $\rho:L\rightarrow Der_{\mathbb{K}}(R)$ is simultaneously an
$R$-module homomorphism and a $\mathbb{K}$-Lie algebra homomorphism and

  * •

$[x,ry]=r[x,y]+\rho(x)(r)y~{}~{}\mbox{ for all}~{}x,y\in L,~{}r\in R.$

######  Definition 2.3.

Lie algebroids over an $a$-space : A Lie algebroid $\mathcal{L}$ over an
$a$-space $(X,\mathcal{O}_{X})$ is a sheaf of
$(\mathbb{K}_{X},\mathcal{O}_{X})$-Lie-Rinehart algebras.

That is $\mathcal{L}$ is a sheaf of $\mathbb{K}$-Lie algebras and
$\mathcal{O}_{X}$-modules equipped with a morphism of
$\mathcal{O}_{X}$-modules
$\mathfrak{a}:(\mathcal{L},[\cdot,\cdot])\rightarrow(\mathcal{D}er_{\mathbb{K}_{X}}(\mathcal{O}_{X}),[\cdot,\cdot]_{c})$,
called the anchor map. The map $\mathfrak{a}$ is also a morphism of sheaves of
$\mathbb{K}$-Lie algebras and satisfying the Leibniz rule:
$[D,fD^{\prime}]=f[D,D^{\prime}]+\mathfrak{a}(D)(f)D^{\prime}$ for every
sections $f\in\mathcal{O}_{X}$ and $D,D^{\prime}\in\mathcal{L}$ .

We denote this Lie algebroid as $(\mathcal{L},[\cdot,\cdot],\mathfrak{a})$ or
simply by $\mathcal{L}$.

######  Remark 2.4.

Lie algebroids over a smooth manifold $X$ can be viewed by this general set up
of Lie algebroids over the $a$-space $(X,C^{\infty})$.

A morphism of Lie algebroids over an $a$-space

$\phi:(\mathcal{L}_{1},[\cdot,\cdot]_{1},\mathfrak{a}_{1})\rightarrow(\mathcal{L}_{2},[\cdot,\cdot]_{2},\mathfrak{a}_{2})$

is a sheaf homomorphism of Lie-Rinehart algebras.

The sheaves of $(\mathbb{K}_{X},\mathcal{O}_{X})$-Lie-Rinehart algebras along
with the sheaf homomorphisms of $(\mathbb{K}_{X},\mathcal{O}_{X})$-Lie-
Rinehart algebras form a category. We call it the category of Lie algebroids
over the $a$-space $(X,\mathcal{O}_{X})$.

######  Theorem 2.5.

Let $(X,\mathcal{O}_{X})$ be an $a$-space. There is a one-to-one
correspondence between vector bundles over $X$ (in the respective category)
and locally free sheaves of $\mathcal{O}_{X}$-modules of finite rank.

In fact, the correspondence mentioned above is an equivalence of categories
(see [18, 19, 22]). The classical cases of Lie algebroids can be viewed now as
Lie algebroids over $a$-spaces which are locally free sheaves of
$\mathcal{O}_{X}$-modules with finite rank. Here, we take the $a$-spaces
$(X,\mathcal{O}_{X})$ as smooth manifold, complex manifold, analytic space,
algebraic variety respectively, all are with their associated structure sheaf.

In case of $X$ is a $C^{k}$-manifold with $k\in\mathbb{N}$, the tangent sheaf
$\mathcal{D}er_{\mathbb{K}_{X}}(\mathcal{O}_{X})$ is trivial. We do not
consider this in our discussion.

######  Note 2.6.

For a finitely generated (regular) commutative algebra $R$ over $\mathbb{K}$
we have the affine scheme $X=Spec(R)$ with its structure sheaf of regular
functions $\mathcal{O}_{X}$ induced by localization of the ring $R$. On
extending the localization ideas from rings to the modules, we find that
$\mathcal{D}er_{\mathbb{K}_{X}}(\mathcal{O}_{X})$ is the induced sheaf by
localization of the $R$-module $Der_{\mathbb{K}}(R)$ with standard
$(\mathbb{K},R)$-Lie-Rinehart algebra structure. It is a locally finitely
presented (free) sheaf of $\mathcal{O}_{X}$-module.

Lie algebroids over schemes (Noetherian, separated or finite type) have been
studied in the literature [2, 4, 1, 9]. But, approaches for proofs of
necessary theorems can be different for the more general set up.

It is well known that by considering the space of global sections, there is
special Lie-Rinehart algebra with finitely generated projective module
structure being associated with a Lie algebroid in the classical case (for
base spaces are smooth manifolds, Stein manifolds and affine varieties) [3].
On the other hand, we recall that for a $(\mathbb{K},R)$-Lie-Rinehart algebra
$L$ (finitely generated projective module), we can associate a Lie algebroid
(locally free sheaf of finite rank) over the affine scheme $Spec(R)$, which is
induced from the localization [22] of $L$ as an $R$-module, whose space of
global sections become $L$.

######  Example 2.7.

For a real smooth manifold (complex manifold) as an $a$-space
$(X,\mathcal{O}_{X})$, the sheaf of smooth (holomorphic) vector fields is
isomorphic to the sheaf of Lie-Rinehart algebras
$(\mathcal{D}er_{\mathbb{K}_{X}}(\mathcal{O}_{X}),[\cdot,\cdot]_{c},id)$. It
is a locally free Lie algebroid of rank equals to dimension of the manifold
$X$.

The sheaf of differential $1$-forms $\Omega^{1}_{X}$ over a Poisson manifold
$X$ has a canonical Lie algebroid structure. For smooth and analytic cases the
details can be found in [7] and [18] respectively.

######  Example 2.8.

A singular foliation on a manifold $(X,\mathcal{O}_{X})$ is a sub-sheaf
$\mathcal{F}$ of the sheaf $\mathcal{T}_{X}$ of $\mathcal{O}_{X}$-modules,
which is $(i)$ stable under the Lie bracket and $(ii)$ locally finitely
generated. It provides a generalized involutive distribution on $X$, forms a
Lie algebroid over $(X,\mathcal{O}_{X})$ (see [12, 7, 18]).

######  Example 2.9.

Let $\mathbb{K}$ be an algebraically closed field (characteristic zero) and
$X=\mathbb{A}^{n}$ be the affine $n$-space. Consider an (irreduciable) affine
algebraic set $Y=V(I)\subset\mathbb{A}^{n}$ with its co-ordinate ring of
regular functions $\mathcal{O}(Y):=\mathbb{K}[x_{1},\dots,x_{n}]/I$, where $I$
is an (prime) ideal in the polynomial algebra $\mathbb{K}[x_{1},\dots,x_{n}]$.
We have $\mathfrak{X}(Y):=Der_{\mathbb{K}}(\mathcal{O}(Y))$, the space of
derivations over $\mathcal{O}(Y)$ as the analogue of space of algebraic vector
fields, which is a $(\mathbb{K},\mathcal{O}(Y))$-Lie-Rinehart algebra and
$\mathfrak{X}^{T}(X):=\\{D\in\mathfrak{X}(X)\mid D(I)\subset I\\}$ as the
analogue of space of algebraic vector fields that are tangent to $Y$, which is
a $(\mathbb{K},\mathcal{O}(X))$-Lie-Rinehart algebra (both of these are
finitely presented module). We can extend this idea by considering it as local
model via sheaf theoretic tools to get sheaf of logarithmic derivations over
an algebraic variety (locally affine).

######  Example 2.10.

Consider the affine schemes $\mathbb{A}^{I}:=Spec(\mathbb{K}[x_{i}]_{i\in I})$
and $\mathbb{A}^{\infty}:=\mathbb{A}^{\mathbb{N}}$ (see [6]). The space of
global sections of the tangent sheaf for the affine scheme $\mathbb{A}^{I}$ is
the free $\mathbb{K}[x_{i}]_{i\in I}$-module spanned by the formal derivations
$\\{\partial_{x_{i}}\\}_{i\in I}$. This provides an example of Lie algebroids
which are quasicoherent $\mathcal{O}_{\mathbb{A}^{\mathbb{N}}}$-module.

###  2.2. Analytic spaces

Let $X=\mathbb{C}^{n}$ be the $n$-dimensional complex manifold and
$\mathcal{O}_{X}$ be the sheaf of holomorphic functions. Let $\mathcal{I}$ be
a coherent ideal-sheaf of $\mathcal{O}_{X}$. The vanishing set
$V(\mathcal{I})$ of $\mathcal{I}$ is a subspace of $X$ which is not
necessarily a submanifold (it may have singularity as well). Let us denote
$Y=V(\mathcal{I})$ and we consider the sheaf of functions
$\mathcal{O}_{Y}:=\mathcal{O}_{X}/{\mathcal{I}}$ as its structure sheaf. Here,
the pair $(Y,\mathcal{O}_{Y})$ is an analytic space (see in [18]).

If $Y$ is not smooth (singular), it has no tangent bundle in general, then we
consider derivations,
$\mathcal{T}_{Y}:=\mathcal{D}er_{\mathbb{C}_{Y}}(\mathcal{O}_{Y})$ is the
tangent sheaf for $(Y,\mathcal{O}_{Y})$. Here
$(\mathcal{T}_{Y},[\cdot,\cdot]_{c},id)$ is a coherent (locally finitely
generated) sheaf of Lie-Rinehart algebras over $(Y,\mathcal{O}_{Y})$.

We consider the generalized involutive distribution (or singular foliation)
given by holomorphic vector fields of $X$ that are tangent to $Y$ [18] as

$\mathcal{T}_{X}(-logY):=\\{D\in\mathcal{T}_{X}:D(\mathcal{I})\subset\mathcal{I}\\}\hookrightarrow\mathcal{T}_{X}$

with canonical Lie algebroid structure. For any two sections
$D,D^{\prime}\in\mathcal{T}_{X}(-logY)$ and a section $g\in\mathcal{I}$ we
have the identity
$[D,D^{\prime}]_{c}(g)=D(D^{\prime}(g))-D^{\prime}(D(g))\in\mathcal{I}$. It is
associated with $\mathcal{T}_{Y}$ via the Lie algebroid epimorphism

$\rho:\mathcal{T}_{X}(-logY)\rightarrow\mathcal{T}_{Y}$

defined by $\rho(D)=\bar{D}$, $\bar{D}([{f}])=[{D(f)}]$ for any sections
$f\in\mathcal{O}_{X},D\in\mathcal{T}_{X}(-logY)$.

Nilpotent cone: Here we take a family of level sets $\\{f^{-1}(c)\mid
c\in\mathbb{C}\\}$ associated to a system of first order homogeneous partial
differential equations

$D_{x}(f)=D_{y}(f)=D_{z}(f)=0.$

For $c=0$ we get the surface with singularity, which is called a nilpotent
cone. It forms an analytic space with a singularity at the origin
$\bar{0}=(0,0,0)$.

One may describe this nilpotent cone through a generalized involutive
distribution generated by the Hamiltonian vector fields in a Poisson manifold
structure on $X:=\mathbb{C}^{3}$. The Poisson algebra
$(\mathcal{O}_{\mathbb{C}^{3}}(\mathbb{C}^{3}),\\{\cdot,\cdot\\})$ defined on
coordinate functions $x,y,z$ as follows.

$\\{x,y\\}=2y,\\{x,z\\}=-2z,\\{y,z\\}=x.$

Then the associated Hamiltonian vector fields are given by

$D_{x}=2y\partial_{y}-2z\partial_{z},D_{y}=-2y\partial_{x}+x\partial_{z},D_{z}=2z\partial_{x}-x\partial_{y}.$

We consider the system of first order homogeneous partial differential
equations

$D_{x}(f)=D_{y}(f)=D_{z}(f)=0.$

Then we find solution of the above system is $f(x,y,z)=x^{2}+4yz$. It
describes a family of level sets $f^{-1}(c)$, is parametrized by
$c\in\mathbb{C}$. For each nonzero values of $c$ we get a $2$-dimensional
submanifold. Now $Y=f^{-1}(0)$ is the vanishing set of an ideal sheaf
$\mathcal{I}\subset\mathcal{O}_{X}$ generated by the single element
$x^{2}+4yz$. Here, the Lie algebroid $\mathcal{T}_{X}(-logY)$ is
$\mathcal{O}_{X}$-linear span of the Hamiltonian vector fields tangent to $Y$,
and $\mathcal{T}_{Y}$ is spanned by $D_{x}|_{Y},D_{y}|_{Y},D_{z}|_{Y}$ as
$\mathcal{O}_{Y}$-module. Moreover, the generators of $\mathcal{T}_{X}(-logY)$
satisfy the equation

$xD_{x}+2zD_{y}+2yD_{z}=0.$

This shows that it is not a locally free $\mathcal{O}_{X}$-module. Here,
$\mathcal{T}_{Y\setminus\\{\bar{0}\\}}$ is a locally free module of rank $2$
over $\mathcal{O}_{Y\setminus\\{\bar{0}\\}}$. However at the origin, this
sheaf is no longer free and the stalk $\mathcal{T}_{Y,\bar{0}}$ is a module of
rank $3$ over the algebra $\mathcal{O}_{Y,\bar{0}}$.

It follows that the tangent sheaf $\mathcal{T}_{Y}$ is not a locally free
sheaf of Lie-Rinehart algebras over the analytic space $(Y,\mathcal{O}_{Y})$
but it is a coherent sheaf generated by the three elements.

Normal crossing: Let $X=\mathbb{C}^{2}$ with global coordinate functions $x,y$
and we consider the analytic space as $(Y,\mathcal{O}_{Y})$ associated to the
vanishing set of the ideal sheaf $\mathcal{I}$ generated by the function $xy$.
Here the subspace $Y$ is the union of points on the axes, known as a normal
crossing divisor. The Lie algebroid $\mathcal{T}_{X}(-logY)$ is
$\mathcal{O}_{X}$-module generated by the derivations $x\partial_{x}$ and
$y\partial_{y}$.

For a point $p$ on $x$-axis or $y$-axis other than the origin
$\bar{0}:=(0,0)$, the stalk $\mathcal{T}_{Y,p}$ is generated by the germs
between one of the vector fields $x\partial_{x}$ or $y\partial_{y}$ at $p$. It
follows that the tangent sheaf $\mathcal{T}_{Y}$ is a locally free
$\mathcal{O}_{Y}$-module of rank $1$ on $Y\setminus\\{\bar{0}\\}$. But the
stalk $\mathcal{T}_{Y,\bar{0}}$ is a module of rank $2$ over
$\mathcal{O}_{Y,\bar{0}}$. Therefore, $\mathcal{T}_{Y}$ is an example for Lie
algebroid which is coherent sheaf but not locally free. Observe that
$\mathcal{T}_{X}(-logY)$ is a locally free sheaf of $\mathcal{O}_{X}$ module
of rank $2$ on $X\setminus Y$. It is equal to $\mathcal{T}_{X}$ on $X\setminus
Y$ and on $Y$ it is isomorphic to $\mathcal{T}_{X}$ but equality not holds.

See Section 4.2.1 for details about non-locally free property of the Lie
algebroids.

##  3\. Universal enveloping algebroid of a Lie algebroid

Here we describe universal enveloping algebroid of a Lie algebroid in terms of
the universal enveloping algebra for a Lie-Rinehart algebra. It is obtained
through sheafification and by focusing on a model object being the sheaf of
differential operators defined on a smooth manifold. Subsequently, we present
a generalized version of Poincare-Birkhoff-Witt theorem (PBW theorem in short
hand notation ) for Lie algebroids over some special $a$-spaces, which is
analogous to its local version of Lie-Rinehart algebras in [20]. First we
recall some basic concepts related to the sheafifications from [19], which
help us to follow notations used in later part of our presentation.

Sheaf associated to a presheaf : Let $\mathcal{F}$ be a presheaf of
$\mathcal{O}_{X}$-modules. At each point $x$, the stalk $\mathcal{F}_{x}$ has
induced algebraic structure from the presheaf. On sheafification of the
presheaf $\mathcal{F}$, we get a sheaf denoted as $\mathcal{F}^{\\#}$ has this
structure along with the universal property. Moreover a section of the
$\acute{e}$tale space associated with $S\in\mathcal{F}(U)$ is given as
$\tilde{S}:U\rightarrow\displaystyle\cup_{x\in U}\mathcal{F}_{x}$ defined by
$\tilde{S}(x)=[S_{x}]$ (the equivalence class of germ of $S$ at $x$). Now for
any section $S^{\prime}\in\mathcal{F}^{\\#}(U)$ there exist an open cover
$\\{U_{i}:i\in I\\}$ of $U$ where $S_{i}\in\mathcal{F}(U_{i})$ for $i\in I$
such that $S^{\prime}|_{U_{i}}=\tilde{S_{i}}$. Here $S_{i}=S_{j}$ on
$U_{i}\cap U_{j}$ (non empty) and the sheaf $\mathcal{F}^{\\#}$ is locally
determined via the presheaf $\mathcal{F}$. Hence, the sheaf
$\mathcal{F}^{\\#}$ become an $\mathcal{O}_{X}$-modules and stalks of
$\mathcal{F}$ and $\mathcal{F}^{\\#}$ are same at each point.

The assignment $\mathcal{F}\mapsto\mathcal{F}^{\\#}$ induces a functor from
the category of presheaves to the category of sheaves over some topological
space. The adjunction of this functor is given by the natural inclusion
functor. Consider the notion ${}^{p}(\cdot)$ as
${}^{p}(\mathcal{F}^{\\#})=\mathcal{F}$ for a presheaf $\mathcal{F}$.

Operations on sheaves: We associate new sheaves using algebraic operations
locally followed by sheafification.

$(i)$ The tensor product
$\mathcal{F}_{1}\otimes_{\mathcal{O}_{X}}\mathcal{F}_{2}$ of any two given
$\mathcal{O}_{X}$-modules $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ is the
sheafification of the canonical presheaf
$\\{U\mapsto\mathcal{F}_{1}(U)\otimes_{\mathcal{O}_{X}(U)}\mathcal{F}_{2}(U)$
for every open set $U\in X$ with componentwise restrictions morphism$\\}$. The
associated presheaf is denoted as
${}^{p}(\mathcal{F}_{1}\otimes_{\mathcal{O}_{X}}\mathcal{F}_{2})$.

$(ii)$ The quotient sheaf $\mathcal{F}_{1}/{\mathcal{F}_{2}}$ for a sheaf
$\mathcal{F}_{1}$ with a subsheaf $\mathcal{F}_{2}$ is the sheafification of
the cannonical preasheaf $\\{U\mapsto\mathcal{F}_{1}(U)/{\mathcal{F}_{2}(U)}$
for every open set $U$ in $X$ with natural restriction morphism$\\}$. The
associated presheaf is denoted as ${}^{p}(\mathcal{F}_{1}/{\mathcal{F}_{2}})$.

There are two other important sheaves associated with sheaves
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ as $\mathcal{O}_{X}$-modules. One is
the direct sum $\mathcal{F}_{1}\oplus\mathcal{F}_{2}$ and the other one is the
sheaf homomorphisms
$\mathscr{H}om_{\mathcal{O}_{X}}(\mathcal{F}_{1},\mathcal{F}_{2})$. These
sheaves are obtained directly by the assignments
$U\mapsto\mathcal{F}_{1}(U)\oplus\mathcal{F}_{2}(U)$ and $U\mapsto
Hom_{\mathcal{O}_{X}(U)}(\mathcal{F}_{1}(U),\mathcal{F}_{2}(U))$ respectively.

For a Lie algebroid $\mathcal{L}$ over $(X,\mathcal{O}_{X})$, we form the
sheaf of tensor algebras $\mathcal{T}_{\mathcal{O}_{X}}\mathcal{L}$ as the
sheafification of the cannonical presheaf: $U\mapsto
T_{\mathcal{O}_{X}(U)}(\mathcal{L}(U))$ of tensor algebras for the
$\mathcal{O}_{X}(U)$-modules $\mathcal{L}(U)$. To get the universal enveloping
algebroid of the Lie algebroid $\mathcal{L}$, we need to consider the quotient
sheaf of this sheaf of tensor algebras (see [21, 18]). A local version of this
quotient sheaf is considered in Section (4.1). We follow here another approach
to define the universal enveloping algebroid.

###  3.1. The sheaf of universal enveloping algebras of Lie-Rinehart algebras

Let $(\mathcal{L},[\cdot,\cdot],\mathfrak{a})$ be a Lie algebroid over any of
the special $a$-space $(X,\mathcal{O}_{X})$. For each open set $U$ of $X$, we
find the universal enveloping algebra
$\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{L}(U))$ of the
$(\mathbb{K},\mathcal{O}_{X}(U))$-Lie-Rinehart algebra $\mathcal{L}(U)$. It
satisfies the compatibility condition that for every open set $V\subset U$
there is a canonical restriction map

$res_{UV}^{\mathcal{U}}:\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{L}(U))\rightarrow\mathcal{U}(\mathcal{O}_{X}(V),\mathcal{L}(V))$

defined by $res_{UV}^{\mathcal{U}}(f$ $\bar{D}_{1}\cdots\bar{D}_{n})=f|_{V}$
$\overline{D_{1}|_{V}}\cdots\overline{D_{n}|_{V}}$ where

$f\in\mathcal{O}_{X}(U),D_{i}\in\mathcal{L}(U),\bar{D}_{i}=\iota_{\mathcal{L}(U)}(D_{i}),i=1,\dots,n$

and
$\iota_{\mathcal{L}(U)}:\mathcal{L}(U)\rightarrow\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{L}(U))$
is the canonical map. Here, $\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{L}(U))$
is an associative unital $\mathbb{K}$-algebra and bimodule over
$\mathcal{O}_{X}(U)$ ([15]).

We find that $\mathcal{U}(\mathcal{O}_{X},\mathcal{L})$ is a presheaf of
associative unital $\mathbb{K}_{X}$-algebras and also a presheaf of left
$\mathcal{O}_{X}$-modules with the assignment
$U\mapsto\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{L}(U))$ and restriction
morphisms are the maps $res_{UV}^{\mathcal{U}}$ for any two open sets $V,U$
with $V\subset U$.

We consider its sheafification and the resulting sheaf is denoted by
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$. All the necessary algebraic
structures on $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ obtained via the
structures induced on stalks of the presheaf
$\mathcal{U}(\mathcal{O}_{X},\mathcal{L})$. Thus, we have the canonical
embedding

$\iota:\mathcal{U}(\mathcal{O}_{X},\mathcal{L})\hookrightarrow\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$

of presheaf of $\mathcal{O}_{X}$-linear $\mathbb{K}$-algebra monorphisms,
defined by $\iota(U)(S)=\tilde{S}$ and $\tilde{S}(x)=S_{x}$ (germ of $S$ at
$x$) for any section $S\in\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{L}(U)),x\in
U$.

Subsequently, we form a sheaf monomorphism
$\iota_{X}:\mathcal{O}_{X}\rightarrow\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$
of associative unital $\mathbb{K}_{X}$ algebras and
$\iota_{\mathcal{L}}:(\mathcal{L},[\cdot,\cdot])\rightarrow(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}),[\cdot,\cdot]_{c})$
a $\mathcal{O}_{X}$-linear sheaf homomorphisms of $\mathbb{K}_{X}$-Lie
algebras.

######  Remark 3.1.

The sheaf of universal enveloping algebras
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ is charecterized by the following
universal property:

Let $\mathcal{A}$ be a sheaf of unital associative $\mathbb{K}_{X}$-algebra
with sheaf homomorphisms $\phi:\mathcal{O}_{X}\rightarrow\mathcal{A}$ of
$\mathbb{K}_{X}$-unital algebras and
$\psi:(\mathcal{L},[\cdot,\cdot])\rightarrow(\mathcal{A},[\cdot,\cdot]_{c})$
of $\mathcal{O}_{X}$-linear $\mathbb{K}_{X}$-Lie algebras such that
$\phi(f)\psi(D)=\psi(fD)$ and $[\psi(D),\phi(f)]_{c}=\phi(\mathfrak{a}(D(f)))$
holds for all sections $f$ of $\mathcal{O}_{X}$ and $D$ of $\mathcal{L}$. Then
there exist a unique presheaf homomorphism of $\mathcal{O}_{X}$-linear
$\mathbb{K}_{X}$-algebras
${}^{p}\widetilde{\psi}:\mathcal{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathcal{A}$.

By the universal property of the sheafification, there exists a unique
$\mathcal{O}_{X}$-linear homomorphism of unital $\mathbb{K}_{X}$-algebras
$\widetilde{\psi}:\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathcal{A}$
such that $\widetilde{\psi}\circ\iota_{X}=\phi$ and
$\widetilde{\psi}\circ\iota_{\mathcal{L}}=\psi$.

The sheaf $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ is called universal
enveloping algebroid of the Lie algebroid $\mathcal{L}$.

###  3.2. Generalized PBW theorem

For an open set $U$ of $X$, the algebra
$\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{L}(U))$ has a canonical filtration of
$\mathcal{O}_{X}(U)$-modules [20, 15]:

$\mathcal{O}_{X}(U)=\mathcal{U}_{(0)}(\mathcal{O}_{X}(U),\mathcal{L}(U))\subset\mathcal{U}_{(1)}(\mathcal{O}_{X}(U),\mathcal{L}(U))\subset\mathcal{U}_{(2)}(\mathcal{O}_{X}(U),\mathcal{L}(U))\subset\cdots.$

Here, $\mathcal{U}_{(n)}(\mathcal{O}_{X}(U),\mathcal{L}(U))$ is spanned by the
elements in $\mathcal{O}_{X}(U)$ and the powers
$i_{\mathcal{L}(U)}(\mathcal{L}(U))^{m}$ for $m=1,2,\dots,n$. Thus, we can
form sub-presheaves of $\mathcal{O}_{X}$-module
$\mathcal{U}_{(n)}(\mathcal{O}_{X},\mathcal{L})$ given by
$\mathcal{U}_{(n)}(\mathcal{O}_{X},\mathcal{L})(U):=\mathcal{U}_{(n)}(\mathcal{O}_{X}(U),\mathcal{L}(U))$.

The presheaf $\mathcal{U}(\mathcal{O}_{X},\mathcal{L})$ of $\mathbb{K}$
algebras has also a natural filtration of presheaves of
$\mathcal{O}_{X}$-modules:

$\mathcal{O}_{X}=\mathcal{U}_{(0)}(\mathcal{O}_{X},\mathcal{L})\subset\mathcal{U}_{(1)}(\mathcal{O}_{X},\mathcal{L})\subset\mathcal{U}_{(2)}(\mathcal{O}_{X},\mathcal{L})\subset\cdots.$

On sheafification, these presheaves yield a filtration of
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$. Let
$\mathscr{U}_{(n)}(\mathcal{O}_{X},\mathcal{L})$ be the associated sheaves of
$\mathcal{O}_{X}$-modules for $n\geq 0$.

The direct sum of the quotient sheaves associated with each of the consecutive
sheaves appear in the above filtration of
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ forms a sheaf of graded algebras.
It is denoted by $gr(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}))$. Therefore,

$gr(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}))=\bigoplus_{n\geq
0}\mathscr{U}_{(n)}(\mathcal{O}_{X},\mathcal{L})/\mathscr{U}_{(n-1)}(\mathcal{O}_{X},\mathcal{L}).$

Note that if a Lie algebroid $\mathcal{L}$ has zero bracket with zero anchor
map then $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ become the sheaf of
symmetric algebras $\mathcal{S}_{\mathcal{O}_{X}}\mathcal{L}$. If for every
$x\in X$, there is an open set $U_{x}$ in $X$ such that $\mathcal{L}(U_{x})$
is a projective left $\mathcal{O}_{X}(U_{x})$-module, then by applying PBW
theorem for the $(\mathbb{K},\mathcal{O}_{X}(U_{x}))$-Lie-Rinehart algebra
$\mathcal{L}(U_{x})$ we get the canonical isomorphisms
$\mathcal{O}_{X}(U_{x})$-algebras

$\theta_{U_{x}}:S_{\mathcal{O}_{X}(U_{x})}(\mathcal{L}(U_{x}))\rightarrow
gr(\mathcal{U}(\mathcal{O}_{X}(U_{x}),\mathcal{L}(U_{x}))).$

This provides a presheaf of $\mathcal{O}_{X}$-algebra homomorphism (locally
isomorphic)

$\theta:S_{\mathcal{O}_{X}}(\mathcal{L})\rightarrow
gr(\mathcal{U}(\mathcal{O}_{X},\mathcal{L})).$

######  Lemma 3.2.

Let $(X,\mathcal{O}_{X})$ be an arbitrary $a$-space and
$\mathcal{F}_{1},\mathcal{F}_{2}$ be two presheaves of
$\mathcal{O}_{X}$-modules with a presheaf homomorphism
$\psi:\mathcal{F}_{1}\rightarrow\mathcal{F}_{2}$ inducing stalkwise
isomorphism at each point of $X$. Then the induced morphism between
corresponding sheaves gives a $\mathcal{O}_{X}$-module isomorphism
$\psi^{\\#}:\mathcal{F}^{\\#}_{1}\rightarrow\mathcal{F}^{\\#}_{2}$.

###### Proof.

Let $s$ be a section of $\mathcal{F}^{\\#}_{1}$, that is
$s\in\mathcal{F}_{1}^{\\#}(U)$ for some open set $U$ in $X$. There exists an
open cover $\\{U_{i}\\}_{i\in I}$ of $U$ with sections
$s_{i}\in\mathcal{F}_{1}(U_{i})$ such that $s|_{U_{i}}=\tilde{s_{i}}$ for each
$i\in I$ and $s(x)=(s_{i})_{x}$ for each $x\in U_{i}$. Then
$\psi^{\\#}(U)(s)|_{U_{i}}=\widetilde{\psi_{U}(s_{i})}$ where
$\tilde{s_{i}}=\iota(U_{i})(s_{i})$ and
$\iota:\mathcal{F}\hookrightarrow\mathcal{F}^{\\#}$ given by
$s\mapsto\tilde{s}$ defined as $\tilde{s}(x)=[s_{x}]$ for a section $s$ of the
presheaf $\mathcal{F}$. Thus
$\psi^{\\#}:\mathcal{F}^{\\#}_{1}\rightarrow\mathcal{F}^{\\#}_{2}$ is a
morphism. We need to check it is a bijection.

One-to-one: Let $U$ be any open subset of $X$ and
$s,s^{\prime}\in\mathcal{F}^{\\#}_{1}(U)$ such that
$\psi^{\\#}(U)(s)-\psi^{\\#}(U)(s^{\prime})=0$. Then stalkwise we get
$\psi_{x}(s_{x}-s^{\prime}_{x})=0$ for every $x\in U$. The stalkwise
isomorphisms provides $s_{x}=s^{\prime}_{x}$ for each $x\in U$. Thus, for each
point $x$ there is an open neighborhood $U_{x}$ of $x$ in $U$ where
$s=s^{\prime}$. Using the local identity property of $\mathcal{F}^{\\#}_{1}$
for this sections $s,s^{\prime}$ on this open cover $\\{U_{x}\\}_{x\in U}$, we
get $s=s^{\prime}$.
Onto: Let $U$ be any open subset of $X$ and $t\in\mathcal{F}^{\\#}_{2}(U)$.
Then considering germs $t_{x}$ of $t$ at each point $x\in U$ and by applying
the inverses of the isomorphisms $\psi_{x}$, we get
$s_{i}\in\mathcal{F}_{1}(U_{x})$ for some open neighborhood $U_{x}$ of $x$
such that $\psi_{y}((s_{i})_{y})=t_{y}$ for all $y\in U_{x}$. Then by the
gluing property of $\mathcal{F}^{\\#}_{1}$ we can form an element
$s\in\mathcal{F}^{\\#}_{1}(U)$ such that $s|_{U_{x}}=\tilde{s_{i}}$, where
$\tilde{s_{i}}(y)=(s_{i})_{y}$ for all $y\in U_{x}$. ∎

######  Lemma 3.3.

Let $(X,\mathcal{O}_{X})$ be an arbitrary $a$-space and
$\mathcal{F}_{1},\mathcal{F}_{2}$ be two presheaf of $\mathcal{O}_{X}$-modules
with a presheaf homomorphism $\psi:\mathcal{F}_{1}\rightarrow\mathcal{F}_{2}$
such that $\psi$ is a locally isomorphism. Then $\psi$ induces stalkwise
isomorphism.

###### Proof.

It is easy to see that on the level of stalks $\psi$ induces morphisms and we
need to check it is a bijection.

One-to-one: Let $s_{1},s_{2}\in(\mathcal{F}_{1})_{x}$ such that $s_{1}\neq
s_{2}$ where $x\in X$. Then there exists $S_{1},S_{2}\in\mathcal{F}_{1}(V)$
for some open set $V\in X$ such that $(S_{1})_{x}=s_{1}$ and
$(S_{2})_{x}=s_{2}$. Since $\psi$ is a locally isomorphism, thus there exist
an open set $U_{x}$ around $x$ such that
$\psi|_{U_{x}}:\mathcal{F}_{1}|_{U_{x}}\rightarrow\mathcal{F}_{2}|_{U_{x}}$ is
an isomorphism. Without loss of generality $V$ can be thought as an open
subset of $U_{x}$. Consider $S^{\prime}_{1}:=\psi(V)(S_{1})$ and
$S^{\prime}_{2}:=\psi(V)(S_{2})$. Our claim is that
$(S^{\prime}_{1})_{x}\neq(S^{\prime}_{2})_{x}$ that is we need to show that
$S^{\prime}_{1}|_{V_{x}}\neq S^{\prime}_{2}|_{V_{x}}$ for every open set
$V_{x}$ around $x$. Suppose not, then there exist an open set $V_{x}$ such
that $S^{\prime}_{1}=S^{\prime}_{2}$ on $V_{x}\subset V$. Consider the one-one
property of the isomorphism
$\psi(V_{x}):\mathcal{F}_{1}(V_{x})\rightarrow\mathcal{F}_{2}(V_{x})$. Then it
contradicts the fact that $S_{1}\neq S_{2}$ on $V_{x}$. Therefore,
$\psi_{x}:(\mathcal{F}_{1})_{x}\rightarrow(\mathcal{F}_{2})_{x}$ is one-one
for every $x\in X$.
Onto: Let $s^{\prime}\in(\mathcal{F}_{2})_{x}$. Then there exist
$S^{\prime}\in\mathcal{F}_{2}(V)$ such that $(S^{\prime})_{x}=s^{\prime}$ for
some $V\subset U_{x}$. By considering onto property of the isomorphism

$\psi(V):\mathcal{F}_{1}(V)\rightarrow\mathcal{F}_{2}(V),$

we get a pre-image of $S^{\prime}$ say $S\in\mathcal{F}_{1}(V)$. We have
$\psi_{x}(S_{x})=s^{\prime}$ by considering the germ $S_{x}$ for $x\in X$. ∎

Next, we proceed with $(X,\mathcal{O}_{X})$ as one of the special $a$-spaces
among smooth manifold, complex manifold, analytic space and algebraic variety.

######  Theorem 3.4.

Generalized PBW theorem : If $\mathcal{L}$ is a locally free Lie algebroid
over $(X,\mathcal{O}_{X})$ then the graded algebra
$gr(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}))$ is isomorphic to the symmetric
algebra $\mathcal{S}_{\mathcal{O}_{X}}\mathcal{L}$ $(\mathcal{O}_{X}$-algebra
isomorphism).

###### Proof.

: If $\mathcal{L}$ is a locally free Lie algebroid over $(X,\mathcal{O}_{X})$,
then for every point $x\in X$ there exist an open neighborhood $U_{x}$ of $x$
in $X$ such that

$\mathcal{L}|_{U_{x}}\cong\mathcal{O}_{X}^{\oplus^{I}}|_{U_{x}}$, where $I$
may be infinite.

Now, we have the PBW presheaf homomorphism of $\mathcal{O}_{X}$-algebras

$\theta:S_{\mathcal{O}_{X}}\mathcal{L}\rightarrow
gr(\mathcal{U}(\mathcal{O}_{X},\mathcal{L}))$

$f\otimes D_{1}\otimes\cdots\otimes
D_{k}\mapsto\frac{1}{k!}f\displaystyle\sum_{\sigma\in
S_{k}}\bar{D}_{\sigma(1)}\cdots\bar{D}_{\sigma(k)}$

where $f$ is a section of $\mathcal{O}_{X}$ and $D_{1},\dots,D_{k}$ are
sections of $\mathcal{L}$. By using local triviality property of
$\mathcal{L}$, on each $U_{x}$ we have $\mathcal{O}_{X}|_{U_{x}}$-algebra
isomorphism on restrictions of the presheaves

$\theta|_{U_{x}}:S_{\mathcal{O}_{X}}\mathcal{L}|_{U_{x}}\rightarrow
gr(\mathcal{U}(\mathcal{O}_{X},\mathcal{L}))|_{U_{x}}.$

It induces a stalkwise isomorphisms of associated presheaves (follows from
Lemma 3.3). Subsequently using the Lemma 3.2, we get the required isomorphism.
∎

######  Remark 3.5.

By compairing $\mathcal{O}_{X}$-modules in degree $1$ of the PBW isomorphism
we get
$\mathcal{L}\cong\mathscr{U}_{1}(\mathcal{O}_{X},\mathcal{L})/{\mathcal{O}_{X}}$.
In turn, we have the short exact sequence (s.e.s) of $\mathcal{O}_{X}$-modules

$0\rightarrow\mathcal{O}_{X}\hookrightarrow\mathscr{U}_{(1)}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathcal{L}\rightarrow
0$.

Infact it is also an s.e.s of Lie algebroids with respect to canonical Lie
algebroid structures. It splits as $\mathcal{O}_{X}$-modules that is
$\mathscr{U}_{(1)}(\mathcal{O}_{X},\mathcal{L})\cong\mathcal{O}_{X}\oplus\mathcal{L}$,
as the Lie algebroid $\mathcal{L}$ is locally free $\mathcal{O}_{X}$-module.
This induces an embedding

$\mathcal{L}\hookrightarrow\mathscr{U}_{(1)}(\mathcal{O}_{X},\mathcal{L})\subset\mathscr{U}(\mathcal{O}_{X},\mathcal{L}).$

Notation : From now on we consider the special type of open cover
$\\{U_{x}\mid x\in X\\}$ of $X$ for a locally free Lie algebroid $\mathcal{L}$
and here the restrictions $\mathcal{L}|_{U_{x}}$ are free
$\mathcal{O}_{X}|_{U_{x}}$-modules for every $U_{x}$.

######  Example 3.6.

For $\mathcal{L}=\mathcal{T}_{X}$ the tangent sheaf on a smooth manifold or
complex manifold $(X,\mathcal{O}_{X})$, the universal enveloping algebroid
$\mathscr{U}(\mathcal{O}_{X},\mathcal{T}_{X})$ is the usual sheaf of regular
differential operators [9, 18, 19].

Let $X$ be an $n$-dimensional $C^{\infty}$-manifold or a Stein manifold. Then
$L:=Der_{\mathbb{K}}(\mathcal{O}_{X}(X))$ is the
$(\mathbb{K},\mathcal{O}_{X}(X))$-Lie-Rinehart algebra of first order
homogeneous differential operators on $X$ and its universal enveloping algebra
$\mathcal{U}(\mathcal{O}_{X}(X),L)$ is isomorphic to the $\mathbb{K}$-algebra
(and $\mathcal{O}_{X}(X)$-module) of differential operators on $X$. The
associated sheaf is denoted as $\mathcal{D}iff(\mathcal{O}_{X})$ or
$\mathcal{D}_{X}$ [9, 19]. In a local coordinate system
$(U,(x_{1},\dots,x_{n}))$, elements of
$\mathcal{U}_{(k)}(\mathcal{O}_{X}(U),Der_{\mathbb{K}}(\mathcal{O}_{X}(U)))$
are of the form

$\displaystyle\sum_{i_{1}+\dots+i_{n}\leq k}f_{i_{1}\dots
i_{n}}(\partial/\partial x_{1})^{i_{1}}\circ\cdots\circ(\partial/\partial
x_{n})^{i_{n}}.$

This approach locally holds for a complex manifold as it has an open Stein
cover [3].

For a Lie algebroid $\mathcal{L}$ over a manifold $(X,\mathcal{O}_{X})$, the
anchor map $\mathfrak{a}:\mathcal{L}\rightarrow\mathcal{T}_{X}$ induce a
cannonical map
$\tilde{\mathfrak{a}}:\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathcal{D}_{X}$.

##  4\. Logarithmic differential operators for principal divisors

In this section, we recall logarithmic derivations associated with principal
divisors in [10, 18, 17] and discuss associated universal enveloping
algebroids.

We use the notation $\langle\\{D_{1},\dots,D_{n}\\}\rangle$ for $R$-module
generated by the elements $D_{1},\dots,D_{n}\in E$, where $R$ is a
$\mathbb{K}$-algebra and $E$ is an $R$-module.

Principal divisors: Let $(X,\mathcal{O}_{X})$ be a complex manifold. A
principal divisor consists of a line bundle $N\overset{\pi}{\rightarrow}X$
equipped with a section $s$ whose zero set $Y:=s^{-1}(0)$ is nowhere dense in
$X$. It corresponds to a principal ideal sheaf $\mathcal{I}$ which is locally
free. If $s|_{U_{\alpha}}=f_{\alpha}$ on some Stein open cover
$\\{U_{\alpha}\\}_{{\alpha}\in I}$ of $X$ then
$\mathcal{I}|_{U_{\alpha}}=\langle f_{\alpha}\rangle$ for ${\alpha}\in I$. In
this case, $\mathcal{I}$ is a locally free $\mathcal{O}_{X}$-module of rank
$1$ and so $Y=V(\mathcal{I})$ is a codimension $1$ submanifold of $X$. We
denote by $N_{Y}$ the normal bundle of $Y$ in $X$. Then $TX|_{Y}\cong TY\oplus
N_{Y}$. The sheaf of sections $\mathcal{N}_{Y}$ of the normal bundle of $Y$ is
locally given by $\langle\\{\nabla(f_{\alpha}|_{Y})\\}\rangle$ on
$V_{\alpha}=Y\cap U_{\alpha}$ with $\alpha\in I$.

More generally, the subspace $Y:=V(\mathcal{I})\subset X$ associated with a
principal ideal sheaf $\mathcal{I}\subset\mathcal{O}_{X}$, considered as a
principal divisor. Similar consideration hold in complex algebraic geometry
context.

Logarithmic derivations : The sheaf $\mathcal{T}_{X}(-logY)$ of derivations
(vector fields) tangent to an analytic subspace (algebraic variety)
$Y=V(\mathcal{I})$ or, the sheaf of derivations of $X$ which preserves the
ideal sheaf $\mathcal{I}$ is called the sheaf of logarithmic derivations
(Section 2.2). It has a canonical Lie algebroid structure over
$(X,\mathcal{O}_{X})$.

######  Example 4.1.

Consider the ideal sheaf $\mathcal{I}=\langle xy\rangle$ and
$Y=V(\mathcal{I})$. Here $(Y,\mathcal{O}_{Y})$ is an analytic space (affine
algebraic set) with a singularity at the origin. It follows that
$\mathcal{T}_{X}(-logY)=\langle\\{x\partial_{x},y\partial_{y}\\}\rangle$ and
the corresponding sheaf of meromorphic $1$-forms with poles along the divisor
$Y$ is $\Omega_{X}^{1}(logY)=\langle\\{dx/x,dy/y\\}\rangle.$

###  4.1. Algebra of logarithmic differential operators :

Next we describe universal enveloping algebroid associated with a Lie
algebroid of logarithmic derivations.

Algebra of differential operators $Diff_{\mathbb{K}}(R)$ for a commutative
$\mathbb{K}$-algebra $R$ appeared in [21] is also denoted as
$\mathcal{D}_{R/{\mathbb{K}}}$ (e.g. [17]). Here we compute its counter part
associated to special divisors, previously described (Section 2.2). For this
we use following identity holds in $Diff_{\mathbb{K}}(R)$: $[D,f]_{c}=D\circ
f-f\circ D=D(f)$ for any $D\in Der_{\mathbb{K}}(R)$ and $f\in R$.

Recall that under additional algebraic condition on $R$, the canonical
$\mathbb{K}$-algebra epimorphism

$Diff_{\mathbb{K}}(R)\rightarrow\mathcal{U}(R,Der_{\mathbb{K}}(R))$

induces an isomorphism. For instance, the algebra $R$ is $C^{\infty}(M)$ or
$\mathbb{K}[x_{1},\dots,x_{n}]$ or any arbitrary regular $\mathbb{K}$-algebra,
where $Der_{\mathbb{K}}(R)$ is a projective $R$-module (see [13] and [21]).

Normal crossing divisors : Here, we recall the normal crossing (Section 2.2)
as an affine algebraic set. We consider $X=\mathbb{C}^{2}$ as an algebraic
variety, $\tilde{R}:=\mathcal{O}_{X}(X)=\mathbb{C}[x,y]$. Suppose the ideal
$I=\langle xy\rangle\subset\mathcal{O}_{X}(X)$. We take the affine algebraic
set $Y:=V(I)$ with its algebra of regular functions
$R:=\mathcal{O}_{Y}(Y)=\mathbb{C}[x,y]/{\langle xy\rangle}$.

The $(\mathbb{C},R)$-Lie-Rinehart algebra
$L:=\mathcal{T}_{X}(-logY)(X)=\langle\\{x\partial_{x},y\partial_{y}\\}\rangle$.
It is the space of logarithmic derivations associated to the divisor $Y$,
denoted sometime by $Der_{\mathbb{C}}(-logI))$. Let $\mathbb{C}\langle
x_{1},x_{2},D_{1},D_{2}\rangle$ denote the $\mathbb{C}$-algebra of
noncommutative polynomials on variables $x_{1},x_{2},D_{1},D_{2}$. Here, we
have a canonical isomorphism

$\mathcal{U}(\tilde{R},L)\cong\frac{\mathbb{C}\langle
x_{1},x_{2},D_{1},D_{2}\rangle}{\langle\\{[x_{i},x_{j}]_{c},[D_{i},x_{j}]_{c}-\delta_{ij}x_{j},[D_{i},D_{j}]_{c};i,j\in\\{1,2\\}\\}\rangle}.$

The normal crossing divisor is a free divisor. Thus,
$\mathcal{D}_{\tilde{R}}(-logI)$-the algebra of logarithmic differential
operators on $X$ associated to the divisor $Y$ is isomorphic to
$\mathcal{U}(\tilde{R},L)$ ([17]). By localizations of the $\tilde{R}$-module
$L$, we can form the $\mathcal{O}_{X}$-module of logarithmic derivations
$\mathcal{L}$ and deduce the sheaf of logarithmic differential operators
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ for the free divisor $Y$
corresponding to the algebra $\mathcal{U}(\tilde{R},L)$.

This approach helps us to determine the Weyl algebra
$\mathscr{U}(\mathcal{O}_{Y},\mathcal{T}_{Y})$ for an analytic space or an
algebraic variety $(Y,\mathcal{O}_{Y})$. But for affine case, i.e., for basic
analytic space or affine variety $Y$ we need only to consider the space of
global sections as Weyl algebra, which is $\mathcal{U}(R,\mathcal{T}_{Y}(Y))$.

Nilpotent cone :  We recall the example from Section 2.2, the nilpotent cone
$Y=V(I)$ in $X=\mathbb{C}^{3}$ where $I=\langle
x^{2}+4yz\rangle\subset\mathcal{O}_{X}(X)=\tilde{R}$ is a principal ideal. The
Hamiltonian vector fields $D_{x},D_{y},D_{z}$ describe a coherent involutive
subsheaf $\mathcal{L}:=\mathcal{T}_{X}(-logY)$ of the tangent sheaf
$\mathcal{T}_{X}$ but not locally free.

The space of global sections $L:=\mathcal{L}(X)=Der_{\mathbb{C}}(-logI)$ is a
$(\mathbb{C},\tilde{R})$-Lie-Rinehart algebra. Here $L$ is a finitely
presented $\tilde{R}$-module generated by $\\{D_{x},D_{y},D_{z}\\}$.

In view of $Y$ as an affine variety inside the affine space $X$, we get

$\tilde{R}=\mathbb{C}[x,y,z],~{}~{}R=\mathbb{C}[x,y,z]/\langle
x^{2}+4yz\rangle.$

Consequently, we have a canonical isomorphism (denote it by $\phi$)

$\mathcal{U}(\tilde{R},L)\cong\frac{\mathbb{C}\langle
x_{1},x_{2},x_{3},D_{1},D_{2},D_{3}\rangle}{\langle\\{[x_{i},x_{j}]_{c},[D_{i},x_{j}]_{c}-\\{x_{i},x_{j}\\},[D_{i},D_{j}]_{c}-D_{\\{x_{i},x_{j}\\}};i,j\in\\{1,2,3\\}\\}\rangle},$

where $\\{x_{i},x_{j}\\}:=\phi^{-1}(\\{\phi(x_{i}),\phi(x_{j})\\})$ and
$D_{\\{x_{i},x_{j}\\}}:=\phi^{-1}(D_{\\{\phi(x_{i}),\phi(x_{j})\\}})$, for
$i,j\in\\{1,2,3\\}$.

The $(\mathbb{C},\tilde{R})$-Lie-Rinehart algebra $Der_{\mathbb{C}}(-logI)$
induces a $(\mathbb{C},R)$-Lie-Rinehart algebra structure on the $R$-module
generated by the set $\\{D_{x}|_{Y},D_{y}|_{Y},D_{z}|_{Y}\\}$. It is equal to
$\mathcal{T}_{Y}(Y)$. Then the Weyl algebra for $Y$ is

$\mathcal{U}(R,Der_{\mathbb{C}}(R))\cong\mathcal{U}(\tilde{R},Der_{\mathbb{C}}(-logI))/{Ker(\psi)}$

where the map

$\psi:\mathcal{U}(\tilde{R},Der_{\mathbb{C}}(-logI))\rightarrow\mathcal{U}(R,Der_{\mathbb{C}}(R))$

is defined on the generators by the canonical restrictions. Also, we get an
analogous result associated with logarithmic differential operators:

$\mathcal{D}_{R/{\mathbb{C}}}\cong\frac{\mathcal{D}_{\tilde{R}}(-logI)}{Ker(\psi)}.$

###  4.2. PBW isomorphism for some non-locally free Lie algebroids :

In this part, we show that generalized PBW isomorphism holds for Lie algebroid
over $(X,\mathcal{O}_{X})$ without the condition of locally free [21, 18].
Here, we consider Lie algebroids for special divisors.

  * •

For a $(\mathbb{K},R)$-Lie-Rinehart algebra $L$, we have the PBW map (also
called the symmetrization map)

$\phi:S_{R}L\rightarrow\mathcal{U}(R,L)$ given by $D_{1}\otimes\cdots\otimes
D_{n}\mapsto\frac{1}{k!}\displaystyle\sum_{\sigma\in
S_{k}}\bar{D}_{\sigma(1)}\cdots\bar{D}_{\sigma(k)}$

where $D_{1},\dots,D_{k}\in L$, is a surjective $R$-linear
$\mathbb{K}$-algebra homomorphism. The morphism $\phi$ induces a surjective
$R$-algebra homomorphism

$\tilde{\phi}:S_{R}L\rightarrow gr(\mathcal{U}(R,L)).$

  * •

The map $\tilde{\phi}$ is an isomorphism when $L$ is, in addition, a
projective $R$-module.

Both of these results are given by G. Rinehart in[20].

  * •

For a locally free Lie algebroid $\mathcal{L}$ over $(X,\mathcal{O}_{X})$, at
each point $x\in X$ the stalk $\mathcal{L}_{x}$ is free
$\mathcal{O}_{X,x}$-module of fixed finite rank. It provides the
$\mathcal{O}_{X}$-algebra isomorphism
$\theta:\mathcal{S}_{\mathcal{O}_{X}}\mathcal{L}\rightarrow
gr(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}))$.

  * •

Suppose $\mathcal{L}$ is a non-locally free Lie algebroid over
$(X,\mathcal{O}_{X})$, whose stalks $\mathcal{L}_{x}$ are free
$\mathcal{O}_{X,x}$-modules of finite ranks and $x\mapsto
rank(\mathcal{L}_{x})$ is not a constant function. Hence, $\mathcal{L}_{x}$ is
a $(\mathbb{K},\mathcal{O}_{X,x})$-Lie-Rinehart algebra and a free
$\mathcal{O}_{X,x}$-module for each $x\in X$. It satisfies Rinehart’s PBW
criteria on the level of stalks. Therefore, the induced map of $\theta$ on
each stalk provides an isomorphism of $\mathcal{O}_{X,x}$-algebras
$\theta_{x}:\mathcal{S}_{\mathcal{O}_{X,x}}\mathcal{L}_{x}\rightarrow
gr(\mathcal{U}(\mathcal{O}_{X,x},\mathcal{L}_{x}))$. As the map $\theta$ is
defined via sheafification of the morphism induces from $\theta_{x}$, the PBW
map is an isomorphism of $\mathcal{O}_{X}$-algebras.

Let $\mathcal{L}_{x}$ be a $(\mathbb{K},\mathcal{O}_{X,x})$-Lie-Rinehart
algebra which is also free $\mathcal{O}_{X,x}$-module for every $x\in X$. In
view of Rinehart’s PBW theorem we have $\mathcal{O}_{X,x}$-algebra isomorphism
by the symmetrization map

$\psi_{x}:\mathcal{S}_{\mathcal{O}_{X,x}}\mathcal{L}_{x}\rightarrow
gr(\mathcal{U}(\mathcal{O}_{X,x},\mathcal{L}_{x}))$

for every $x\in X$.

The generalized PBW morphism
$\theta:\mathcal{S}_{\mathcal{O}_{X}}\mathcal{L}\rightarrow
gr(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}))$ is surjective and induces
stalkwise isomorphisms of $\mathcal{O}_{X,x}$-algebras

$\theta_{x}:\mathcal{S}_{\mathcal{O}_{X,x}}\mathcal{L}_{x}\rightarrow
gr(\mathcal{U}(\mathcal{O}_{X,x},\mathcal{L}_{x}))$

for each $x\in X$. We have $\psi_{x}=\theta_{x}$ for all $x\in X$.

####  4.2.1. On PBW isomorphism for some examples

We show that PBW isomorphism holds for the Lie algebroid of logarithmic
derivations $\mathcal{L}:=$$\mathcal{T}_{X}(-logY)$ and tangent sheaf
$\mathcal{T}_{Y}$ associated to $Y$ the nilpotent cone and the normal crossing
divisor respectively (defined in Section 2.2).

The PBW isomorphism for nilpotent cone $Y$:

  * •

The equation $xD_{x}+2zD_{y}+2yD_{z}=0$ is satisfied by the logarithmic
derivations $D_{x},D_{y},D_{z}$ associated to $Y$. Therefore,
$\mathcal{L}(U_{p})=\mathcal{T}_{X}(-logY)(U_{p})$ is a free
$\mathcal{O}_{X}(U_{p})$-module of rank $2$ on open neighborhood $U_{p}$ of
$p$ in $X$ with $p\in X\setminus\\{\bar{0}\\}$. For $p=\bar{0}$ we get that
$(\mathcal{L})_{\bar{0}}$ is a free $\mathcal{O}_{X,\bar{0}}$-module of rank
$3$. Consequently, $\mathcal{L}$ is a Lie algebroid with stalks
$\mathcal{L}_{x}$ is of rank $2$ or $3$ as a free $\mathcal{O}_{X,x}$-module
when $x\neq\bar{0}$ or $x=\bar{0}$ respectively. In this case, we get PBW
isomorphism for the Lie algebroid $\mathcal{L}$.

  * •

Next, we consider the tangent sheaf $\mathcal{T}_{Y}$ of the analytic space
$(Y,\mathcal{O}_{Y})$, i.e.
$\mathcal{D}er_{\mathbb{C}_{Y}}(\mathcal{O}_{Y})=\langle\\{[D_{x}|_{Y}],[D_{y}|_{Y}],[D_{z}|_{Y}]\\}\rangle$.
For $p\in Y\setminus\\{\bar{0}\\}$ and an open set $U_{p}$ containing $p$, we
consider an open set $V_{p}:=Y\cap U_{p}$ in $Y$ containing $p$. Here the
restriction, $\mathcal{T}_{Y}|_{V_{p}}$ is a free
$\mathcal{O}_{Y}|_{V_{p}}$-module of rank $2$. Hence, $\mathcal{T}_{Y,p}$ is a
free $\mathcal{O}_{Y,p}$-module of rank $2$. But, for $p=\bar{0}$ with an open
neighborhood $V$ of $p$ in $Y$, the space of derivations $\mathcal{T}_{Y}(V)$
cannot be generated by two generators. In fact, $\mathcal{T}_{Y,\bar{0}}$ is a
free $\mathcal{O}_{Y,\bar{0}}$-module of rank $3$. We get the PBW isomorphism
for the Lie algebroid of the tangent sheaf $\mathcal{T}_{Y}$.

The PBW isomorphism for normal crossing divisor $Y$:

  * •

Let $p\in X\setminus Y$ and $U_{p}$ be an open set containing $p$ in
$X\setminus Y$. Now $\mathcal{T}_{X}(-logY)|_{U_{p}}$ is given by the
restriction $\mathcal{T}_{X}|_{U_{p}}$ as $\mathcal{O}_{X}|_{U_{p}}$-modules.
For $q\in\\{x-axis\\}\setminus\\{\bar{0}\\}$, we have
$U_{q}:=X\setminus\\{x=0\\}$ is an open subset containing $q$. Here
$\partial_{y}\notin\mathcal{T}_{X}(-logY)(U_{q})$ on $U_{q}$ as $\frac{1}{y}$
is not defined on $U_{q}$. We find that $\mathcal{T}_{X}(-logY)|_{U_{q}}$ is a
free $\mathcal{O}_{X}|_{U_{q}}$-module of rank $2$. Similarly, for a point
$q\in\\{y-axis\\}\setminus\\{\bar{0}\\}$, there exists an open set $U_{q}$
containing $q$ with the analogous property. Also, stalk at $\bar{0}$ for
$\mathcal{T}_{X}(-logY)$ is of rank $2$ and spanned by the stalks of the
elements in the generating set. Thus, $\mathcal{T}_{X}(-logY)$ is a locally
free $\mathcal{O}_{X}$-module of rank $2$. It is important to note here that
for the singular space $Y$, associated Lie algebroid $\mathcal{L}$ is locally
free, is called a free divisor. The generalized PBW map for $\mathcal{L}$
provides an isomorphism.

  * •

Next, we consider the Lie algebroid $\mathcal{T}_{Y}$. For an open set
containing $q$ say $V_{q}:=Y\cap U_{q}$, we get
$\mathcal{T}_{Y}|_{V_{q}}=\langle\\{(x\partial_{x})|_{Y}\\}\rangle$. Thus,
$\mathcal{T}_{Y}|_{V_{q}}$ is a free $\mathcal{O}_{Y}|_{V_{q}}$-module of rank
$1$. Similarly $\mathcal{T}_{Y}|_{V_{p}}$ is a free
$\mathcal{O}_{Y}|_{V_{p}}$-module of rank $1$. But, the stalk
$\mathcal{T}_{Y,\bar{0}}=\langle\\{(x\partial_{x})|_{Y},(y\partial_{y})|_{Y}\\}\rangle$
is a $\mathcal{O}_{Y,\bar{0}}$-module of rank $2$. Therefore we cannot apply
generalized PBW theorem for $\mathcal{T}_{Y}$, since it is not locally free
$\mathcal{O}_{Y}$-module but its stalks at any point $q$ is free
$\mathcal{O}_{Y,q}$-module with rank either $1$ or $2$. In this case also we
get the PBW isomorphism.

##  5\. $\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebras

The notation $\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebra is considered in
this section as a sheaf of generalized bialgebra structure starting with a
sheaf of unital associative $\mathbb{K}_{X}$-algebra which extends the sheaf
$\mathcal{O}_{X}$. As an example, we consider universal enveloping algebroid
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ for a Lie algebroid $\mathcal{L}$
on $(X,\mathcal{O}_{X})$. On an open set we get back the
$R/{\mathbb{K}}$-bialgebra structure defined on the universal enveloping
algebra. For this $R/{\mathbb{K}}$-bialgebra one can recover the Lie-Rinehart
algebra by considering the space of primitive elements (see [15]). In complex
and algebraic geometry set up, this bialgebra structure appeared in [1, 4, 9],
but we do not find construction involving the space of primitive elements
through which we can recover the Lie algebroid.

###  5.1.  $\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebras :

Let $(X,\mathcal{O}_{X})$ be an $a$-space. Suppose $\mathcal{A}$ is a sheaf of
unital associative $\mathbb{K}_{X}$-algebra extending the sheaf
$\mathcal{O}_{X}$ i.e., $\mathcal{O}_{X}$ is a subsheaf of algebras of
$\mathcal{A}$. We may view $\mathcal{A}$ as a sheaf of
$\mathcal{O}_{X}-\mathcal{O}_{X}$-bimodules. Let us recall the notation that
${}^{p}(\mathcal{A}\otimes_{\mathcal{O}_{X}}\mathcal{A})$ is a presheaf of
left $\mathcal{O}_{X}$-modules. We consider the presheaf homomorphism

$\psi:~{}^{p}(\mathcal{A}\otimes_{\mathcal{O}_{X}}\mathcal{A})\rightarrow\mathcal{H}om_{\mathbb{K}_{X}}(\mathcal{O}_{X},~{}^{p}(\mathcal{A}\otimes_{\mathcal{O}_{X}}\mathcal{A}))$

defined by $\psi(U)(a\otimes b)(f)=a\otimes(bf-fb)$ where
$f\in\mathcal{O}_{X}(U)$ and $a,b\in\mathcal{A}(U)$, for an open subset $U$ of
$X$.

The sheafification of this presheaf morphism $\psi$ yields the sheaf of
$\mathbb{K}_{X}$-algebras expressed as

$\mathcal{A}\bar{\otimes}_{\mathcal{O}_{X}}\mathcal{A}:=\mathscr{K}er(\psi^{\\#})=(\mathcal{K}er~{}\psi)^{\\#}=(^{p}(\mathcal{A}\bar{\otimes}_{\mathcal{O}_{X}}\mathcal{A}))^{\\#}.$

######  Definition 5.1.

A tuple $(\mathcal{A},\Delta,\epsilon)$ is called
$\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebra if the following conditions hold.

  * •

$\mathcal{A}$ is a sheaf of unital associative $\mathbb{K}_{X}$-algebra
extending the sheaf $\mathcal{O}_{X}$;

  * •

$\mathcal{A}$ is equipped with the morphism of sheaf of
$\mathcal{O}_{X}$-modules the comultiplication
$\Delta:\mathcal{A}\rightarrow\mathcal{A}\otimes_{\mathcal{O}_{X}}\mathcal{A}$
and counit $\epsilon:\mathcal{A}\rightarrow\mathcal{O}_{X}$;

  * •

$\Delta(\mathcal{A})$ is a subsheaf of the sheaf of $\mathcal{O}_{X}$-modules
$\mathcal{A}\bar{\otimes}_{\mathcal{O}_{X}}\mathcal{A}$.

  * •

$\Delta(1)=1\otimes 1$ and $\epsilon(1)=1$, where $1$ is the unit of
$\mathcal{O}_{X}$;

  * •

$\Delta(ab)=\Delta(a)\Delta(b)$, $\epsilon(ab)=\epsilon(a\epsilon(b))$ for any
two sections $a,b$ in $\mathcal{A}$.

######  Definition 5.2.

Let $(\mathcal{A}_{1},\Delta_{1},\epsilon_{1})$ and
$(\mathcal{A}_{2},\Delta_{2},\epsilon_{2})$ be two
$\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebras. A morphism

$\psi:(\mathcal{A}_{1},\Delta_{1},\epsilon_{1})\rightarrow(\mathcal{A}_{2},\Delta_{2},\epsilon_{2})$

between the sheaves of $\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebras is a
$(a)$ morphism of sheaves of unital $\mathbb{K}_{X}$-algebras, and
$(b)$ morphism of sheaves of coalgebras in the category of left
$\mathcal{O}_{X}$-modules.

It follows that the $\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebras and their
morphisms forms a category, we call it the category of
$\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebras.

######  Example 5.3.

The universal enveloping algebroid $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$
as $\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebra:

Let $(\mathcal{L},[\cdot,\cdot],\mathfrak{a})$ be a Lie algebroid over one of
the special $a$-space $(X,\mathcal{O}_{X})$. Consider the presheaf
$U\mapsto(\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{L}(U)),\Delta_{U},\epsilon_{U})$
of $\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebra. From the presheaf
$\mathcal{U}(\mathcal{O}_{X},\mathcal{L})$, we get a presheaf of associative
unital $\mathbb{K}$-algebras and left $\mathcal{O}_{X}$-module structure on
$\mathcal{U}(\mathcal{O}_{X},\mathcal{L})\bar{\otimes}_{\mathcal{O}_{X}}\mathcal{U}(\mathcal{O}_{X},\mathcal{L}):=$
${}^{p}(\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\bar{\otimes}_{\mathcal{O}_{X}}\mathscr{U}(\mathcal{O}_{X},\mathcal{L}))$.
Thus, we have a canonical presheaf homomorphisms

${}^{p}\Delta:\mathcal{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathcal{U}(\mathcal{O}_{X},\mathcal{L})\bar{\otimes}_{\mathcal{O}_{X}}\mathcal{U}(\mathcal{O}_{X},\mathcal{L})$

with ${}^{p}\Delta(U)=\Delta_{U}$ and
${}^{p}\epsilon:\mathcal{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathcal{O}_{X}$
with ${}^{p}\epsilon(U)=\epsilon_{U}$.

Hence, $(\mathcal{U}(\mathcal{O}_{X},\mathcal{L}),^{p}\Delta,^{p}\epsilon)$ is
a presheaf of $\mathcal{O}_{X}/\mathbb{K}_{X}$-bialgebras.

It follows that
$\mathcal{U}(\mathcal{O}_{X},\mathcal{L})\bar{\otimes}_{\mathcal{O}_{X}}\mathcal{U}(\mathcal{O}_{X},\mathcal{L})\hookrightarrow\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\bar{\otimes}_{\mathcal{O}_{X}}\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$.
By the universal property of the sheafification we get

$\Delta=(^{p}\Delta)^{\\#}:\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\bar{\otimes}_{\mathcal{O}_{X}}\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$

a morphism between sheaves of associative unital $\mathbb{K}$-algebras and
left $\mathcal{O}_{X}$-modules,

$\epsilon=(^{p}\epsilon)^{\\#}:\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathcal{O}_{X}$

morphism between sheaves of $\mathcal{O}_{X}$-modules where $\Delta\circ
i=(\cdot)^{\\#}\circ$ ${}^{p}\Delta$ and $\epsilon\circ i=$ ${}^{p}\epsilon$.
Consequently, $(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}),\Delta,\epsilon)$ is
a sheaf of $\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebras.

Observe that $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ is a cocommutative
$\mathcal{O}_{X}$-coalgebra in the sense that there is a natural cocommutative
coassociative comultiplication
$\Delta:\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\otimes_{\mathcal{O}_{X}}\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$,
and counit
$\epsilon:\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathcal{O}_{X}$
respectively, which are locally given by the following formulas (using the
Sweedler convention) [5]

$\Delta(f)=f\otimes 1=1\otimes f$,

$\Delta(\bar{D})=\bar{D}\otimes 1+1\otimes\bar{D}$,

$\Delta(D^{\prime}D^{\prime\prime})=\sum
D^{\prime}_{(1)}D^{\prime\prime}_{(1)}\otimes
D^{\prime}_{(2)}D^{\prime\prime}_{(2)}$,

$\epsilon(D)=D(1)$,

for a section $f$ of $\mathcal{O}_{X}$, for a section $D$ of $\mathcal{L}$ and
for sections $D^{\prime},D^{\prime\prime}$ of
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$.

Remark: In [9], the notion of free Lie algebroid $\mathcal{P}_{X}$ is
constructed for tangent sheaf $\mathcal{T}_{X}$. It forms a locally free sheaf
of Lie-Rinehart algebras over $(X,\mathcal{O}_{X})$ of infinite rank. The
universal enveloping algebroid
$\mathscr{U}(\mathcal{O}_{X},\mathcal{P}_{X})=:\mathbb{D}_{X}$ is described as
sheaf of non commutative differential operators on $X$. It is presented
therein as a $\mathcal{O}_{X}$-bialgebra, can be viewed as
$\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebra defined above.

###  5.2. The dual jet space of a Lie algebroid :

The notion of jet algebroid is defined as the dual of the universal enveloping
algebroid:

$\mathscr{J}(\mathcal{O}_{X},\mathcal{L}):=\mathscr{H}om_{\mathcal{O}_{X}}(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}),\mathcal{O}_{X})$.

It is sheafification of the presheaf associated to jet algebras
$\mathcal{J}(\mathcal{O}_{X}(U),\mathcal{L}(U))$ of the
$(\mathbb{K},\mathcal{O}_{X}(U))$-Lie-Rinehart algebra $\mathcal{L}(U)$ (see
[1, 4, 11]). If $\mathcal{L}$ is locally free $\mathcal{O}_{X}$-module of
constant rank, we can view $\mathscr{J}(\mathcal{O}_{X},\mathcal{L})$ as
projective limit of $n$-jets of $\mathcal{L}$, that is

$\mathscr{J}(\mathcal{O}_{X},\mathcal{L})=\underset{n}{\varprojlim}\mathscr{J}^{n}(\mathcal{O}_{X},\mathcal{L})$
(as $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})=\underset{n}{\varinjlim}$
$\mathscr{U}_{(n)}(\mathcal{O}_{X},\mathcal{L}))$,

where space of $n$-jets of $\mathcal{L}$ is defined as
$\mathscr{J}^{n}(\mathcal{O}_{X},\mathcal{L}):=\mathscr{H}om_{\mathcal{O}_{X}}(\mathscr{U}_{(n)}(\mathcal{O}_{X},\mathcal{L}),\mathcal{O}_{X})$.
By definition, $\mathscr{J}(\mathcal{O}_{X},\mathcal{L})$ is complete and
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ is cocomplete with respect to the
canonical PBW filtration.

One can dualize the structures on $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$.
The product on $\mathscr{J}(\mathcal{O}_{X},\mathcal{L})$ is induced from the
coproduct of $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ on each space of
sections, which is locally defined by

$(\phi_{1}\phi_{2})(D):=\phi_{1}(D_{(1)})\phi_{2}(D_{(2)})$

for sections $\phi_{1},\phi_{2}\in\mathscr{J}(\mathcal{O}_{X},\mathcal{L})$
and a section $D\in\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$. By
cocommutativity of $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$, this defines a
commutative algebra structure on $\mathscr{J}(\mathcal{O}_{X},\mathcal{L})$.
The unit for this multiplication is locally given by the left counit
$\epsilon:\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathcal{O}_{X}$,
since

$(\epsilon\phi)(D)=\epsilon(D_{(1)})\phi(D_{(2)})=\phi(\epsilon(D_{(1)})D_{(2)})=\phi(D)$

for a section $\phi$ in $\mathscr{J}(\mathcal{O}_{X},\mathcal{L})$ and a
section $D$ in $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$. The left and right
$\mathcal{O}_{X}$-module structure on
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ provides
$\mathcal{O}_{X}-\mathcal{O}_{X}$-bimodule structure on
$\mathscr{J}(\mathcal{O}_{X},\mathcal{L})$.

Observe now that the product on $\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$
descends to a sheaf homomorphism
$m:\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\otimes_{\mathcal{O}_{X}}\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$.
We can therefore dualize the product to obtain a coproduct
$\Delta^{*}:\mathscr{J}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathscr{J}(\mathcal{O}_{X},\mathcal{L})\otimes_{\mathcal{O}_{X}}\mathscr{J}(\mathcal{O}_{X},\mathcal{L})$
locally defined as

$\phi(DD^{\prime})=:\Delta^{*}(\phi)(D\otimes
D^{\prime})=\phi_{(1)}(D\phi_{(2)}(D^{\prime}))$

Associativity of the multiplication implies that $\Delta^{*}$ is
coassociative. The counit for this coproduct is given by
$\epsilon^{*}:\phi\mapsto\phi(1_{\mathscr{U}(\mathcal{O}_{X},\mathcal{L})})$.
It follows that
$(\mathscr{J}(\mathcal{O}_{X},\mathcal{L}),\Delta^{*},\epsilon^{*})$ is
$\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebra.

PBW Theorem for jet algebroid : For a locally free Lie algebroid $\mathcal{L}$
over $(X,\mathcal{O}_{X})$ of finite rank, say $r$, locally on some open
neighborhood $U_{x}$ of $x$ we get

$\mathscr{J}(\mathcal{O}_{X},\mathcal{L})(U_{x})\cong\mathcal{O}_{X}(U_{x})[[w_{1},\dots,w_{r}]]$

(since $\mathscr{J}(\mathcal{O}_{X},\mathcal{L})$ is commutative
$\mathcal{O}_{X}$-algebra and $\mathcal{L^{*}}$ is locally free
$\mathcal{O}_{X}$-module of rank $r$ and whose local basis on $U_{x}$ is
$\\{w_{1},\dots,w_{r}\\}$). Thus,
$\mathscr{J}(\mathcal{O}_{X},\mathcal{L})\cong\widehat{\mathcal{S}_{\mathcal{O}_{X}}\mathcal{L^{*}}}$
(this sheaf of symmetric algebras is formally completed with respect to the
degree) as $\mathcal{O}_{X}$-algebras [5, 1].

######  Example 5.4.

Enveloping algebra as a generalized bialgebra: Let $R$ be an associative
unital $\mathbb{K}$-algebra. Then the enveloping algebra of $R$ is denoted by

$R^{e}:=R\otimes_{\mathbb{K}}R^{op}$

has a canonical left bialgebroid (or $R/{\mathbb{K}}$-bialgebra) structure
(see[11]).

As a sheaf theoretical generalization we get a
$\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebra. Here, we consider
$\mathcal{O}^{e}_{X}$ as the sheafification of the presheaf given by

$U\mapsto\mathcal{O}_{X}(U)\otimes\mathcal{O}_{X}(U)^{op}$

where the restriction map is defined by the component-wise restrictions.

The multiplication
$m:\mathcal{O}_{X}^{e}\otimes_{\mathbb{K}_{X}}\mathcal{O}_{X}^{e}\rightarrow\mathcal{O}_{X}^{e}$
locally expressed as

$m((a_{1}\otimes b_{1}),(a_{2}\otimes b_{2}))=a_{1}a_{2}\otimes b_{2}b_{1}$

for sections $a_{1},a_{2}$ in $\mathcal{O}_{X}$ and $b_{1},b_{2}$ in
$\mathcal{O}_{X}^{e}$. The unit
$\eta=i\circ\eta_{1}:\mathbb{K}_{X}\rightarrow\mathcal{O}^{e}_{X}$ is given by
the compositions of the canonical maps
$\eta_{1}:\mathbb{K}_{X}\rightarrow\mathcal{O}_{X}$ and
$i:\mathcal{O}_{X}\hookrightarrow\mathcal{O}^{e}_{X}$.

The comultiplication and counit are locally defined as follows:

$\Delta(a\otimes b)=(a\otimes 1)\otimes_{\mathcal{O}_{X}}(1\otimes b)$

and $\epsilon(a\otimes b)=ab$ for sections $a,b$ in $\mathcal{O}_{X}^{e}$.

More generally, if we consider any $\mathbb{K}_{X}$-algebra $\mathcal{A}$, the
sheafification of the presheaf of enveloping algebras
$\mathcal{A}^{e}:=\mathcal{A}\otimes\mathcal{A}^{op}$ forms a natural
$\mathcal{A}/{\mathbb{K}_{X}}$-bialgebra structures whose local descriptions
is as given above.

###  5.3. Primitive filtration of a
$\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebra :

For a given $\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebra $\mathcal{A}$, it
follows that $\bar{\mathcal{A}}:=$ $\mathscr{K}er$ $\epsilon$ is a subsheaf of
$\mathcal{O}_{X}$-modules and
$\mathcal{A}=\mathcal{O}_{X}\oplus\bar{\mathcal{A}}$ as direct sum of
$\mathcal{O}_{X}$-modules. Moreover, the $\mathcal{O}_{X}$-submodule,
$\bar{\mathcal{A}}$ is a subsheaf of $\mathbb{K}_{X}$-algebras equipped with
cocommutative coassociative coproduct

$\bar{\Delta}:\bar{\mathcal{A}}\rightarrow\bar{\mathcal{A}}\otimes_{\mathcal{O}_{X}}\bar{\mathcal{A}}$

is locally given by $\bar{\Delta}(s)=\Delta(s)-(s\otimes 1+1\otimes s)$ for a
section $s$ in $\bar{\mathcal{A}}$. Consequently, $\bar{\mathcal{A}}$ is a
non-counital $\mathcal{O}_{X}$-coalgebra.

Here, the bialgebra $\mathcal{A}$ can be reconstructed from
$\mathcal{O}_{X}$-coalgebra $\bar{\mathcal{A}}$ with the coproduct
$\bar{\Delta}$ and the structure of sheaf of $\mathbb{K}_{X}$-algebras on
$\bar{\mathcal{A}}$. Suppose $\bar{\Delta}^{(n)}$ the iterated coproduct
between $\mathcal{O}_{X}$-modules where

$\bar{\Delta}^{(1)}=\bar{\Delta}~{}~{}\mbox{and}~{}~{}\bar{\Delta}^{(n)}=(\bar{\Delta}\otimes\mbox{Id}_{\bar{\mathcal{A}}})\circ\bar{\Delta}^{(n-1)}~{}~{}\mbox{for}~{}~{}n\geq
2.$

If $\bar{\mathcal{A}}_{n}=$ $\mathscr{K}$er $\bar{\Delta}^{(n)}$ then we get a
filtration (of $\mathcal{O}_{X}$-modules $\bar{\mathcal{A}}$) written as
follows:

${0}=\bar{\mathcal{A}}_{0}\subset\bar{\mathcal{A}}_{1}\subset\bar{\mathcal{A}}_{2}\subset\cdots.$

It is refered as the primitive filtration of $\bar{\mathcal{A}}$. We write
${\mathcal{A}}_{n}=\mathcal{O}_{X}\oplus\bar{\mathcal{A}}_{n}$ for $n\geq 0$.
The subsheaf of $\mathcal{O}_{X}$-modules $\bar{\mathcal{A}}_{1}$ is called
(sub)sheaf of primitive elements and is also denoted by
$\mathscr{P}(\mathcal{A})$. The bialgebra $\mathcal{A}$ is said to be
cocomplete if $\mathcal{A}=\cup_{n\geq 0}{\mathcal{A}}_{n}$. It is called
locally graded free if each of the sheaf of subquotients

${\mathcal{A}}_{n+1}/{{\mathcal{A}}_{n}}={\bar{\mathcal{A}}}_{n+1}/{{\bar{\mathcal{A}}}_{n}}$

is a locally free $\mathcal{O}_{X}$-modules. For this we have
$\mathscr{P}(\mathcal{A})$ is a locally free $\mathcal{O}_{X}$-module We say
that $\mathcal{A}$ is of finite type if $\mathscr{P}(\mathcal{A})$ is locally
free $\mathcal{O}_{X}$-module of finite rank.

######  Remark 5.5.

The universal enveloping algebra of a Lie algebroid over a smooth manifold is
cocomplete, graded projective and finite type (see [15]). Here, we consider
the presheaf $\mathcal{U}(\mathcal{O}_{X},\mathcal{L})$ for a Lie algebroid
$\mathcal{L}$ with $\mathcal{L}$ is a locally free left
$\mathcal{O}_{X}$-module. It follows that the presheaf of natural filtration
and primitive filtration coincides on each special open set $U_{x}$ for the
presheaf $\mathcal{U}(\mathcal{O}_{X},\mathcal{L})$. There is an isomorphism
of $\mathcal{O}_{X,x}$-modules between the induced stalks of the two types of
presheaves. On application of sheafification functor we obtain sheaf
isomorphism between the filtrations of the sheaf
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$. The universal enveloping algebroid
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ is cocomplete, locally graded free.
Furthermore, if $\mathcal{L}$ is of finite rank then
$\mathscr{U}(\mathcal{O}_{X},\mathcal{L})$ is of finite type.

###  5.4. Sheaf of Lie-Rinehart algebras of primitive elements :

Let $\mathcal{A}$ be a $\mathcal{O}_{X}/\mathbb{K}_{X}$-bialgebra. We want to
construct its associated Lie algebroid
$(\mathscr{P}(\mathcal{A}),[\cdot,\cdot]_{c},\mathfrak{a})$ over an $a$\-
space $(X,\mathcal{O}_{X})$.

The counit map $\epsilon:\mathcal{A}\rightarrow\mathcal{O}_{X}$ induces sheaf
homomorphism
$\rho:\mathcal{A}\rightarrow\mathscr{E}nd_{\mathbb{K}_{X}}(\mathcal{O}_{X})$
of $\mathbb{K}_{X}$-algebras. This map is locally expressed as
$\rho(s)(f)=\epsilon(s.f)$ for a section $s\in\mathcal{A}$ and a section
$f\in\mathcal{O}_{X}$. Thus, for each open set $U$ of $X$ we get the
$(\mathbb{K},\mathcal{O}_{X}(U))$-Lie-Rinehart algebra
$(\mathcal{P}(\mathcal{A}(U)),[\cdot,\cdot]_{c},\mathfrak{a}_{U})$ consisting
with primitive elements of $\mathcal{O}_{X}/{\mathbb{K}}$-bilagebra
$\mathcal{A}(U)$. The anchor map

$\mathfrak{a}_{U}:(\mathcal{P}(\mathcal{A}(U)),[\cdot,\cdot]_{c})\rightarrow(Der_{\mathbb{K}}(\mathcal{O}_{X}(U)),[\cdot,\cdot]_{c})$

is defined by $\mathfrak{a}_{U}(D)(f)=\epsilon(U)(D\cdot f)$ for
$f\in\mathcal{O}_{X}(U)$ and $D\in\mathcal{P}({\mathcal{A}(U)})$.

Now, for a pair of open sets $V,U$ in $X$ with $V\subset U$ we have the
restriction morphism
$res_{UV}^{\mathcal{A}}:\mathcal{A}(U)\rightarrow\mathcal{A}(V)$. It induces a
morphism $res_{UV}^{\mathscr{P}(\mathcal{A})}$ between the
$(\mathbb{K},\mathcal{O}_{X}(U))$ Lie-Rinehart algebras
$(\mathcal{P}(\mathcal{A}(U)),[\cdot,\cdot]_{c},\mathfrak{a}_{U})$ and
$(\mathbb{K},\mathcal{O}_{X}(V))$ Lie-Rinehart algebra
$(\mathcal{P}(\mathcal{A}(V)),[\cdot,\cdot]_{c},\mathfrak{a}_{V})$. Then we
get $res_{UV}^{\mathcal{T}}\circ\mathfrak{a}_{U}=\mathfrak{a}_{V}\circ
res_{UV}^{\mathscr{P}(\mathcal{A})}$. Thus, it follows that

$\mathscr{P}(\mathcal{A}):{Open}_{X}\rightarrow\mathbb{K}$-Lie Algebras

$U\mapsto(\mathcal{P}(\mathcal{A}(U)),[\cdot,\cdot]_{c},\mathfrak{a}_{U})$

forms a sheaf of Lie-Rinehart algebras with the restriction morphism
$res_{UV}^{\mathscr{P}(\mathcal{A})}$ (the sheaf structure on
$\mathscr{P}(\mathcal{A})$ induces canonically from the sheaf $\mathcal{A}$).

Thus $(\mathscr{P}(\mathcal{A}),[\cdot,\cdot]_{c},\mathfrak{a})$ is a Lie
algebroid over $(X,\mathcal{O}_{X})$ with anchor map

$\mathfrak{a}:(\mathscr{P}(\mathcal{A}),[\cdot,\cdot])\rightarrow(\mathcal{D}er_{\mathbb{K}_{X}}(\mathcal{O}_{X}),[\cdot,\cdot]_{c})$

is defined by $\mathfrak{a}(U)=\mathfrak{a}_{U}$ on the space of sections.

Primitive elements for Weyl algebra associated with normal crossing: Consider
the $\mathbb{K}$-algebra $\tilde{R}:=\mathbb{K}[x,y]$. The
$(\mathbb{K},\tilde{R})$-Lie-Rinehart algebra of derivations
$Der_{\mathbb{K}}(\tilde{R})=\langle\\{\partial_{x},\partial_{y}\\}\rangle$.
Thus the Weyl algebra of $\tilde{R}$ is

$\mathcal{D}iff(\tilde{R})\cong\mathcal{U}(\tilde{R},Der_{\mathbb{K}}(\tilde{R}))$

forms a $\tilde{R}/{\mathbb{K}}$-bialgebra. Consider the $\mathbb{K}$-algebra
$R:=\mathbb{K}[x,y]/\langle xy\rangle$. The associated space of differential
operators $\mathcal{D}iff(R)$ and the associated universal enveloping algebra
$\mathcal{U}(R,Der_{\mathbb{K}}(R))$ with
$Der_{\mathbb{K}}(R)=\langle\\{x\partial_{x},y\partial_{y}\\}\rangle$, forms a
$R/{\mathbb{K}}$-bialgebras.

For the first case $Der_{\mathbb{K}}(\tilde{R})$ is a projective (in fact
free) $\tilde{R}$-module of rank $2$, by using Cartier-Milnor-Moore theorem
for Lie-Rinehart algebra [15], we get the space of primitive elements of
$\mathcal{D}iff(\tilde{R})$ is
$\mathcal{P}(\mathcal{U}(\tilde{R},Der_{\mathbb{K}}(\tilde{R})))\cong
Der_{\mathbb{K}}(\tilde{R})$. But, this approach will not work for the second
case, since there $Der_{\mathbb{K}}(R)$ is not a projective module over the
base algebra $R$.

Here we compute the space of primitive elements
$\mathcal{P}(\mathcal{D}iff(R))$. First note that
$Der_{\mathbb{K}}(R)\subset\mathcal{P}(\mathcal{D}iff(R))$. Here
$\Delta_{R}(x\partial_{x})=x\partial_{x}\otimes_{R}1+1\otimes_{R}x\partial_{x}$
and similarly
$\Delta(y\partial_{y})=y\partial_{y}\otimes_{R}1+1\otimes_{R}y\partial_{y}$.
Consider the second order differential operator $x\partial_{x}\circ
y\partial_{y}$, then $\Delta(x\partial_{x}\circ
y\partial_{y})=x\partial_{x}\circ
y\partial_{y}\otimes_{R}1+1\otimes_{R}x\partial_{x}\circ y\partial_{y}$ (as
$y\partial_{y}\otimes_{R}x\partial_{x}=0=x\partial_{x}\otimes_{R}y\partial_{y})$
and thus $\mathcal{D}iff_{(2)}(R)\setminus
R\subset\mathcal{P}(\mathcal{D}iff(R))$ where $\mathcal{D}iff_{(2)}(R)$ is the
second filtered $R$-module for the natural filtration of $\mathcal{D}iff(R)$
(like for the case $\mathcal{U}(R,Der_{\mathbb{K}}(R))$). Note that
$y\partial_{y}\circ x\partial_{x}=x\partial_{x}\circ
y\partial_{y}-[x\partial_{x},y\partial_{y}]\in\mathcal{P}(\mathcal{U}(R,Der_{\mathbb{K}}(R))$.
Consequently, $Der_{\mathbb{K}}(R)\subsetneq\mathcal{P}(\mathcal{D}iff(R))$
and thus, $Der_{\mathbb{K}}(R)$ is non-isomorphic to
$\mathcal{P}(\mathcal{U}(R,Der_{\mathbb{K}}(R))$.

This idea can be extended to its associated complex geometric situation.

##  6\. Generalized Cartier-Milnor-Moore Theorem

Let us fix an special $a$-space $(X,\mathcal{O}_{X})$ and consider the
following three categories.

  * •

$\mathcal{C}$ be the category of Lie algebroids over $(X,\mathcal{O}_{X})$,

  * •

$\bar{\mathcal{D}}$ denotes the category of presheaves of
$\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebras, and

  * •

$\mathcal{D}$ denotes the category of
$\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebras.

###  6.1. $\mathscr{U}(\mathcal{O}_{X},-)$ or simply $\mathscr{U}$ as a
functor :

Following notations in previous sections, the assignment
$\mathcal{L}\mapsto\mathcal{U}(\mathcal{O}_{X},\mathcal{L})$ defines a
covariant functor

$\mathcal{U}:\mathcal{C}\rightarrow\bar{\mathcal{D}}.$

Then we define the functor $\mathscr{U}:\mathcal{C}\rightarrow\mathcal{D}$ by
composition with the sheafification functor

$(\cdot)^{\\#}:\bar{\mathcal{D}}\rightarrow\mathcal{D}.$

Here we proceed as follows for morphisms, in order to define the composition
of the functor $\mathcal{U}$ followed by sheafification. Let
$\phi:(\mathcal{L}_{1},[\cdot,\cdot]_{1},\mathfrak{a}_{1})\rightarrow(\mathcal{L}_{2},[\cdot,\cdot]_{2},\mathfrak{a}_{2})$
be a morphism in $\mathcal{C}$. We find a morphism $\widetilde{{}^{p}\phi}$ in
the category $\bar{\mathcal{D}}$ where

$\widetilde{{}^{p}\phi}:\mathcal{U}(\mathcal{O}_{X},\mathcal{L}_{1})\rightarrow\mathcal{U}(\mathcal{O}_{X},\mathcal{L}_{2})$

is defined by $\widetilde{{}^{p}\phi}(U)=\widetilde{\phi(U)}$ for every open
set $U$ of $X$. Note that $\widetilde{{\phi(U)}}$ is obtained by evaluating
the functor $\mathcal{U}(\mathcal{O}_{X}(U),-)$ on the morphism $\phi(U)$.

Finally we obtain the morphism of sheaves of
$\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebras denoted by
${\widetilde{\phi}}:\mathscr{U}(\mathcal{O}_{X},\mathcal{L}_{1})\rightarrow\mathscr{U}(\mathcal{O}_{X},\mathcal{L}_{2})$
where $\widetilde{\phi}=(\cdot)^{\\#}\circ\widetilde{{}^{p}\phi}$.

###  6.2. $\mathscr{P}(-)$ or simply $\mathscr{P}$ as a functor :

Following notations in previous sections, we show that the assignment
$\mathcal{A}\mapsto\mathscr{P}(\mathcal{A})$ defines a covariant functor

$\mathscr{P}:\mathcal{D}\rightarrow\mathcal{C}.$

We have seen before that if $\mathcal{A}$ is an object in the category
$\mathcal{D}$ then $\mathscr{P}(\mathcal{A})$ is an object in the category
$\mathcal{C}$. Let $\psi:\mathcal{A}\rightarrow\mathcal{\tilde{A}}$ be a
morphism in the category $\mathcal{D}$. Then, on each open set $U$ of $X$, we
have the morphism of $(\mathbb{K},\mathcal{O}_{X}(U))$-Lie-Rinehart algebras

$\phi_{U}:=\psi(U)|_{\mathcal{P}(\mathcal{A}(U))}:\mathcal{P}(\mathcal{A}(U))\rightarrow\mathcal{P}(\mathcal{\tilde{A}}(U)),$

which are compatible with the restriction morphisms.

Thus we obtain a morphism of Lie algebroids
$\phi:\mathscr{P}(\mathcal{A})\rightarrow\mathscr{P}(\mathcal{\tilde{A}})$
defined by $\phi(U)=\phi_{U}$.

Next, we use results from [15] as the local descriptions.

######  Note 6.1.

The canonical maps (3.1)

$\iota_{\mathcal{L}}:\mathcal{L}\rightarrow\mathscr{U}(\mathcal{O}_{X},\mathcal{L})~{}~{}\mbox{and}~{}~{}\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathscr{P}(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}))$

obtained from the functor $\mathscr{P}$, provides a morphism
$\alpha_{\mathcal{L}}:\mathcal{L}\rightarrow\mathscr{P}(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}))$
in the category $\mathcal{C}$ (see Example 5.3) via compositions of the maps.
On each open set $U$, we have the canonical homomorphism
$\alpha_{\mathcal{L}(U)}:\mathcal{L}(U)\rightarrow\mathcal{P}(\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{L}(U)))$
compatible with restriction morphism, and sheafification provides the map
$\alpha_{\mathcal{L}}$.

In addition, if $\mathcal{L}$ is locally free $\mathcal{O}_{X}$-module, the
morphism $\alpha_{\mathcal{L}}$ become an isomorphism i.e.,
$\mathcal{L}\cong\mathscr{P}(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}))$.

######  Note 6.2.

For an object $\mathcal{A}$ in the category $\mathcal{D}$, the inclusion map
$\mathscr{P}(\mathcal{A})\hookrightarrow\mathcal{A}$ provides a morphism

$\beta_{\mathcal{A}}:\mathscr{U}(\mathcal{O}_{X},\mathscr{P}(\mathcal{A}))\rightarrow\mathcal{A}$

in $\mathcal{D}$ by using the universal property of the universal enveloping
algebroid (Remark 3.1)
$\mathscr{U}(\mathcal{O}_{X},\mathscr{P}(\mathcal{A}))$. On each open set $U$,
we have the canonical homomorphism
$\beta_{\mathcal{A}(U)}:\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{P}(\mathcal{A}(U)))\rightarrow\mathcal{A}(U)$
compatible with restriction morphism, provides the map $\beta_{\mathcal{A}}$

In addition, if $\mathcal{A}$ is cocomplete locally graded free
$\mathcal{O}_{X}$-module, the morphism $\beta_{\mathcal{A}}$ become an
isomorphism i.e.,
$\mathcal{A}\cong\mathscr{U}(\mathcal{O}_{X},\mathscr{P}(\mathcal{A}))$.

###  6.3. Cartier-Milnor-Moore Theorem for Lie algebroids over $a$-spaces

Here we consider a generalized version of Cartier-Milnor-Moore Theorem (CMM
theorem in short hand notation) in the sheaf theoretic terminology.

We denote by $C$ the category of $(\mathbb{K},R)$-Lie-Rinehart algebras and
the category of $R/{\mathbb{K}}$-bialgebras by $D$. In terms of the notations
previously used, we consider the next result.

######  Theorem 6.3.

  1. (1)

The functor $\mathscr{U}$ is left adjoint to the functor $\mathscr{P}$.

  2. (2)

The functors $\mathscr{U}$ and $\mathscr{P}$ restrict to an equivalence
between the full subcategory $\mathcal{C^{\prime}}$ of locally free Lie
algebroids over $(X,\mathcal{O}_{X})$ and the full subcategory
$\mathcal{D^{\prime}}$ of cocomplete locally graded free
$\mathcal{O}_{X}/\mathbb{K}_{X}$-bialgebras.

###### Proof.

(1) Let $\phi:\mathcal{L}\rightarrow\mathscr{P}(\mathcal{A})$ be a morphism in
the category $\mathcal{C}$ of Lie algebroids over $(X,\mathcal{O}_{X})$. For a
pair of open sets $V,U$ in $X$ where $V\subset U$ we have the morphisms
$\phi(U)$ and $\phi(V)$ in the category $C$ with
$res_{UV}^{\mathscr{P}(\mathcal{A})}\circ\phi(U)=\phi(V)\circ
res_{UV}^{\mathcal{L}}$.

As a byproduct of the universal enveloping algebra functor we get morphisms

$\widetilde{\phi(U)}:\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{L}(U))\rightarrow\mathcal{A}(U)~{}~{}\mbox{and}~{}~{}\widetilde{\phi(V)}:\mathcal{U}(\mathcal{O}_{X}(V),\mathcal{L}(V))\rightarrow\mathcal{A}(V)$

with the compatibility condition
$res_{UV}^{\mathcal{A}}\circ\widetilde{\phi(U)}=\widetilde{\phi(V)}\circ
res_{UV}^{\mathcal{U}}$. Then

$\widetilde{\phi}:\mathcal{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathcal{A}$

is a morphism between presheaf of $\mathcal{O}_{X}/\mathbb{K}_{X}$-bialgebras,
where $\widetilde{\phi}(U):=\widetilde{\phi(U)}$ for every open set $U$ of
$X$. By the universal property of sheafification we find a morphism
$\bar{\phi}:\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathcal{A}$ in
$\mathcal{D}$ where $\bar{\phi}$ is locally given by the map
$\widetilde{\phi}$.

On the other hand, suppose
$\psi:\mathscr{U}(\mathcal{O}_{X},\mathcal{L})\rightarrow\mathcal{A}$ is a
morphism in $\mathcal{D}$. Then for any pair of open sets $V,U$ of $X$ where
$V\subset U$ we have the morphisms $\psi(U)$ and $\psi(V)$ in the category $D$
with $res_{UV}^{\mathcal{A}}\circ\psi(U)=\psi(V)\circ res_{UV}^{\mathscr{U}}$.

The primitive elements functor gives (using Note 6.1) the morphisms in $C$
written as $\phi_{U}:\mathcal{L}(U)\rightarrow\mathcal{P}(\mathcal{A}(U))$ and
$\phi_{V}:\mathcal{L}(V)\rightarrow\mathcal{P}(\mathcal{A}(V))$ respectively.

It follows by using the functor $\mathscr{P}$ that
$res_{UV}^{\mathscr{P}(\mathcal{A})}\circ\phi_{U}=\phi_{V}\circ
res_{UV}^{\mathcal{L}}$.

Thus, we can form $\phi:\mathcal{L}\rightarrow\mathscr{P}(\mathcal{A})$, a
morphism in $\mathcal{C}$ where $\phi(U):=\phi_{U}.$ Consequently, we get a
bijection of the sets
$\mathscr{H}om_{\mathcal{C}}(\mathcal{L},\mathscr{P}(\mathcal{A}))$ and
$\mathscr{H}om_{\mathcal{D}}(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}),\mathcal{A})$.

For an object $\mathcal{L}$ in the category $\mathcal{C}$, we have the
canonical morphism

$\alpha_{\mathcal{L}}:\mathcal{L}\rightarrow\mathscr{P}(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}))$

in $\mathcal{C}$ (Note 6.1).

Let $\phi:\mathcal{L}_{1}\rightarrow\mathcal{L}_{2}$ be a morphism in category
$\mathcal{C}$. We find the sheaf homomorphisms $\alpha_{\mathcal{L}_{1}}$,
$\alpha_{\mathcal{L}_{2}}$ and
$\widetilde{\phi}_{res}:=\mathscr{P}(\tilde{\phi})$, which are morphisms in
$\mathcal{C}$ such that
$\tilde{\phi}_{res}\circ\alpha_{\mathcal{L}_{1}}=\alpha_{\mathcal{L}_{2}}\circ\phi$.

It defines unit of the adjunction defined by a natural transformation

$\alpha:Id_{\mathcal{C}}\rightarrow\mathscr{P}\circ\mathscr{U}$ where
$\alpha_{\mathcal{L}}:\mathcal{L}\mapsto\mathscr{P}(\mathscr{U}(\mathcal{O}_{X},\mathcal{L}))$.

Analogously for a morphism $\psi:\mathcal{A}_{1}\rightarrow\mathcal{A}_{2}$ in
the category $\mathcal{D}$, we have compatible sheaf homomorphisms
$\beta_{\mathcal{A}_{1}},\beta_{\mathcal{A}_{2}}$ (Note 6.1) and

$\widetilde{\psi_{res}}:=\mathscr{U}\circ\mathscr{P}~{}(\psi).$

As a result, the counit of the adjunction is given by a natural transformation

$\beta:\mathscr{U}\circ\mathscr{P}\rightarrow Id_{\mathcal{D}}$ where
$\beta_{\mathcal{A}}:\mathscr{U}(\mathcal{O}_{X},\mathscr{P}(\mathcal{A}))\mapsto\mathcal{A}$.

The underlying presheaf homomorphisms satisfy the triangular identities on the
space of sections of objects $\mathcal{L}$ in $\mathcal{C}$ and $\mathcal{A}$
in $\mathcal{D}$ respectively. Here we have

$\mathcal{P}(\beta_{\mathcal{A}(U)})\circ\alpha_{\mathcal{P}(\mathcal{A}(U))}=id_{\mathcal{P}(\mathcal{A}(U))}$

and
$\beta_{\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{L}(U))}\circ\mathcal{U}(\mathcal{O}_{X}(U),\alpha_{\mathcal{L}(U)})=id_{\mathcal{U}(\mathcal{O}_{X}(U),\mathcal{L}(U))}$.

By sheafification, it induces morphism in $\mathcal{C}$ and $\mathcal{D}$
respectively satisfying

$\mathscr{P}(\beta_{\mathcal{A}})\circ\alpha_{\mathscr{P}(\mathcal{A})}=id_{\mathscr{P}(\mathcal{A})}$
and
$\beta_{\mathscr{U}(\mathcal{O}_{X},\mathcal{L})}\circ\mathscr{U}(\mathcal{O}_{X},\alpha_{\mathcal{L}})=id_{\mathscr{U}(\mathcal{O}_{X},\mathcal{L})}$.

(2) Now, we have a special open cover $\\{U_{x}\\}_{x}$ of $X$ for which
$\mathcal{L}|_{U_{x}}$ is free as left $\mathcal{O}_{X}|_{U_{x}}$-module. For
any open set $V$ inside a special open set $U_{x}$, we get the isomorphism

$\alpha_{\mathcal{L}(V)}:\mathcal{L}(V)\rightarrow\mathcal{P}(\mathcal{U}(\mathcal{O}_{X}(V),\mathcal{L}(V)))$

in $C$. This is also compatible with the restriction maps and induces stalk-
wise isomorphisms. Thus, $\alpha_{\mathcal{L}}$ is an isomorphism (using Note
6.1, Lemma 3.2) for $\mathcal{L}$ in the full subcategory of locally free Lie
algebroids over $(X,\mathcal{O}_{X})$. We get the unit of the adjunction
$\alpha:Id_{\mathcal{C^{\prime}}}\rightarrow\mathscr{P}\circ\mathscr{U}$ is a
natural isomorphism, or,
$Id_{\mathcal{C^{\prime}}}\cong\mathscr{P}\circ\mathscr{U}$.

In this case too, we have the special open cover $\\{U_{x}\\}_{x}$ of $X$ for
which $\mathcal{A}(V)$ is cocomplete graded free
$\mathcal{O}_{X}(V)/\mathbb{K}$-bialgebra for any open set $V$ inside some
$U_{x}$. Thus, we get the isomorphism

$\beta_{{\mathcal{A}}(V)}:\mathcal{U}(\mathcal{O}_{X}(V),\mathcal{P}(\mathcal{A}(V)))\rightarrow\mathcal{A}(V)$

in $D$, compatible with restriction maps. It induces stalk-wise isomorphism
and provides $\beta_{\mathcal{A}}$ is an isomorphism (using Note 6.1, Lemma
3.2) for $\mathcal{A}$ in the full subcategory of cocomplete graded free
$\mathcal{O}_{X}/{\mathbb{K}_{X}}$-bialgebras. Thus, the counit
$\beta:\mathscr{U}\circ\mathscr{P}\rightarrow Id_{\mathcal{D^{\prime}}}$ of
the adjunction is a natural isomorphism, or, $\mathscr{U}\circ\mathscr{P}\cong
Id_{\mathcal{D^{\prime}}}$.

∎

######  Corollary 6.4.

The functors $\mathscr{U}$ and $\mathscr{P}$ restrict to an equivalence
between the full subcategory of locally free Lie algebroids over
$(X,\mathcal{O}_{X})$ of finite ranks and the full subcategory of cocomplete
locally graded free $\mathcal{O}_{X}/\mathbb{K}_{X}$-bialgebras of finite
type.

######  Remark 6.5.

One may work on locally free Lie algebroids $\mathcal{L}$ (of finite rank)
over noetherian separated schemes (or schemes of finite types) over a field of
characteristic zero [2, 9]. The PBW theorem and the CMM theorem also hold in
this case. On each affine pieces $U_{i}$, the
$(\mathbb{K},\mathcal{O}_{X}(U_{i}))$-Lie-Rinehart algebra
$\mathcal{L}(U_{i})$ is projective $\mathcal{O}_{X}(U_{i})$-module and the
stalks ${\mathcal{L}}_{x}$ are free $(\mathbb{K},\mathcal{O}_{X,x})$-Lie-
Rinehart algebras for every $x\in X$ (using Lemma 5.1.3. from[22]). We find
the PBW isomorphisms of $\mathcal{O}_{X}(U_{i})$-algebras
$S_{\mathcal{O}_{X}(U_{i})}\mathcal{L}(U_{i})\overset{\theta_{U_{i}}}{\longrightarrow}\mbox{gr}(~{}\mathcal{U}(\mathcal{O}_{X}(U_{i}),\mathcal{L}(U_{i}))~{}),$
which are compatible with restrictions. It provides PBW isomorphisms for
$\mathcal{L}_{x}$ on the level of stalks and also the generalized PBW
isomorphism.

Moreover, if we consider a locally free Lie algebroid of infinite rank, then
the PBW isomorphisms on the level of stalks provides the generalized PBW
isomorphism.

Acknowledgements:
We acknowledge the research support from C. S. I. R fellowship grant $2015$
and SERB, DST, Government of India, MATRICS -Research grant $MTR/2018/000510$.
We sincerely thank the anonymous referee for making several useful and
important suggestions, which helped to improve the presentation and clarity of
the article.

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Ashis Mandal and Abhishek Sarkar
Department of Mathematics and Statistics,
Indian Institute of Technology, Kanpur 208016, India.
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>