Categorical Perspective on Quantization of Poisson Algebra

We propose a generalization of quantization as a categorical way. For a fixed Poisson algebra quantization categories are defined as subcategories of R-module category with the structure of classical limits. We construct the generalized quantization categories including matrix regularization, strict deformation quantization, prequantization, and Poisson enveloping algebra, respectively. It is shown that the categories of strict deformation quantization, prequantization, and matrix regularization with some conditions are categorical equivalence. On the other hand, the categories of Poisson enveloping algebra is not equivalent to the other categories.


Introduction
Noncommutative geometry is regarded as one of the key concepts for formulating the quantum gravity theory or non-perturbative string theory. There are many ways to construct noncommutative geometry, including deformation quantization, geometric quantization, C * -algebra, matrix regularization, and so on. To find the best approach for quantum gravity or other physics, a unified perspective and a more general formulation containing the existing quantization models would be useful.
In this article, we define a generalized quantization of Poisson manifolds as a subcategory of the category of modules over a ring. It is shown that matrix regularization, strict deformation quantization, and prequantization are included in the generalization, and each pair of them are equivalent categories under some conditions described later. In addition, universal enveloping algebra derived from Poisson manifolds is also formulated as the generalized quantization method for Poisson manifolds.
In preparation for the following sections, we review several definitions of noncommutative geometries or quantizations.
Dirac introduces the quantization rule as replacing the Poisson brackets by commutators. When we regard quantization as a mapˆfrom functions to operators acting linearly on a Hilbert space, it is generally considered that the axioms for quantization mapsˆsatisfy the following conditions: (1) (H1 + H2) = H1 +Ĥ2. (2) (λH) = λĤ, λ ∈ R. Let us consider the definition of matrix regularization of a symplectic manifold (M, ω). Matrix regularization [17] has evolved from the ideas of Berezin-Toeplitz quantization [10,34], Fuzzy space [25], and so on. We employ the definition by [4]. Definition 1.1. Let N1, N2, . . . be a strictly increasing sequence of positive integers and be a realvalued strictly positive decreasing function such that limN→∞ N (N ) converges. Let T k be a linear map from C ∞ (M ) to N k × N k Hermitian matrices for k = 1, 2, . . .. If the following conditions are satisfied, then we call the pair (T k , ) a C 1 -convergent matrix regularization of (M, ω).

lim
where is the operator norm, ω is a symplectic form on M and { , } is the Poisson bracket induced by ω.
Formal deformation quantization is defined as follows [12,13,19,31]. where is a noncommutative parameter. A star product is defined on F by such that the product satisfies the following conditions.
2. C k is a bidifferential operator.
3. C0 and C1 are defined as where {f, g} is the Poisson bracket.
Several variations of the deformation quantization with some minor changes from this definition exist. In this definition, algebra is treated as a set of formal power series of smooth functions. For an arbitrary Poisson manifold, there exists a deformation quantization [19]. The formal deformation quantization has been widely adopted. However, it is difficult to regard the deformation quantization as a theory of physics when the theory remains formal. We thus employ a strict deformation quantization introduced by Rieffel [32,33]. There are various definitions of this quantization. We use a definition similar to that of [20] in this article. Before defining the strict deformation quantization, therefore, we will consider the definition of strict quantization in [20]. Definition 1.3. Let A0 be a Poisson algebra which is densely contained in the self-adjoint part C 0 R of an abelian C * -algebra C 0 . Let I be a subset of real numbers which contains 0. Strict quantization of the Poisson algebra A is a family of maps (Q : A0 → C R ), where 1. C is a C * -algebra with an associative product × and a C * -norm . For a, b ∈ C R which is the self-adjoint part of C , 2. ∀ ∈ I, Q : A0 → C R is R-linear and Q 0 is just the inclusion map such that (a) For f ∈ A0, the map → Q (f ) is continuous.
We define a strict deformation quantization based on this strict quantization. Definition 1.4. If ∀ ∈ I, Q (A0) is a subalgebra of C R and Q is injective, a strict quantization Q is called a strict deformation quantization.
Especially, a strict deformation quantization of a Poisson manifold M is defined by a Poisson subalgebra A0(M ) of C ∞ (M ) composed of bounded functions. In this article, we use not formal but strict deformation quantization (A0(M ), Q ).
The prequantization is defined as follows. (See for example [9,36]. For the general Poisson manifolds case, Vaisman established the geometric quantization [23].) Definition 1.5. Let M be a Poisson manifold. The prequantization is a correspondence between a real smooth function Hi ∈ C ∞ (M ) and a Hermitian operator Q(H) =Ĥ acting on a Hilbert space H, such that 4.1 = Id, (1 is a constant 1 and Id is an identity operator.) The following theorems are important to understand the prequantization through the concrete construction of the quantization mapˆ. Theorem 1.6. Let (M, ω) be a prequantizable symplectic manifold, L be a line bundle called prequantum line bundle, and S be the subset of smooth square integrable sections of L with compact support. There exist a quantization map C ∞ (M ) → Op(S) : f →f satisfying the above 1-4. Here, Op(S) is a set of linear operators acting on S. The map is constructed concretely: where X f is a Hamilton vector field of f and ∇X f s is a covariant derivative of the section s along X f .
In this article, we put forward a new framework of quantization including the above quantization theories by using categorical methods. In Section 2, the category of quantization of Poisson algebras is defined. In Sections 3, 4 and 5, we introduce categories including matrix regularization, strict deformation quantization, and prequantization. We show that they are the categories of quantization of the Poisson algebras. In Section 6, we study the universal enveloping algebra derived from Poisson algebras. It is found that the categories including matrix regularization, strict deformation quantization and prequantizaton are equivalent categories under some conditions in Section 7. In Section 8, we summarize all results and make some remarks. In addition, we discuss applications of the quantization category to physics.

Quantization Category
In this section, we give a generalization of quantization as a subcategory of the category of modules. We call this subcategory "quantization category" and show that some quantization theories are included in this category in the following sections. A limit in the category theory corresponds to a classical limit by considering a sequence of morphisms as quantization maps. Before introducing the quantization category, we define a category P(A(M )).  (2) ∀Mi ∈ ob(P), at least a morphism Ti ∈ P(A(M ), Mi) exists. We call Ti a quantization map.
by noncommutative complex parameter (T k ) for all quantization maps T k .
(5) ∀A, B ∈ ob(P), if TAB ∈ P(A, B) is an algebraic isomorphism, then there exist T −1 AB ∈ P(B, A) which satisfies We call the subcategory P(A(M )) the pre-Q category.
Note that the Lie bracket Below we denote P(A(M )) by P for simplicity.
Definition 2.2. A map χ : ob(P) → R is defined by the absolute maximum of : We call the map χ a noncommutative character.
Let us define an index category of P.
. Note that J • is uniquely determined by P and χ. Each connected component J α is also an index category.
for all i, j ∈ ob(J α ) and f k ∈ J α (i, j).
Note that, since J α is a connected category F (J α ) is also connected. However, objects which are connected in P are not always connected from the condition (ii) of Definition 2.3.
Proof. From the condition (1) of Definition 2.1, A(M ) is a candidate for the limit of all (J α , F α ).
The definition of the limit for (J, F ) used in this article can be found, for example, in Chapter 5 in [24] and [22]. The limit M∞ is usually written as a combination with projections π like (M∞, π). We often denote (M∞, π) as M∞ for short.
We can now define the quantization category.
We denote Q(P, J • , F • , χ) by Q for short. Note that all morphisms are linear maps since Q ⊂ RMod. In general, therefore, a quantization map Ti is not a ring homomorphism.
We show that this definition includes matrix regularization, deformation quantization, prequantization, and Poisson enveloping algebra in the following sections.

Matrix Regularization
In this section, we construct a quantization category to include matrix regularization of Definition 1.1.
If there exists some algebraic isomorphism Tiso in the morphism of PMR, then T −1 iso also exists. The consistency of this definition is confirmed in the proofs of the following lemmas. Proof. Let us denote Tij as Proof. From the definition of PMR, the conditions (1), (2) and (3) of Definition 2.1 are satisfied, where (T k ) = (N k ). For PMR to be a category, the consistency Tij • T jk = T ik should be satisfied for all i, j and k such that Ni ≤ Nj ≤ N k . We show that there is no contradiction even if T ik is simply defined as a composition of Tij • T jk . For adjacent objects M atN i and M atN i+1 , let us define T i(i+1) so that T i(i+1) Ti+1 = Ti for every i. Since Ni < Ni+1, T 11 i(i+1) in Lemma 3.2 exists. So the existence of this T i(i+1) for all i is guaranteed. We define Tij by the ordered product as follows.
These Tij trivially satisfy Tij • T jk = T ik for every Ni ≤ Nj ≤ N k at least if we put every T 22 ij = 0. Now, we check that they do not contradict the condition Ti = Tij • Tj. For the block matrix ik is satisfied, then PMR is a consistently pre-Q category. These morphisms Ti, Tij for i, j = 1, 2, · · · can be specifically configured in the case of Berezin-Toeplitz quantization as follows. We denote L ⊗k by L k for all k ≥ 1, and an algebra of smooth sections of L k by C ∞ (M, L k ). The quantum space at k given as with respect to the inner product ·, · k , and Π k be the orthogonal projector from L 2 (M, L k ) to H k . The Berezin-Toeplitz operator is defined as That is, T k is a linear map from the commutative algebra C 0 (M ) to a matrix algebra.
when k goes to infinity, where 2n = dim M .
From this theorem, Tij is constructed as follows. Let |n k be an orthonormal basis that diagonalizes T k , i.e., ∀T k = Π k = dim H k n |n k k n|. For dim Hj ≥ dim Hi, For the Berezin-Toeplitz quantization map T k , the following theorem is known.
where T stands for the operator norm and T ∞ stands for the uniform norm.
Thus, the Berezin-Toeplitz quantization map T k is the homomorphism of algebra between C ∞ (M ) and End(H k ) when k → ∞.
Since {Ni} is a strictly increasing sequence, FMR is a diagram which satisfies Definition 2.4 from a sequence in JMR is a quantization category of the matrix regularization with the only limit M at∞ of (JMR, FMR).
Proof. From the Lemma 3.3, PMR exists as a pre-QMR category. A limit (M∞, π) of FMR is given as the following commutative diagram for any i, j. This quantization category QMR can be further minimized. Proof. Since the diagram of QMR M at N i is given by this corollary is derived in the same way of Theorem 3.7.

Deformation Quantization
In the following, we consider the strict deformation quantization. In other words, we consider not (F, * ) but (A0(M ), Q ).
(These examples are shown in [32,33].) If there exists some algebraic isomorphism Tiso in the morphism of PDQ, then there also exists inverse PDQ is a pre-QDQ category.
Proof. The diagram of PDQ is given by Proof. The proof is given in a completely parallel manner as the proof of Theorem 3.7. From Lemma 4.2, PDQ is a pre-Q category. We consider a limit M∞ of FDQ as the following commutative diagram.
Thus, the limit of FDQ is C 0 . Trivially, the quantization conditions are satisfied on C 0 , so QDQ is a quantization category of the strict deformation quantization.
The quantization category QDQ can be further minimized in a manner similar to QMR.

Corollary 4.4.
For the quantization category QDQ, a restriction QDQ C is given as follows: where is fixed. Similarly, the morphisms are restricted to Q 0 , Q , T 0 .

.1)
Then QDQ C is a quantization category of the strict deformation quantization with the limit C 0 .

Prequantization
It is possible to construct a quantization category including the prequantization whose definition is given in Definition 1.5 Ifˆis a algebraic isomorphism, then the inverse ofˆexists as a morphism of PP Q. Lemma 5.2. PP Q is a pre-QP Q category.
Proof. The diagram of PDQ is given by where π =ˆ. Trivially, PP Q is a category, and the Definition 2.1 is also satisfied by the definition of PP Q.
There is only one quantization mapˆ, so we obtain χ(End(Γ hol (M, L))) = | |. This χ gives the index category J • P Q and a set of diagrams F • P Q as follows. For J • P Q := {JP Q}, JP Q is given as a category of only one object, and we denote it by A. For Proof. From Lemma 5.2, QP Q is a pre-Q category. We consider a limit (M∞, π) of FP Q as the following commutative diagram.
Thus, the limit of FP Q is (A(M ), π). It is trivial that the quantization conditions are satisfied on A(M ), so QDQ is a quantization category of the deformation quantization.

Poisson Enveloping Algebra
In this section, we describe the relationship between the quantization category and Poisson enveloping algebra.
First, we will review the definition of the Poisson enveloping algebra.
Let X be an arbitrary algebra such that there exist an algebra homomorphism α from (A, ·) to X and a Lie homomorphism β from (A, { , }) to X such that Then, the enveloping algebra P e n of Pn and the Weyl algebra A2n are isomorphic. From Example 6.2 and so on, we are convinced that the Poisson enveloping algebra is a kind of quantization. where Mi for i = 1, 2, · · · , are Lie algebras such that the following diagram commutes.

Mi
Morphisms of Penv are given by where Π1 is the identity forgetting the { , } structure and Π2 is the identity, and the homomorphisms and Lie homomorphisms are as follows. These homomorphisms and Lie homomorphisms hi :U → Mi, satisfy (6.1) − (6.5), respectively. The other morphisms are simply defined by a composition such that Penv is a category. For all quantization maps T k , whose domain is A, this codomain is equipped with the Lie bracket [ , ] k as the commutator such that If there exists some algebraic isomorphism Tiso in the morphism of Penv, then there also exists an inverse T −1 iso of Tiso. Lemma 6.4. Penv is a pre-Penv category.
Proof. This lemma follows immediately from the Definition 6.3.  Proof. From the Lemma 6.4, Penv is a pre-Q category. We consider a limit (M∞, π) of Fenv as the following commutative diagram.

Mi
Note that there exist only morphisms that satisfy the condition (ii) of the Definition 2.3. The only candidate for a domain of Hom(·, Fenv(i)) for all i ∈ ob(Jenv) is A. Thus, a limit of Fenv is (A, Hom(A, Fenv(i))).
Because the quantization map to A is the identity, the quantization conditions are satisfied on A trivially, so Qenv is a quantization category of the Poisson enveloping algebra.

Categorical Equivalence
In this section, we discuss the equivalence of the quantization categories appearing in Section 3-5.
Let QMR M at N i and QDQ C be those appearing in Corollary 3.8 and Corollary 4.3. The following proposition is trivially obtained. This proposition can be immediately extended to the following proposition.
For the F and G, let us show that F G id Q P Q and GF id (Q M R ) . The natural transformation θ : F G → id Q P Q is trivial. The diagrams of θ : GF → id (Q M R ) is given by  [34]). Therefore, this is the case of the above Proposition 7.3.
Proposition 7.5. If A0(M ) and C 0 is isomorphic in QDQ C , then QDQ C and QP Q are equivalent categories.
Proof. From Proposition 7.1 and 7.3, Proposition 7.6. Suppose that QDQ and QMR have more than three objects each pair of which are not isomorphic, respectively. Then, the quantization category QP Q is not categorically equivalent to either QDQ or QMR.
Proof. Suppose that QP Q and QDQ are equivalent categories. In other words, functors F : QP Q → QDQ and G : QDQ → QP Q exist, such that F G id Q DQ and GF id Q P Q . Then there are functors X, Y ∈ ob(QDQ) such that  L)). Let Z be a object of QDQ which is not isomorphic to X or Y . Note that there is a morphism from Z to X or from X to Z. First we will consider the case that there is a morphism from X to Z. The functor G must be G(Z) = A(M ) or G(Z) = End(Γ hol (M, L)).
is naturally isomorphic. However, arbitrary object Z is not isomorphic to X and Y . This is a contradiction. Similarly, the case that there is a morphism from Z to X also has a contradiction.
(ii) Let us consider the case that A(M ) and End(Γ hol (M, L)) are linear isomorphic.
Then X and Y must be isomorphic. There exists Z ∈ ob(QDQ) such that Z and X are not isomorphic. If F G id Q DQ then at least a natural transformation of one of the following four diagrams is isomorphic. This is also a contradiction. Thus, QP Q is not categorically equivalent to QDQ. The case of QMR is proved in the same way.
Proof. We consider the following diagram.

Mi
Suppose that Qenv and QDQ are equivalent categories, i.e., for functors F : Qenv → QDQ and G : QDQ → Qenv, F G id Q DQ and GF id Qenv . Then there exists X, Y ∈ ob(QDQ) such that F (A(M )) = X, α • Π1 and β • Π2 must be mapped to QDQ(F (A(M )), F (U )) by F : Since the morphisms of QDQ or QP Q are unique, for arbitrary functor G : QDQ → Qenv. However, α • Π1 = β • Π2 in Qenv. This is a contradiction. Thus, Qenv is not categorically equivalent to QDQ. The other cases for both QDQ and QP Q are proved in the same way.

Conclusions and Discussions
In this article, we discuss a category of quantization of Poisson manifolds or Poisson algebras as a subcategory of RMod, but its objects are commutative and noncommutative algebras. We define the quantization category as a generalization of quantizations of the Poisson algebra, and show that this category contains categories of some known quantizations of the Poisson algebra. We also discuss relationships between the categories of various types of quantizations. The pre-Q category P is defined by choosing a fixed Poisson algebra A(M ) and algebras Mi of A(M )'s quantized algebras as objects, and choosing quantization linear maps Ti : A(M ) → Mi and linear maps between each Mi as morphisms. If a morphism is an algebraic isomorphism, then its inverse is also a morphism. Each quantization map Ti has a noncommutative parameter (Ti). The character χ(Mi) is introduced as the maximum absolute value of (Ti). The index category J • and a set of functors F • are determined by the noncommutative character χ to consider the classical limit. In addition to these structures, we defined the quantization category as being equipped with Lie homomorphisms of the algebra between Poisson brackets in A(M ) and Lie brackets in the limit determined by F • of J • . As concrete examples, the quantization category of matrix regularization (including Berezin-Toeplitz quantization), strict deformation quantization, prequantization and Poisson enveloping algebra are constructed. The equivalence or non-equivalence of these categories is also discussed. In particular, we show the equivalence of the quantization category of matrix regularization and the quantization category of strict deformation quantization when N = . In addition, we show that equivalence between QDQ C , QMR M at N i and QP Q under a condition that the quantization map T from A(M ) to the limit M∞ and Q 0 from A0(M ) to the limit M∞ are an isomorphism, respectively. For example, this condition is satisfied for compact Kähler manifolds in the case of Berezin-Toeplitz quantization. On the other hand, it is shown that the quantization category of Poisson enveloping algebra is not equivalent to the other quantization categories.
We have focused on the quantization category which contains one quantization procedure. However, we define the quantization category such that it is possible to include multiple types of quantization theories. For example, if the union of index categories J • 1 and J • 2 of two quantization categories Q1 and Q2 with the same Poisson algebra A(M ) is empty, a category consisting of the sum of Q1 and Q2 is also a quantization category. Here, the category made up by this summation is a category whose object set is the union of the object sets of Q1 and Q2, and its set of morphisms is created from the union of morphisms of Q1 and Q2 and adding composite maps to the union so that the whole becomes a category. For example, a category created by the sum of QMR and QDQ is a quantization category. One of the future tasks will be to examine the concrete construction and to study properties of such a category made up of the sum of quantization categories. It is also necessary to consider the sums of more complex categories whose J • 1 and J • 2 are not disjoint. Such researches should be done as a next step.
In this paper, we formulated a quantization category by adopting (1), (2) and (3) among the conditions by Dirac enumerated at the beginning of this article. However, we can choose other combinations. Therefore, the quantization category studied in this paper might be an example of a series of quantization categories that have a variety of quantization conditions. The task of investigating such a large area of quantization categories remains for future work.
Finally, we will consider potential applications of the quantization category to physics. The universe we live is classically described as a vector bundle. The base manifold is a Riemannian manifold. The fibers are for the electromagnetic field, non-Abelian gauge fields, matter fields and so on. In the case of the particle physics given by a Hamiltonian formulation of mechanics, a cotangent bundle over the Riemannian manifold is its geometry. The cotangent bundle is the Poisson manifold. When we consider M a cotangent bundle over a Riemannian manifold, then the A(M ) in Q corresponds with a classical physics. In that case, the character χ or should be chosen as the Planck constant or energy scale. To make concrete predictions or to clarify physical properties, we have to import further structures into the quantization category. The findings of several previous studies might be useful when introducing physical structures. For example, Ojima and Takeori [30] studied the correspondence between classical and quantum physics, which is called Micro-macro duality by categorical approach Alternatively, categorical approaches to quantum mechanics have shown how to describe the fine structure of physics as functors. [1,8] As an example, let us consider the IKKT matrix model or noncommutative gauge theory in the context of QMR [3,11,18]. In Section 3, we considered matrix regularization. The classical IKKT matrix model is regarded as a matrix regularization of the type IIB string theory with the Schild gauge. The bosonic part of the Lagrangian is given as where X µ is a map from a parameter space to a world sheet and {X µ , X ν } is the Poisson bracket on the world sheet. Using a determinant of the world sheet metric g and Levi-Civita symbol ab , the Poisson bracket is defined as The quantization map Ti of the quantization category of matrix regularization in Section 3 maps this Poisson bracket to a commutator [X µ , X ν ] := X µ X ν − X ν X µ , up to higher order terms of . Then the Lagrangian is also obtained as [X µ , X ν ][X µ , X ν ], at the limit M at∞. This matrix model is also regarded as a noncommutative U (1) gauge theory on noncommutative Euclidean space at the limit of the quantization category of matrix regularization. In this context, type IIB string theory with Schild gauge is defined on the object A(M ) and the IKKT matrix model is defined on the object M at∞.
In this article, only the ordinary Poisson structure has been considered for quantization. However, there are many other types of classical mechanics, such as Nambu mechanics [27]. To attack the quantization problem of the membrane theory, quantization of the Nambu bracket has been shown to be an effective approach. For this purpose, the Nambu bracket should be replaced with the Lie 3 bracket by the quantization morphism [2,5,6,7,14,15]. The category of quantization defined in this article is naturally generalized to such a quantization type. This would be a suitable problem to address in the next stage of investigation.
The quantization category proposed in this article involves many basic or applied problems, including pure mathematical and physical problems, as mentioned above. All of these should be solved in the future.