# On an Algebraical Computation of the Tensor and the Curvature for 3-Web

Authors: Thomas B. Bouetou
Journal: ISRN Geometry
 (2012)

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## Abstract

The algebraic methods are used in the web geometry, in particular in the 3-web. Along the line, we suggest a new, alternative algebraic method for computation of the quantities∇l1ajki, ∇l2ajki, and djklmi by means of the embedding of local loops into Lie groups.


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## Body

## 1. Introduction

Web geometry is one of the fields of mathematics which springs from two different fields of mathematics, namely, projective differential geometry and nomography. It was derived mostly from projective differential geometry. Initially, projective differential geometry mainly consisted of the study of projective properties of curves and surfaces inℝ3, that is, of their differential properties that are invariant up to homographies. Web geometry studied the properties of (curves and) surfaces in ordinary euclidian space that are invariant up to isometric transformations. Gauss and other mathematicians have shown the usefulness of the first and second fundamental forms in the study of surfaces. They also brought to light the relevance of derived concepts, such as the principal, asymptotic, and conjugated directions. When considering the integral curves of these tangent direction fields, the mathematicians of the 19th century were considering what they called 2-nets of lines on surfaces, that is, the data of 2 families of curves, or in more modern terms, 2-webs. It is when they tried to generalize these constructions to the projective differential geometry that some 3-nets projectively attached to surfaces in ℝ3 quite naturally made their appearance, Darboux introduced a 3-web named after him in [1]. These webs were useful at that time because they encoded properties of the surfaces under study. Thomsen in [2] shows that a surface area in ℝ3 is isothermally asymptotic if and only if its Darboux 3-web is hexagonal. At that time, the study of 3-web on surfaces from the point of view of projective differential geometry was on the agenda. Thomsens result has this particular feature of characterizing the geometric-differential property of being isothermally asymptotic by a closedness property of more topological nature that is (or not) verified by a configuration traced on the surface itself. It is this feature which struck some mathematicians and led to the study of webs at the beginning of the 1930s. The development of geometry of fiber bundles and foliations stimulates the interest for new investigation of three webs [3–17]. In [18–27], the techniques were developed for webs using the intrinsic geometry structure. In this investigation, we propose to give another approach of computation of some classical relations, using the technique of the projective space. Our approach is based on the embedding of a smooth loop into a Lie group, by means of a closed subgroup. This transports the geometric problem into an abstract algebraic problem, where the 3-web is seen as a homogeneous space coset in a generic position. Using this technique the computation of the tensor structure of local loop is made easier. Therefore, we give an application of the computation of the well-known tensor. We use algebraic methods to compute the relations ∇1lajki, ∇2lajki, and djklmi. The paper is organized as follows. In Section 2, we derive the analytic representation of the law of composition of local smooth loops, embedding in Lie groups. In Section 3, we evaluate tensor structure of a smooth analytic loop. In Section 4, we look at the tensor structure of a smooth local loop, embedding in Lie group. In Section 5 we applied our method to compute ∇2lajki and ∇1lajki. In Section 6 we deal with the computation of the tensor djklmi=∇2mbjkli. The last section is devoted to the hexagonal loops.


## 2. Analytic Representation of Law of Composition of Local Smooth Loops, Embedding in Lie Groups

Let〈G,·,e〉 be a local Lie group and let H be its local closed subgroup. Denote by 𝔊 and 𝔥 their corresponding Lie algebra and Lie subalgebra, and let Q be a smooth space section of left coset GmodH passing through e the unit element of G(e∈G).The composition law(2.1)×̲:Q×Q⟶Q,(x,y)⟼x×̲y=∏Q(x⋅y),
where ∏Q:G→Q is the projection on Q parallel to the subgroup H, defines in Q a structure of a local loop, that is, 〈Q,×̲,e〉-loop [25, 28–36].Let us map the tangent spaceTeQ with the vector subspace V⊂G such that TeQ=V. Then 𝔊=V∔𝔥 since the submanifolds Q and H are transversal in the Lie group G.Let us introduce the mappingϕ(2.2)ϕ:V⟶h,ξ⟼ϕ(ξ),
defined by the condition exp(ξ+ϕ(ξ))∈Q (for every vector ξ∈V, in the neighborhood of O, and the map ϕ is well defined).Thenϕ(O)=O and(2.3)ϕ(ξ)=R(ξ,ξ)+S(ξ,ξ,ξ)+o(3),
where (2.4)R:V×V⟶h,S:V×V×V⟶h
are bilinear and trilinear symmetric maps. A base 〈e1,e2,…,eN〉 is fixed in 𝔊 such that 〈e1,e2,…,en〉 generates V, that is, V=〈e1,e2,…,en〉 and 〈en+1,en+2,…,eN〉 generates 𝔥:𝔥=〈en+1,en+2,…,eN〉. Introduce in the local Lie group G the following normal coordinates: the coordinate on the submanifold Q which is the projection from expV, that is, for all x∈Q, x=(xi)i=1,n¯, this means exp(xiei+ϕ(xiei))=x∈Q.Introduce the map(2.5)Q⟶V,x⟼x¯=xiei.
Then the condition written before is equivalent to (2.6)x¯+ϕ(x¯)=x∈Q.
In what follows, we will compute the constructed coordinates, fixed on the submanifold Q.It is known that the law of composition in a Lie groupG(·) has the following representation up to the fourth order in the normal coordinates:(2.4′)a⋅b=a+b+12[a,b]+112[a,[a,b]]+112[b,[b,a]]-148[b,[a,[a,b]]]-148[a,[b,[a,b]]]+o(4).Consider the coordinate representation of the law of composition×̲, for y:x=(x¯) and y=(y¯) in Q. We have(2.7)(x×̲y)¯=x¯+y¯+K(x¯,y¯)+L(x¯,x¯,y¯)+M(x¯,y¯,y¯)+P(x¯,x¯,x¯,y¯)+Q(x¯,x¯,y¯,y¯)+U(x¯,y¯,y¯,y¯)+o(4).(Our notations are similar to the notations of the work [24]).Denote the right side in (2.7) by z=(z¯). Then, for its computation, we obtain the following: (2.8)exp⁡(z¯+ϕ(z¯))=exp⁡(x¯+ϕ(x¯))⋅exp⁡(y¯+ϕ(y¯))h,
where h is an element from 𝔥, and indeed we have h=h(x¯,y¯).The following proposition holds.Proposition 2.1.
We have(2.9)K(x¯,y¯)=12∏[x¯,y¯],
where ∏[x¯,y¯] is the projection of the commutator [x¯,y¯] on V parallel to the subalgebra 𝔥(2.10)h(x,y)=-12[x¯,y¯]+12∏[x¯,y¯]+2R(x¯,y¯)+o(2).Proof.
we use the formulae (2.8). Comparing the terms from V and 𝔥 and considering only the terms of the first order, we obtain that
(2.11)z¯=x¯+y¯∈V,h=o∈h.
For computing the term of the second order, we denote
(2.12)z¯=x¯+y¯+K(x¯,y¯)∈V,h=N(x¯,y¯)∈h,
from (2.8) and considering (2.4) and (2.4′), we have
(2.13)x¯+y¯+K(x¯,y¯)+R(x¯,x¯)+R(y¯,y¯)+2R(x¯,y¯)=x¯+y¯+N(x¯,y¯)+R(x¯,x¯)+R(y¯,y¯)+12[x¯,y¯],
then by comparing term from V and 𝔥 and noting that
(2.14)12[x¯,y¯]=12∏[x¯,y¯]+(12[x¯,y¯]-12∏[x¯,y¯]),
hence(2.15)K(x¯,y¯)=12∏[x¯,y¯],h(x,y)=-12[x¯,y¯]+12∏[x¯,y¯]+2R(x¯,y¯).Corollary 2.2.
From Proposition2.1, it follows that
(2.16)(x×̲y)¯=x¯+y¯+12∏[x¯,y¯]+o(2).Proposition 2.3.
One can show that(2.17)L(x¯,x¯,y¯)=-16∏[x¯,[x¯,y¯]]+12∏[R(x¯,x¯),y¯]+14∏[x¯,∏[x¯,y¯]]+∏[x¯,R(x¯,y¯)],M(x¯,y¯,y¯)=13∏[y¯,[y¯,x¯]]+12∏[x¯,R(y¯,y¯)]-14∏[y¯,∏[y¯,x¯]]+∏[y¯,R(x¯,y¯)],h(x¯,y¯)=-12[x¯,y¯]+12∏[x¯,y¯]+2R(x¯,y¯)+R(x¯,∏[x¯,y¯])+3S(x¯,x¯,y¯)+16Λ[x¯,[x¯,y¯]]-14Λ[x¯,∏[x¯,y¯]]-12Λ[R(x¯,x¯),y¯]-Λ[x¯,R(x¯,y¯)]+R(y¯,∏[x¯,y¯])+3S(x¯,y¯,y¯)-13Λ[y¯,[y¯,x¯]]+14Λ[y¯,∏[y¯,x¯]]-12Λ[x¯,R(y¯,y¯)]-Λ[y¯,R(x¯,y¯)]+0(3),
where Λ:𝔊→𝔥 is the projection on 𝔥 parallel to V.Proof.
The proof is based on the direct computation. Denote that(2.18)z¯=x¯+y¯+12[x¯,y¯]+L(x¯,x¯,y¯)+M(x¯,y¯,y¯),h(x¯,y¯)=-12[x¯,y¯]+12∏[x¯,y¯]+2R(x¯,y¯)+E(x¯,x¯,y¯)+F(x¯,y¯,y¯).
From (2.8) with the consideration of (2.4) and (2.4′), we obtain the following:
(2.19)L(x¯,x¯,y¯)+M(x¯,y¯,y¯)+R(x¯,∏[x¯,y¯])+R(y¯,∏[x¯,y¯])+S(x¯,x¯,x¯)+3S(x¯,y¯,y¯)+3S(x¯,x¯,y¯)+S(y¯,y¯,y¯)+⋯=112[x¯,[x¯,y¯]]+112[y¯,[y¯,x¯]]+E(x¯,x¯,y¯)+F(x¯,y¯,y¯)+S(x¯,x¯,x¯)+S(y¯,y¯,y¯)+12[R(x¯,x¯),y¯]+12[x¯,R(y¯,y¯)]+14[x¯+y¯,∏[x¯,y¯]]-14[x¯+y¯,[x¯,y¯]]+[x¯+y¯,R(x¯,y¯)]+⋯.
Then by comparing term fromV and 𝔥 in the last identity, we obtain the requirement for L(x¯,x¯,y¯), M(x¯,y¯,y¯) and h(x¯,y¯) in addition
(2.20)E(x¯,x¯,y¯)=R(x¯,∏[x¯,y¯])+3S(x¯,x¯,y¯)+16Λ[x¯,[x¯,y¯]]-14Λ[x¯,∏[x¯,y¯]]-12Λ[R(x¯,x¯),y¯]-Λ[x¯,R(x¯,y¯)],F(x¯,y¯,y¯)=R(y¯,∏[x¯,y¯])+3S(x¯,y¯,y¯)-13Λ[y¯,[y¯,x¯]]+14Λ[y¯,∏[y¯,x¯]]-12Λ[x¯,R(y¯,y¯)]-Λ[y¯,R(x¯,y¯)].Corollary 2.4.
One can obtain that(2.21)(x×̲y)¯=x¯+y¯+12∏[x¯,y¯]-16∏[x¯,[x¯,y¯]]+12∏[R(x¯,x¯),y¯]+14∏[x¯,∏[x¯,y¯]]+∏[x¯,R(x¯,y¯)]+13∏[y¯,[y¯,x¯]]+12∏[x¯,R(y¯,y¯)]-14∏[y¯,∏[y¯,x¯]]+∏[y¯,R(x¯,y¯)]+o(3).
For the computation of terms of the fourth order, denote that(2.22)z¯=(2.21)+P(x¯,x¯,x¯,y¯)+Q(x¯,x¯,y¯,y¯)+U(x¯,y¯,y¯,y¯),
and for h to take terms of the third order
(2.23)P(x¯,x¯,x¯,y¯)+Q(x¯,x¯,y¯,y¯)+U(x¯,y¯,y¯,y¯)=[x¯+R(x¯,x¯)+S(x¯,x¯,x¯)]⋅[y¯+R(y¯,y¯)+S(y¯,y¯,y¯)]⋅(-12Λ[x¯,y¯]+2R(x¯,y¯)+E(x¯,x¯,y¯)+F(x¯,y¯,y¯)+⋯),
in the fourth order one needs to compute only the term in V. Conducting the reasoning as in the previous cases one obtains that
(2.24)P(x¯,x¯,x¯,y¯)+Q(x¯,x¯,y¯,y¯)+U(x¯,y¯,y¯,y¯)modh=⁡={x¯+R(x¯,x¯)+S(x¯,x¯,x¯)+y¯+R(y¯,y¯)+S(y¯,y¯,y¯)+12[x¯,y¯]+12[x¯,R(y¯,y¯)]+12[R(x¯,x¯),R(y¯,y¯)]+12[x¯,S(y¯,y¯,y¯)]+12[S(x¯,x¯,x¯),y¯]+112[x¯,[x¯,y¯]]+112[x¯,[x¯,R(y¯,y¯)]]+112[y¯,[y¯,x¯]]+112[y¯,[y¯,R(x¯,x¯)]]-148[y¯,[x¯,[x¯,y¯]]]-148[x¯,[y¯,[x¯,y¯]]]+⋯}⋅(-12Λ[x¯,y¯]+2R(x¯,y¯)+E(x¯,x¯,y¯)+F(x¯,y¯,y¯)+⋯12Λ[x¯,y¯])modh=12∏[x¯,E(x¯,x¯,y¯)]+12∏[x¯,F(x¯,y¯,y¯)]+12∏[y¯,E(x¯,x¯,y¯)]+12∏[y¯,F(x¯,y¯,y¯)]-18∏[∏[x¯,y¯],[x¯,y¯]]+12∏[∏[x¯,y¯],R(x¯,y¯)]+112∏[x¯,[x¯,-12Λ[x¯,y¯]+2R(x¯,y¯)]]+112∏[y¯,[y¯,-12Λ[x¯,y¯]+2R(x¯,y¯)]]+112∏[x¯,[y¯,-12Λ[x¯,y¯]+2R(x¯,y¯)]]+112∏[y¯,[x¯,-12Λ[x¯,y¯]+2R(x¯,y¯)]]+12∏[x¯,S(y¯,y¯,y¯)]+12∏[S(x¯,x¯,x¯),y¯]+112∏[x¯,[x¯,R(y¯,y¯)]]+112∏[y¯,[y¯,R(x¯,x¯)]]-148∏[y¯,[x¯,[x¯,y¯]]]-148∏[x¯,[y¯,[x¯,y¯]]].
All the equalities in the above expression are modulo 𝔥.Then the following proposition holds.Proposition 2.5.
We have the following:(2.25)P(x¯,x¯,x¯,y¯)=-12∏[y¯,S(x¯,x¯,x¯)]+112∏[x¯,[x¯,-112Λ[x¯,y¯]+2R(x¯,y¯)]]+12∏[x¯,E(x¯,x¯,y¯)],U(x¯,y¯,y¯,y¯)=12∏[x¯,S(y¯,y¯,y¯)]+112∏[y¯,[y¯,-112Λ[x¯,y¯]+2R(x¯,y¯)]]+12∏[y¯,F(x¯,y¯,y¯)],Q(x¯,x¯,y¯,y¯)=12∏[y¯,E(x¯,x¯,y¯)]+12∏[x¯,F(x¯,y¯,y¯)]-18∏[∏[x¯,y¯],[x¯,y¯]]+12∏[∏[x¯,y¯],R(x¯,y¯)]+112∏[x¯,[y¯,-12Λ[x¯,y¯]+2R(x¯,y¯)]]+112∏[y¯,[x¯,-12Λ[x¯,y¯]+2R(x¯,y¯)]]+112∏[x¯,[x¯,R(y¯,y¯)]]+112∏[y¯,[y¯,R(x¯,x¯)]]-148∏[y¯,[x¯,[x¯,y¯]]]-148∏[x¯,[y¯,[x¯,y¯]]].Corollary 2.6.
We have the following:(2.26)(x×̲y¯)=x¯+y¯+12∏[x¯,y¯]-16∏[x¯,[x¯,y¯]]+12∏[R(x¯,x¯),y¯]+14∏[x¯,∏[x¯,y¯]]+∏[x¯,R(x¯,y¯)]+13∏[y¯,[y¯,x¯]]+12∏[x¯,R(y¯,y¯)]-14∏[y¯,∏[y¯,x¯]]+∏[y¯,R(x¯,y¯)]+P(x¯,x¯,x¯,y¯)+Q(x¯,x¯,y¯,y¯)+U(x¯,y¯,y¯,y¯)+0(4),
where P(x¯,x¯,x¯,y¯), Q(x¯,x¯,y¯,y¯), and U(x¯,y¯,y¯,y¯) are from (2.25).


## 3. Tensor Structure of a Smooth Analytic Loop

Let〈Q,×,e〉 be a smooth analytic loop with the neutral element e. In a standard way, see [26] on the Cartesian product Q×Q, we introduce the structure of a three-web W such that the submanifold in the view of {a}×Q is a vertical foliations (a∈Q), Q×{b} is a horizontal foliations (b∈Q) and the set {(a,b):a×b=c=conts} foliations of the third family (c∈Q). In the coordinate (x1,x2,…,xn,y1,y2,…,yn), the indicated foliations are described by the system of differential 1-form [18, 19, 21, 23, 28, 37–40](3.1)ω1i=o,ω2i=o,ω3i=ω1i+ω2i=o,
where
(3.2)ω1i=Pαidxα,ω2i=Qβidyβ,Pαi(x,y)=∂μi∂xα,Qβi(x,y)=∂μi∂yβ,μi(x,y)=(x×y)i.In the space of a 3-WebW, introduce the so-called Chern canonical connection ∇=(∇1,∇2) [24, 38].The indicated connection is described by(3.3)ωjk=Γijkω1i+Γjlkω2j,Γijk=-P̃iαQ̃jβ∂2μk∂xα∂yβ,
where P̃iα and Q̃jβ are inverse matrices for Piα and Qjβ, respectively, in terms of the following structural equations:
(3.4)dω1k=ω1l∧ωlk+aijkω1i∧ωlj,dω2k=ω2l∧ωlk-aijkω2i∧ω2j,dωjk=ωji∧ωik+bjlmkω1l∧ω2m,
where
(3.5)aijk=-12∂2μk∂xα∂yβ(P̃iαQ̃jβ-P̃jαQ̃iβ),bjlmk=(-∂3μk∂xα∂xβ∂yγP̃jβ+∂3μk∂xα∂yβ∂yγQ̃jβ)P̃lαQ̃mγ-Γpmk∂2μp∂xα∂xβP̃lαP̃jβ+Γlpk∂2μp∂yα∂yβP̃jαQ̃mβ-ΓpmkΓljp+ΓlpkΓjmp.The Chern connection in the 3-Web associated to the loop〈Q,×,e〉 admits an alternative description in terms of antiproduct of the loop Q by itself [31, 33]. In the set Q×Q, introduce the covering loopuscular structure, by denoting for any pair X=(x,x′), Y=(y,y′), A(u,v)(3.6)L(X,A,Y)=((x(u∖yv))/v,u∖((uy′/v)x′)).Then the Chern connection coincides with the connection tangent to the covering loopuscular structure [33].In particular, for any tensor fieldΩ(u,v), in the space of 3-web W=Q×Q(3.7)∇1iΩ(u=e,v=e)=∂∂ui[{[L(u,e)(e,e)]*,(e,e)}-1Ω(u,e)]|u=e,(3.8)∇2iΩ(u=e,v=e)=∂∂vi[{[L(e,v)(e,e)]*,(e,e)}-1Ω(e,v)]|v=e.The value in the point(e,e) of the 3-Web W=Q×Q to the loop 〈Q,×,e〉 fundamental tensor field ajki, bjkli and their corresponding derivations ∇1i, ∇2i are called the tensors structure of the loop. The structure tensor of the smooth loop 〈Q,×,e〉 is defined uniquely by its construction up to isomorphism [24, 28, 29, 38].Proposition 3.1 (see [17, 38]).
The following relations hold(3.9)∇1lajki=b[j|l|k]i,∇2lajki=b[jk]li.For the proof of the proposition, it is sufficient to consider the first differential expression of the system (3.4).Introduce the notation(3.10)cjklmi=∇1mbjkli|(e,e),djklmi=∇2mbjkli|(e,e).And consider Proposition2.3. The law of composition (×) of the smooth local loop 〈Q,×,e〉 in the coordinate x=(x¯) centralized at the point e is given by
(3.11)(x×y)¯=x¯+y¯+K(x¯,y¯)+L(x¯,x¯,y¯)+M(x¯,y¯,y¯)+P(x¯,x¯,x¯,y¯)+Q(x¯,x¯,y¯,y¯)+U(x¯,y¯,y¯,y¯)+o(4).Consider〈Q,×,e〉 as a coordinate loop of the 3-Web W, defined in the neighborhood of the point (e,e) of the manifold Q×Q. Then in conformity with [24, 37], the basic tensor of the web can be expressed in terms of coefficient of the decomposition of the loop in the following way:
(3.12)a(x¯,y¯)=-K(x¯,y¯),b(x¯,y¯,z¯)=-B(y¯,x¯,z¯),(3.13)c(x¯,y¯,z¯,t¯)=(4Q-6P)(y¯,t¯,x¯,z¯)+a(t¯,b(x¯,y¯,z¯))+a(y¯,b(x¯,t¯,z¯))-b(x¯,a(t¯,y¯),z¯)+a(2L(y¯,t¯,x¯),z¯)-2L(a(x¯,y¯),t¯,z¯)-2L(y¯,a(x¯,t¯),z¯)-2L(y¯,t¯,a(x¯,z¯)),d(x¯,y¯,z¯,t¯)=(4Q-6P)(y¯,x¯,z¯,t¯)-a(b(x¯,y¯,z¯),t¯)-a(b(x¯,y¯,t¯),z¯)+b(x¯,y¯,a(z¯,t¯))+a(y¯,2M(x¯,z¯,t¯))-2M(a(y¯,x¯),z¯,t¯)-2M(y¯,a(z¯,x¯),t¯)-2M(y¯,z¯,a(t¯,x¯)),
where (3.14)B(x¯,y¯,z¯)=2L(x¯,y¯,z¯)-2M(x¯,y¯,z¯)-K(x¯,K(y¯,z¯))+K(K(x¯,y¯),z¯).


## 4. Tensor Structure of a Smooth Local Loop, Embedding in Lie Group

Let〈Q,×,e〉 be a local smooth loop, the embedding in the Lie group G as a section of left coset GmodH, where H is a closed subgroup in G. In what follows, we will consider that 〈Q,×,e〉 is referred to the normal coordinates X=(x¯).Proposition 4.1.
The following relations holds(4.1)a(x¯,y¯)=-12∏[x¯,y¯],(4.2)b(x¯,y¯,z¯)=-12∏[[x¯,y¯],z¯]+12∏[∏[x¯,y¯],z¯]-2∏[R(x¯,y¯),z¯].Proof.
The first relation follows from Proposition2.1 and the relation (3.12). In the relation (3.14), we have
(4.3)B(x¯,y¯,z¯)=2L(x¯,y¯,z¯)-2M(x¯,y¯,z¯)-K(x¯,K(y¯,z¯))+K(K(x¯,y¯),z¯),
and from Proposition 2.3 we have
(4.4)2L(x¯,y¯,z¯)=-16∏[x¯,[y¯,z¯]]+∏[R(x¯,y¯),z¯]+14∏[x¯,∏[y¯,z¯]]-16∏[y¯,[x¯,y¯]]+∏[x¯,R(y¯,z¯)]+∏[y¯,R(x¯,z¯)]+14∏[y¯,∏[x¯,z¯]],2M(x¯,y¯,z¯)=13∏[y¯,[z¯,x¯]]+∏[x¯,R(y¯,z¯)]-14∏[y¯,∏[z¯,x¯]]+13∏[z¯,[y¯,x¯]]+∏[y¯,R(x¯,z¯)]+∏[z¯,R(x¯,y¯)]-14∏[z¯,∏[y¯,x¯]],
Furthermore,(4.5)K(x¯,K(y¯,z¯))=14∏[x¯,∏[y¯,z¯]],K(K(x¯,y¯),z¯)=14∏[∏[x¯,y¯],z¯].
Substituting these expressions inB(x¯,y¯,z¯), we obtain that
(4.6)B(x¯,y¯,z¯)=-12∏[[x¯,y¯],z¯]+12∏[∏[x¯,y¯],z¯]+2∏[R(x¯,y¯),z¯],
but from (3.12) we have b(x¯,y¯,z¯)=-B(y¯,x¯,z¯). Hence,
(4.7)b(x¯,y¯,z¯)=-12∏[[x¯,y¯],z¯]-12∏[∏[x¯,y¯],z¯]-2∏[R(x¯,y¯),z¯].
LetΩ be one of the structural tensor of the loop Q, and consider the expression of the fundamental tensor field Ω(u,v) in the space of three-web W=Q×Q. Then Ω=Ω(u=e,v=e) and for ∇1iΩ(u=e,v=e), ∇2iΩ(u=e,v=e), the formulae obtained in (3.7) hold.
Consider the computation of∇1iΩ(u=e,v=e), the value of the tensor field Ω(u,v) for v=e can be seen as the structure of the smooth local loop 〈Q,×u,u〉, where (4.8)x×uy=x×(u∖y).
As a result, ∇ is transported from TuQ in TeQ by means of the inverse transformation Ru, which coincide with the structure of the tensor Ωũ and the smooth local loop 〈Q,·u,e〉 with the operation
(4.9)x⋅uy=u∖((u×x)×y).
So that
(4.10)∇1iΩ(u=e,v=e)=∂Ωũ∂ui|u=e,
in addition the law of composition (4.9) allows an intuitive algebraic interpretation in terms of the enveloping Lie group G.
Consider the sectionQu′=Q·u-1 of the coset space G/Hũ, where  Hũ=u·H·u-1, u∈Q and the map
(4.11)Ψu:Q⟶Qu′x⟼(u×x)×u-1.
Denote by (*u) the law of composition in Qu′, so that
(4.12)a*ub=∏u′(ab),
where ∏u′:G→Qu′ is the projection on Qu′ parallel to Hũ. The following proposition hold.Proposition 4.2.
The mapΨu:Q→Qu′ is an isomorphism of the smooth loops 〈Q,·u,e〉 and 〈Qu′,*u,e〉.Proof.
Leta=Ψux, b=Ψuy, and a*ub=Ψuz, where x,y,z∈Q.
Then(4.13)a*ub=∏u′(ab)=∏u′((u×x)⋅u-1⋅(u×y)⋅u-1),(a*ub)×u⋅h⋅u-1=(u×x)u-1⋅(u×y)⋅u-1.
Multiplying byu we obtain that
(4.14)(a*ub)×u⋅h=(u×x)×y.
Applying the projection to the last equality, we obtain that
(4.15)(a*ub)×u=(u×x)×y.
Furthermore,(4.16)(a*ub)×u=(Ψuz)×u=(u×z)⋅u-1×u=(u×x)×y.
Then z=u∖(u×x)×y and
(4.17)(a*ub)=(Ψux)*u(Ψuy)=Ψuz=Ψu{u∖(u×x)×y}=Ψu(x⋅uy).
Therefore Ψu(x·uy)=(Ψux)*(Ψuy). Hence, here is the result. Similarly we establish that
(4.18)∇2iΩ(u=e,v=e)=∂Ωṽ̃∂vi|υ=e,
where Ω̃̃ correspond to the structure tensor of the local loop 〈Q,1/v,e〉 with the composition law
(4.19)x1vy=(x×(y×v))/v.
The law of composition (4.19) allows us to find an algebraic interpretation in terms of the enveloping Lie group G.Let us introduce in consideration the subgroupHv′′=vHv-1 where v∈Q. The following proposition holds.Proposition 4.3.
We have the following:(4.20)x1vy=∏v′′(xy)
for all x,y∈Q, where
(4.21)∏v′′:G→Qis the projection onQparallel toHv′′.Proof.
In the Lie groupG, we have xy=(x⊥y)×vhv-1 which is equivalent to xy·v=(x⊥y)×vh. Applying ∏ to the last formula, we get the following:
(4.22)x×(y×v)=(x⊥y)×v.
Therefore, x⊥y=x×(y×v)/v.


## 5. Application: Computation of∇2lajki and ∇1lajki

(I) Computation of∇2lajki
Foru∈Q, introduce the map
(5.1)Adu:G⟶G,x⟼uxu-1.
Let u=expζ, where ζ∈Q and g∈H. Then
(5.2)Adu(g)=ugu-1=Ad(exp⁡⁡ζ)(g)=exp⁡⁡(adζ(g))=g+[ζ,g]+o(ζ)
and g+[ζ,g]+o(ζ)∈Hu′′, where Hu′′=uHu-1.
Let∏u′′:𝔊→TeQ be the projection on TeQ parallel to 𝔥u′′ and exp𝔥u′′=Hu′′.
By fixingξ,η from 𝔊, we find that
(5.3)[ξ,η]=∏[ξ,η]+h1,(5.4)[ξ,η]=∏u′′[ξ,η]+h2,
where h1∈𝔥 and h2∈𝔥u′′. From (5.2) we obtain that h2 has the form h2=h1+ĥ(ζ)+[ζ,h1]+o(ζ), where ĥ(ζ)∈𝔥u′′. From (5.3) and (5.4), it follows that
(5.5)∏u′′[ξ,η]=[ξ,η]-h2=∏[ξ,η]-ĥ(ζ)-[ζ,h1]+o(ζ)=∏[ξ,η]-∏[ζ,h1]+o(ζ).
But from (5.3), we have h1=[ξ,η]-∏[ξ,η]. It follows that
(5.6)∏u′′[ξ,η]=∏[ξ,η]-∏[ζ,[ξ,η]]+∏[ζ,∏[ξ,η]]+o(ζ)=∏[ξ,η]+∏[[ξ,η],ζ]-∏[∏[ξ,η],ζ]+o(ζ).
Denote byau′′(ξ,η)=-(1/2)∏u′′[ξ,η]. Then
(5.7)au′′(ξ,η)=a(ξ,η)-12∏[[ξ,η]]+12∏[∏[ξ,η],ζ].
Finally we have(5.8)∇2lajkiξjηkζl=ddt(aexp⁡⁡tζ′′(ξ,η))|t=0=-12∏[[ξ,η]]+12∏[∏[ξ,η],ζ].
We obtain a result in conformity with Proposition3.1 and the relation (4.2) indeed, from the relation (4.2)
(5.9)b(ξ,η,ζ)=-12[[ξ,η],ζ]+12∏[∏[ξ,η],ζ]-2∏[R(ξ,η),ζ].
From which we find that
(5.10)12[b(ξ,η,ζ)-b(η,ξ,ζ)]=-12∏[[ξ,η],ζ]+12∏[∏[ξ,η],ζ],
so that ∇2lajki=b[jk]li.(II) Computation of∇1lajki
Let us introduce the map(5.11)Ψu:Q⟶Qu′,x⟼(u×x)u-1.
ThendΨu|e:TeQ→TeQu′. Then the following proposition holds.Proposition 5.1.
The map defined from the tangent spaceTeQ to tangent space TeQu′ is defined as follows:
(5.12)dΨu|e:TeQ⟶TeQu′,ξ⟼ξ+12[u,ξ]+12∏[u,ξ]+2R(u,ξ)+o(u).Proof.
For the proof of this proposition, using the notion from Section2 and the relation (2.8), we have u×ξ=(u·ξ)·h but from Proposition 1.4, we have
(5.13)h(u,ξ)=-12[u,ξ]+12[u,ξ]+2R(u,ξ)+o(u).
Thus,(5.14)u×ξ=(u⋅ξ)⋅h=u+ξ+12[u,ξ]+12∏[u,ξ]+2R(u,ξ)+o(u),(u×ξ)×u-1=u+ξ+12∏[u,ξ]+2R(u,ξ)-u-12[ξ,u]+o(u)=ξ+12∏[u,ξ]+12[u,ξ]+2R(u,ξ)+o(u).
Let∏ũ:𝔊→TeQ′ be the projection on TeQ′ parallel to 𝔥ũ′ where exp𝔥ũ′=uHu-1.
Then we obtain the following:(5.15)ω+h1=ω′+h1′+[u,h1′]
with ω∈TeQ, h1∈𝔥, ω′∈TeQ′, h1′∈𝔥. For the computation of ω′=ω′(u,ω). From Proposition 5.1, we have
(5.16)ω+h1=ω̃+12∏[u,ω̃]+12[u,ω̃]+2R(u,ω̃)+h1′+[u,h1′]+o(u),
where ω̃∈TeQ, so that
(5.17)ω̃+12∏[u,ω̃]+12[u,ω̃]+2R(u,ω̃)=ω′.
It follows that
(5.18)ω=ω̃+∏[u,ω̃]+[u,h1′],h1=h1′+termswithu,
from which
(5.19)ω̃=ω-∏[u,ω]-[u,h1′],h1′=h1+termwithu.
Then substituting in ω′ the expression from ω̃, we obtain that
(5.20)ω′=ω-∏[u,ω]-∏[u,h1]+12∏[u,h1]+12[u,ω]+2R(u,ω)+o(u)=ω+12[u,ω]-12∏[u,ω]-∏[u,h1]+2R(u,ω)+o(u),
from which we find that
(5.21)∏ũ(ω+h1)=ω′=ω+12[u,ω]-12[u,ω]+2R(u,ω)-∏[u,h1].
Now let us compute that(5.22)aũ(ξ,η)=-12(dΨ)-1∏ũ[dΨξ,dΨη],
where ξ,η∈TeQ,
(5.23)(dΨ)-1∏ũ[dΨξ,dΨη]=(dΨ)-1∏ũ[ξ+12[u,ξ]+12∏[u,ξ]+2R(u,ξ),η+12[u,η]+12∏[u,η]+2R(u,η)]=(dΨ)-1∏ũ{[ξ,η]+12[ξ,[u,η]]+12[ξ,∏[u,η]]+2[ξ,R(u,η)]-12[η,[u,ξ]]-12[η,∏[u,ξ]]-2[η,R(u,ξ)]}=(dΨ)-1{∏[ξ,η]+12∏[ξ,[u,η]]+12∏[ξ,∏[u,η]]+2∏[ξ,R(u,η)]-12∏[η,[u,ξ]]-12∏[η,∏[u,ξ]]-2∏[η,R(u,ξ)]+12[u,∏[ξ,η]]-12[u,∏[ξ,η]]+2R(u,∏[ξ,η])-∏[u,[ξ,η]]+∏[u,∏[ξ,η]]}=∏[ξ,η]+12[ξ,[u,η]]+12∏[ξ,∏[u,η]]+2∏[ξ,R(u,η)]-12∏[η,[u,ξ]]-12∏[η,∏[u,ξ]]-2∏[η,R(u,ξ)]-∏[u,[ξ,η]]=∏[ξ,η]+12[ξ,[η,u]]-12∏[ξ,∏[η,u]]+2∏[ξ,R(u,η)]-12∏[η,[ξ,u]]+12∏[η,∏[ξ,u]]-2∏[η,R(u,ξ)],
where
(5.24)aũ(ξ,η)=-12∏[ξ,η]-14∏[[ξ,u],η]+14∏[∏[ξ,u],η]-∏[R(u,ξ),η]+14∏[[η,u],ξ]-14∏[∏[η,u],ξ]+∏[R(u,η),ξ].
From this last equation, it follows that(5.25)∇1lajkiξjηkζl=ddtaexp⁡tζ̃(ξ,η)|t=0=-14∏[[ξ,ζ],η]+14∏[∏[ξ,ζ],η]-∏[R(ξ,ζ),η]+14∏[[η,ζ],ξ]-14∏[∏[η,ζ],ξ]+∏[R(η,ζ),ξ].
We obtain a result in conformity with Proposition3.1 and the relation (4.2) indeed from the formulae (4.2), it follows that
(5.26)12[b(ξ,ζ,η)-b(η,ζ,ξ)]=12{-12∏[[ξ,ζ],η]+12∏[∏[ξ,ζ],η]-2∏[R(ξ,ζ),η]+12∏[[η,ζ],ξ]-12∏[∏[η,ζ],ξ]+2∏[R(η,ζ),ξ]}=-14∏[[ξ,ζ],η]+14∏[∏[ξ,ζ],η]-∏[R(ξ,ζ),η]+14∏[[η,ζ],ξ]-14∏[∏[η,ζ],ξ]+∏[R(η,ζ),ξ].
Therefore,(5.27)∇1lajki=b[j|j|k]i.


## 6. Computation of the Tensordjklmi=∇2mbjkli

Denote thatu·R(η,η)·u-1 by Ru′′(η,η). For the computation of djklmi let us firstly compute Ru′′(η,η).The following proposition holds.Proposition 6.1.
We have the following:(6.1)Ru′′(η,η)=R(η,η)+∏[u,R(η,η)]+0(u,η2).The proof of this proposition is from Section1. It is clear that ξ+ϕ(ξ)∈Q, and from Section 4  hu′′=h1+[u,h1]+0(u), where h1∈h. Furthermore η+Ru′′(η,η)∈Q but Ru′′(η,η)∈hu′′ that is why Ru′′(η,η) can be represented as Ru′′(η,η)=h1+[u,h1]+0(u), where h1=Ru′′(η,η)-[u,Ru′′(η,η)]+0(u). Let us write η+Ru′′(η,η) as
(6.2)η+Ru′′(η,η)={(η+∏[u,Ru′′(η,η)])+(Ru′′(η,η)-[u,Ru′′(η,η)])+([u,Ru′′(η,η)]-∏[u,Ru′′(η,η)])},putη+∏[u,Ru′′(η,η)]=ξ then
(6.3)ϕ(ξ)=Ru′′(η,η)-[u,Ru′′(η,η)]+[u,Ru′′(η,η)]-∏[u,Ru′′(η,η)]=Ru′′(η,η)-∏[u,Ru′′(η,η)]+o(u),
from the relation (2.4) we have ϕ(ξ)=R(ξ,ξ)+S(ξ,ξ,ξ)+0(3).Therefore by comparing the term on the right hand sides of the last two relations, we obtain that(6.4)Ru′′(η,η)=R(η,η)+∏[u,R(η,η)]+0(u,η2).Let∏u′′:𝔊→V=TeQ be the projection of 𝔊 to V parallel to 𝔥u′′. Then we obtain the following: (6.5)ξ+h̃=ξ̃+h1+[u,h1],
where ξ,ξ̃∈V and h̃,h1∈𝔥 for the search of ξ̃=ξ̃(ξ,u), we have(6.6)ξ+h̃=ξ̃+h1+∏[u,h1]+([u,h1]-∏[u,h1]),
where
(6.7)ξ=ξ̃+∏[u,h1],h̃=h1+[u,h1]-∏[u,h1]=h1+termswithu.
From these two equalities, we obtain that(6.8)ξ̃=ξ-∏[u,h̃]+0(u).
Hence (6.9)∏u′′(ξ+h̃)=ξ-∏[u,h̃].We pass now to the computation ofdjklmi.From (4.2) it follows that(6.10)b(ξ,η,ζ)=-12∏[[ξ,η],ζ]+12∏[∏[ξ,η],ζ]-2∏[R(ξ,η),ζ],
that is why (6.11)bu′′(ξ,η,ζ)=-12∏u′′[[ξ,η],ζ]+12∏u′′[∏u[ξ,η],ζ]-2∏u′′[Ru′′(ξ,η),ζ].From (6.9) it follows that(6.12)-12∏u′′[[ξ,η],ζ]=-12∏[[ξ,η],ζ]+12∏[u,[[ξ,η],ζ]]-12∏[u,∏[[ξ,η],ζ]].Furthermore,(6.13)12∏u′′[∏u′′[ξ,η],ζ]=12∏u′′[∏[ξ,η],ζ]-12∏u′′[∏[u,[ξ,η]],ζ]+12∏u′′[∏[u,∏[ξ,η]],ζ]=12∏[∏[ξ,η],ζ]-12∏[u,[∏[ξ,η],ζ]]+12∏[u,∏[∏[ξ,η],ζ]]+o(u).
Finally from (6.1) and (6.9), it follows that
(6.14)-2∏u′′[Ru′′(ξ,η),ζ]=-2∏u′′[Ru(ξ,η),ζ]-2∏u′′[∏[u,R(ξ,η)],ζ]=-2∏[R(ξ,η),ζ]+2∏[u,[R(ξ,η),ζ]]-2∏[u,∏[R(ξ,η),ζ]]-2∏[∏[u,R(ξ,η)],ζ]+o(u)
from (6.12), (6.13), and (6.14), it follows that
(6.15)d(ξ,η,ζ,τ)=∇2mbjkli|(e,e)ξjηkζlτm=ddt(bexp⁡⁡tτ′′(ξ,η,ζ))|t=0=12∏[τ,[[ξ,η],ζ]]-12∏[τ,∏[[ξ,η],ζ]]-12∏[τ,[∏[ξ,η],ζ]]+12∏[τ,∏[∏[ξ,η],ζ]]-12∏[∏[τ,[ξ,η]],ζ]+12∏[∏[τ,∏[ξ,η]],ζ]+2∏[τ,[R(ξ,η),ζ]]-2∏[τ,∏[R(ξ,η),ζ]]-2∏[∏[τ,R(ξ,η)],ζ].
In the theory of 3-Web [17, 37, 39], the following relation is known: (6.16)djk[lm]i=-bjkpialmp.
Let us verify that:
(6.17)12(d(ξ,η,ζ,τ)-d(ξ,η,τ,ζ))=14∏[τ,[[ξ,η],ζ]]-14∏[ζ,[[ξ,η],τ]]-14∏[τ,∏[[ξ,η],ζ]]+14∏[ζ,∏[[ξ,η],τ]]-14∏[τ,[∏[ξ,η],ζ]]+14∏[ζ,[∏[ξ,η],τ]]+14∏[τ,∏[∏[ξ,η],ζ]]-14∏[ζ,∏[∏[ξ,η],τ]]-14∏[∏[τ,[ξ,η]],ζ]+14∏[∏[ζ,[ξ,η]],τ]+14∏[∏[τ,∏[ξ,τ]],ζ]-14∏[∏[ζ,∏[ξ,η]],τ]+∏[τ,[R(ξ,η),ζ]]-∏[ζ,[R(ξ,η),τ]]-∏[τ,∏[R(ξ,η),ζ]]+∏[ζ,∏[R(ξ,η),τ]]-∏[∏[τ,R(ξ,η)],ζ]+∏[∏[ζ,R(ξ,η)],τ]=-14∏[[ξ,η],[ζ,τ]]+14∏[∏[ξ,η],[ζ,τ]]-∏[R(ξ,η),[ζ,τ]].In addition, considering that(6.18)[ζ,τ]=∏[ζ,τ]+([ζ,τ]-∏[ζ,τ]).One obtain that(6.19)12(d(ξ,η,ζ,τ)-d(ξ,η,τ,ζ))=-14∏[[ξ,η],∏[ζ,τ]]+14∏[∏[ξ,η],∏[ζ,τ]]-∏[R(ξ,η),∏[ζ,τ]].From relations (4.1) and (4.2), it follows that
(6.20)b(ξ,η,a(ζ,τ))=12b(ξ,η,∏[ζ,τ])=-14∏[[ξ,η],∏[ζ,τ]]+14∏[∏[ξ,η],∏[ζ,τ]]-∏[R(ξ,η),∏[ζ,τ]].
Hence djk[lm]i=-bjkpialmp.


## 7. Hexagonal Loops

The analytic hexagonal 3-Web and their corresponding loops can be characterize by the following condition:(7.1)b(jkl)i=0,
where b(ξ,η,ζ)=-(1/2)∏[[ξ,η],ζ]+(1/2)∏[∏[ξ,η],ζ]-2∏[R(ξ,η),ζ], that is, way, b(jkl)i=0 is equivalent to the following condition:(7.2)∏[R(ξ,η),ζ]+∏[R(η,ζ),ξ]+∏[R(ζ,ξ),η]=0,
which can be written as follows:
(7.3)σξηζ∏[R(ξ,η),ζ]=0,
where σξηζ is the cyclic sum for ξ,η,ζ.We have furthermore, for the hexagonal three webs the following relation:(7.4)d(jkl)mi=0.
Considering (6.15) and (7.2), one obtain that(7.5)σξηζ∏{[τ,[R(ξ,η),ζ]]-[∏[τ,R(ξ,η)],ζ]}=0,
where σξηζ is the cyclic sum for ξ,η,ζ.



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*Source: 804051-2012-04-08.xml*