{'fraction_non_alphanumeric': 0.05543585861526815, 'fraction_numerical': 0.01694520468176789, 'mean_word_length': 3.6154259607632357, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 18, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Pfaffian Systems from Twistor Fibrations

Canonical twistor fibrations lead to Pfaffian systems by means of their superhorizontal distribution. The aim of this note is to identify explicitly the Pfaffian systems of five or less variables that arise in this way in terms of the classification given by A.Awane and M.Goze.


Introduction
A Pfaffian system on an n-dimensional differentiable manifold is a collection of 1-forms which are linearly independent at each point. Geometrically this system gives rise to a subbundle of the tangent space, that is a distribution. Conversely a distribution induces an equivalence class of Pfaffian systems (see [1], [3], [6] or [9]). For n ≤ 5 such systems were investigated in detail by E. Cartan.
On the total space of any canonical twistor fibration (see [2] or [5]), there is a natural "superhorizontal" distribution, which is important in the theory of harmonic maps. The total space is a flag manifold, and a corresponding Pfaffian system on the big cell of the flag manifold can be constructed by using Lie-theoretic local coordinates to express integral curves of the distribution. The equations for these holomorphic curves are solved explicitly in [4].
In this paper we identify explicitly all Pfaffian systems with n ≤ 5 which arise from twistor fibrations in this way, in terms of the known classification. It is of interest to consider the significance of these twistor Pfaffian systems, and we discuss briefly one aspect here, namely the Lie algebra of infinitesimal symmetries. Cartan observed that the Lie algebra of the exceptional group G 2 can be characterized as the symmetry algebra of such a system. However, all the other systems with n ≤ 5 arising from twistor fibrations have infinite-dimensional symmetry algebras. In the simplest example, we present an explanation for this phenomenon.
The paper is organized as follows. In section 2 we describe the basic definitions and theorems concerning Pfaffian systems. In addition, the classification of low dimensional Pfaffian systems from [1] is reproduced at the end of this section (tables 1-3). Section 3 reviews some facts from structure theory of Lie algebras as well as the concept of twistor fibration. The construction of Pfaffian systems from twistor fibrations is explained in section 4. Subsections 4.1 -4.3 contain detailed computations for three representative examples: π : F 3 (C) → CP 2 , π : SO(5)/U (1) × SO(2) → SO(5)/SO(2) × SO(3) and π : G 2 /U (2) → G 2 /SO (4). The complete list of results is given in table 4. Finally in the last section we discuss the Lie algebra of infinitesimal symmetries.

Pfaffian systems
In this section we will review some basic concepts and facts on Pfaffian systems theory (see [3]). Let M be an n−dimensional differentiable manifold. Let T M and T * M denote the tangent and cotangent bundle respectively.
In the following Ω k (M ) will denote the space of all differentiable k forms on A (local) Pfaffian system of rank r is a set: where the r differential 1-forms are defined on an open subset U of M and are required to be linearly independent at each point.
Conversely, a distribution give rise to an equivalence class of Pfaffian systems, since the annihilator subbundle of D, with fibers is locally spanned by r linearly independent 1-forms i.e. D ⊥ p = span{(ω 1 ) p . . . (ω r ) p } ⊂ T * p M . The ideal of Γ( * T * M ) generated by the system is: At a point p ∈ M the characteristic space of I(D) is defined as: Its annihilator C(I(D)) p := A(I(D)) ⊥ p ⊂ T * p M , called the retracting subspace at p, is the smallest subspace of T * p M such that * (C(I(D)) contains a finite set S of elements generating I(D) as an ideal. The dimension of C(I(D)) p is called the class of S at p.
A symmetry of S is a local biholomorphism φ on U which preserves D (or S), i.e. φ * (D) = D (or φ * (S) = S). An infinitesimal symmetry is a vector field X over U which generates a local one parameter transformation group of symmetries of S. Namely X satisfies: [X, Γ(D)] ⊆ Γ(D), or equivalently: L X ω i ≡ 0 mod S with ω i ∈ S. The set of all infinitesimal symmetries of S is denoted L(S).
Then the characteristic space A(I(D)) = p∈U A(I(D)) p of I(D) can be described (cf. [6]) in terms of infinitesimal symmetries as: A submanifold N of M is called an integral manifold of S (or D) if T p N ⊆ D p for any p ∈ N . Equivalently an integrable manifold is given by an immersion i : N −→ M such that i * (α) = 0 for all α ∈ I(D). A natural problem in Pfaffian systems theory is the classification of systems with constant rank, which requires the study of invariants. The class and the rank are fundamental invariants that entirely determine the local classification of completely integrable systems by the Frobenius theorem.
For a nonintegrable Pfaffian system S of rank r and corresponding distribution D, we define the subbundle D ′ which has fibers spanned by all elements of Notice that D p ⊂ D ′ p . A representative S ′ of the equivalence class of Pfaffian systems corresponding to this distribution is called (with abuse of notation) the derived system of S. This leads to the notion of higher derived systems. The k-th derived system S k is defined successively by: It follows that the rank and the class of S k are invariants of such a Pfaffian systems. S is called regular if all derived systems are of constant rank. For a regular system S, there exists µ ∈ Z ≥0 such that: S µ is always 0 or completely integrable; It is the smallest completely integrable subsystem of S. (See [10], [1]).
The basic idea to achieve the classification of nonintegrable Pfaffian systems of constant rank is to look for completely integrable subsystems of S and use the defined invariants to determine a local model of the system.
In [1] the equivalence classes of low dimensional (n ≤ 5) Pfaffian systems of constant rank and class is presented by giving local models of those systems. Nevertheless this is a partial classification, since for the cases S 11 5 (f ) and S 15 5 (f ) the local equivalence classes are not specified. For the sake of reference this classification is reproduced in the following tables. Table 1. Pfaffian systems of dimension 3. Table 2. Pfaffian systems of dimension 4.

Twistor fibrations
The aim of this section is to review the twistor fibration concept. We start by recalling some facts from the structure theory of Lie algebras.
Let g be a Lie algebra of a compact semisimple Lie group with Cartan subalgebra t. Let g C be the complexification of g, and t C that of t.
The Lie algebra g C has then a root decomposition into the Cartan subalgebra and the root spaces: Here ∆ denotes the set of roots of g C . This is a finite subset of (t C ) * .
It is also necessary to recall the existence of a subset Π = {α 1 . . . α r } (r = dim C t C ) of ∆ such that every root can be expressed uniquely as α = l i=1 n i α i where the n i are integers, either all nonnegative (α is then a positive root) or all non-positive (α is a negative root). Such a set is called set of simple roots for ∆. Each subset I = {α i1 , . . . , α is } of Π defines on ∆ a height function n I by n I (α) = i∈I n i .
According to this any parabolic subalgebra of g C can be expressed as: Given a parabolic subalgebra p I , there exists a unique element This element is called the canonical element of p I . This element defines an involutive automorphism τ ξ = Ad(exp(πξ)) of g (the complex linear extension to g C ) called the canonical involution for p I .
The facts mentioned above are intimately related with the concept of generalized flag manifolds, which are homogeneous spaces of the form G/H where G is a compact Lie group and H is the centralizer of a torus in G. We have a natural isomorphism G C /P I ∼ = G/H where G C is the complexification of G and Let G/H ∼ = G C /P I be a generalized flag manifold and let ξ be the canonical element of the Lie algebra p I of P I with canonical automorphism τ ξ . Then the following is true: Here G σg denotes the set of fixed points of σ g and G 0 σg its identity component. For details we refer to [5], [2] and [7].

Pfaffian systems from twistor fibrations
Now we are in a position to describe how Pfaffian systems and twistor fibrations are related. The key is to consider a certain subbundle of the tangent bundle of the generalized flag manifold of the twistor fibration.
Let G C /P I be a generalized flag manifold. Recall that as an homogeneous space its tangent bundle is isomorphic to G C × PI g C /p I . An explicit isomorphism f : G C × PI g C /p I → T G C /P I is given by: Definition 2. The subbundle SH of T G C /P I which corresponds to This concept arises from the study of harmonic maps that come from twistor fibrations.
We shall construct a particular Pfaffian system corresponding to the superhorizontal distribution by expressing integral curves of the distribution in Lie theoretic local coordinates. The equations for integral curves of this distribution are solved expliclitly in [4], (see page 561).
It is well known that local coordinates for the generalized flag manifold, G C /P I are provided by the "big cell " [exp i>0 g I i ] ∼ = C m . Locally therefore we can construct any complex integral curve Φ : C −→ G C /P I by giving a holomorphic map C : C −→ i>0 g I i and setting Φ = ρ(exp C), where ρ denotes the projection G C −→ G C /P I .
The condition for this integral curve to be tangent to the superhorizontal distribution, i.e. dΦ ∂ ∂z ∈ SH, is that: In other words, (exp −C)(exp C) ′ should have zero component in i≥2 g I i .
Using the formula for the derivative of the exponential map one has: Applying this to (3), the resultant expression determines n differential equations in m functions, C ′ i = F (C 1 , . . . , C i−1 ) (i ≥ 2) for an integral curve of the superhorizontal distribution, where n = i≥2 dim g I i and m = i≥1 dim g I i . These equations can be written in local coordinates corresponding to i>0 g I i as 1-forms of m-variables giving in this way a Pfaffian system S on C m with rank(S) = n.

Proposition 1. The Pfaffian system S just obtained satisfies the following properties:
• class(S) = i≥1 dim g I i = m. • The first derived system S ′ of S corresponds to: Proof. Notice first that dim A(I(SH)) = 0 is equivalent to A(I(SH)) = L(S) ∩ Γ(SH) = 0 , i.e. if X ∈ Γ(SH) satisfies then X = 0. We show that this is the case. Take X ∈ SH satisfying (4). Then X is expressed as X = [(g, [Z])] with g ∈ G C and [Z] ∈ g I 1 /p I . Now under the isomorphism f , the condition (4) is equivalent to having Z ∈ g I 1 such that [Z, g I 1 ] ∈ g I 1 . But since [g I i , g I j ] ∈ g I i+j we must have Z = 0.
Similarly, SH ′ = ker(S ′ ) is spanned by: which under the isomorphism f is equivalent to: Now we concentrate entirely on the case when m ≤ 5 and compute all Pfaffian systems that arises from twistor fibrations of semisimple Lie algebras by using the above construction and identifying the obtained system -after a suitable change of local coordinates -with a model in the classification of low dimensional Pfaffian system mentioned in section 1 (tables 1, 2 and 3).
Theorem (2) collects these results in a detailed list. Since the proof is purely computational, in order to illustrate how the calculation is carried out we present three representative examples: Let G = SU 3 , then G C = SL 3 C with Lie algebra sl 3 C consisting of all endomorphisms of C 3 with zero trace. Let t C = {H ∈ sl 3 C |H = diag(a 1 , a 2 , a 3 )} be the corresponding Cartan subalgebra.
The set of roots is then given by: where r i denotes the linear functional r i : t C → C defined by r i (H) = a i . In this case the set of simple roots is Π = {α 1 = r 1 − r 2 , α 2 = r 2 − r 3 } and the corresponding root spaces are: where E ij stands for the matrix whose (i, j) entry is 1 and all others 0.
We have g = exp πξ = diag(e √ −1π , 1, e − √ −1π ) and therefore the set of fixed points of σ g is: Here since H ⊂ G σg , setting K = G σg we can write: Thus in this case the twistor fibration is precisely π : F 3 (C) → CP 2 . Now, the integral curve is constructed defining the map C : C → 2 i=1 g I i as: After applying the condition (3) we obtain the following ordinary differential equation: In terms of local coordinates of exp 2 i=0 g I i this becomes the change of coordinates identifies this Pfaffian system with:

π : SO(5)/U(1) × SO(2) → SO(5)/SO(2) × SO(3).
Let G = SO (5). By definition the corresponding Lie algebra is given by: After choosing a basis: where {e i } 5 i=1 denotes the canonical basis of C 5 , a suitable matrix representation is obtained: In this case the set of roots is: ∆ = {±(r i ± r j ), ±r k | 1 ≤ i, j, k ≤ 2, i = j} with simple roots Π = {α 1 = r 1 − r 2 , α 2 = r 2 }, and the root spaces are: Setting I = Π, we have: The map C : C → i>0 g I i can be defined as: In the same way as above, calculating (exp −C)(exp C) ′ under the condition (3), we obtain the following system of differential equations: In terms of local coordinates these are: dt 1 − 1 2 xdt 2 + 1 2 t 2 dx = 0 and dy + t 1 dt 2 −t 2 dt 1 + 1 3 t 2 xdt 2 − 1 3 t 2 2 dx = 0. The associated Pfaffian system S = {ω 1 , ω 2 } is given by: Notice that S is not completely integrable and that the derived system is S ′ = {ω 2 } since dω 2 ∧ ω 1 ∧ ω 2 = 0. Moreover, since dω 2 ∧ ω 2 = 0 we have class(S ′ ) = 3 and the system must be equivalent to S 7 4 . We shall find an explicit change of coordinates. First, ω 2 can be expressed as: to obtain: On the other hand: hence the change of coordinates: shows that the Pfaffian system is : In this example we consider G 2 = {X ∈ SO(7, R) | X(v × w) = Xv × Xw}, with Lie algebra: As for the last example, in order to obtain a matrix representation for SO(7) we choose a basis: Then under this representation any element of g C 2 can be expressed as:  In this case the set of roots is: with simple roots Π = {α 1 , α 2 }, where α 1 = r 3 and α 2 = r 2 − r 3 . Here the root spaces are: Taking I = {α 1 } we have : Thus the map C : C → i>0 g I i takes the form: This time the condition of the integral curve to the superhorizontal distribution gives the following system of differential equations: After expressing them in terms of local coordinates we set Then the induced Pfaffian system is S = {ω 1 , ω 2 , ω 3 } which is not completely integrable. Now, for i = 2, 3 we have that: Therefore, the derived system is S ′ = {ω 2 , ω 3 }. Moreover since dω 2 = −3ω 1 ∧dx 2 and dω 3 = 3ω 1 ∧ dx 1 we have class(S ′ ) = 5, and then the system S is identified to be S 11 5 (f ). Table 4.
Low dimensional Pfaffian systems that arise from twistor fibrations.
Lie group decomposition of T 1,0 G/H Pfaffian system

Remark:
Since so 5 is isomorphic to sp 2 and so 6 to su 4 , the corresponding Pfaffian systems must be equivalent. However since in each case the details of the calculation are different and for the sake of confirmation their description is also included in the above list.
The remainder of this note presents an example of how the twistor Pfaffian systems discussed above are related to the corresponding Lie algebra of infinitesimal symmetries, which in general is infinite dimensional.
It is well known that g C ⊆ L(S), because the superhorizontal distribution is a G C −invariant distribution, i.e. G C acts on G C /P I by left translations preserving SH. More precisely, for any X ∈ g C the infinitesimal symmetries of SH are given in a canonical way by the vector fields: with [g] ∈ B, the big cell of G C /P I .
More generally, K. Yamaguchi in [10] considered examples of regular differential systems, which turn out to agree with the Pfaffian systems arising from twistor fibrations. For any such system S, the main result of [10] asserts that L(S) ∼ = g C except for the following three cases: (1) g C = g I −1 ⊕ g I 0 ⊕ g I 1 .
Notice that all the examples presented in table 4 belong to one of these categories with the exception of S 11 5 (z 2 5 ), the Cartan case.
The fact that the Pfaffian system S 2 3 on C 3 originates from two different twistor fibrations (see table 4) give us a natural explanation of the fact that dim L(S 2 3 ) = ∞. In fact, as we have seen in 3.1, the twistor fibration with big cell C 3 has a superhorizontal distribution generated by {∂ 1 , ∂ 2 + w 1 ∂ 3 } and therefore a Pfaffian system S = {dw 3 − w 1 dw 2 }, which is equivalent to S 2 for a basis of the lie algebra sl 3 C, a set of 8 vector fields is obtained. The Lie algebra L ′ spanned by this set is a subalgebra of the infinitesimal symmetries of S 2 3 . Similarly, for the fibration π : Sp 2 /Sp 1 × U 1 ∼ = CP 3 → Sp 2 /Sp 1 × Sp 1 ∼ = S 4 , we have SH = {∂ 1 − w 2 ∂ 3 , ∂ 2 + w 1 ∂ 3 } and S = {dw 3 + w 2 dw 1 − w 1 dw 2 } which under the change of coordinates (z 1 , z 2 , z 3 ) = (w 3 − w 1 w 2 , 2w 2 , w 1 ) also corresponds to S 2 3 . In the same way as above, we can construct a Lie subalgebra L ′′ of L(S 2 3 ) spanned by 10 vector fields by means of (7).
By direct calculation we find that the Lie algebra L ′ , L ′′ generated by L ′ and L ′′ is an infinite dimensional Lie subalgebra of L(S 2 3 ). In a future paper we shall discuss further ramifications of this observation.