arXiv:math/0310097v1 [math.DG] 7 Oct 2003
On an Algebraical computation of the tensor and the curvature for 3-Webs Thomas B. Bouetou∗ ´ Ecole Nationale Sup´erieure Polytechnique, B.P. 8390 Yaound´e, Cameroun e-mail:
[email protected] February 1, 2008
Abstract. We suggest a new, alternative algebraic method for com1
2
putation the quantities ∇l aijk , ∇l aijk and dijklm by means of the embedding of local loops into Lie groups. Keywords: Homogeneous smooth Loops, Lie groups, Lie algebras, 3-Webs, Chern connection AMS subject classification 2000: 11E57, 14C21, 19L99, 20N05, 22E60, 22E67, 32C22, 53A60 Introduction The development of geometry of fiber bundles and foliations stimulates the interest for new investigation of three-Webs. In [3, 5, 10] the techniques was developed for webs using the intresic 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 yield. Therefore we give an application of the computation of the well known tensor We use 1
2
algebraic methods to compute the relations ∇l aijk , ∇l aijk and dijklm . ∗ currently at UMR 5030 (CNRS), D´ epartement des Sciences Math´ ematiques Universit´ e Montpellier II Case courrier 051-Place Eug´ ene Bataillon 34095 Montpellier CEDEX 05, France e-mail:
[email protected]
1
1
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 unity element of G(e ∈ G). The composition law: × : Q × Q −→ Q Y (x, y) 7−→ x × y = (x · y), Q
Q
where Q : G → Q is the projection on Q parallel to the subgroup H, defines in Q a structure of a local loop, i.e < Q, ×, e >-loop [8, 12, 15] . Let us map the tangent space Te Q with the vector subspace V ⊂ G. Then G = V ∔ h since the submanifolds Q and H are transversal in the Lie group G. Let us introduce the mapping φ: φ : V −→ h ξ 7−→ 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). Then φ(O) = O and φ(ξ) = R(ξ, ξ) + S(ξ, ξ, ξ) + o(3) where R : V × V −→ h S : V × V × V −→ h
(1.1)
are bilinear and trilinear symmetric maps. A base < e1 , e2 , ...., eN > is fixed in G such that < e1 , e2 , ...., en > generate V i.e. V =< e1 , e2 , ...., en > and < en+1 , en+2 , ...., eN > generate 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 exp V , that is for all x ∈ Q, x = (xi )i=1,n , this mean exp(xi ei + φ(xi ei )) = x ∈ Q Introduce the map Q −→ V x 7−→ x = xi ei . Then the condition written before is equivalent to x + φ(x) = x ∈ Q. 2
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 group G(·) has the following representation up to the fourth order in the normal coordinates: 1 1 1 a · b = a + b + [a, b] + [a, [a, b]] + [b, [b, a]] 2 12 12 1 1 − [b, [a, [a, b]]] − [a, [b, [a, b]]] + o(4). (1.1)′ 48 48 Consider the coordinate representation of the law of composition ×;for y: x = (x) and y = (y) in Q. We have:
(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)
(1.2)
(Our notation are similar to the notations of the work [7]). Denote the right side in (1.2) by z = (z). Then for its computation we obtain the equation exp(z + φ(z)) = exp(x + φ(x)) · exp(y + φ(y))h
(1.3)
where h is and element from h in deed we have h = h(x, y). The following proposition holds: Proposition 1.1 We have: K(x, y) =
1Y [x, y] 2
Q where [x, y] is the projection of the commutator [x, y] on V parallel to the subalgebra h. 1 1Y h(x, y) = − [x, y] + [x, y] + 2R(x, y) + o(2). 2 2 Proof: we use the formulae (1.3). Comparing the terms from V and h and considering only the terms of first order we obtain: z =x+y ∈V h = o ∈ h. For computing the term of second order we denote z = x + y + K(x, y) ∈ V 3
h = N (x, y) ∈ h from (1.3) and considering (1.1) and (1.1)’ we have:
1 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)+ [x, y] 2 then by comparing term from V and h and noting that: 1 1Y 1 1Y [x, y] = [x, y] + ( [x, y] − [x, y]) 2 2 2 2
hence
K(x, y) =
1Y [x, y] 2
1 1Y h(x, y) = − [x, y] + [x, y] + 2R(x, y) 2 2 Corollary 1.1: from the proposition (1.1) it follows that (x × y) = x + y + Proposition 1.2 One can show: L(x, x, y) = −
M (x, y, y) =
1Y [x, y] + o(2) 2
Y 1Y 1Y 1Y Y [x, [x, y]]+ [R(x, x), y]+ [x, [x, y]]+ [x, R(x, y)] 6 2 4
Y 1Y 1Y 1Y Y [y, [y, x]] + [x, R(y, y)] − [y, [y, x]] + [y, R(x, y)] 3 2 4
Y 1 1Y [x, y] + 2R(x, y) + R(x, [x, y]) + 3S(x, x, y)+ h(x, y) = − [x, y] + 2 2 Y 1 1 1 Λ[x, [x, y]] − Λ[x, [x, y]] − Λ[R(x, x), y] − Λ[x, R(x, y)]+ 6 4 2
Y 1 [x, y]) + 3S(x, y, y) − Λ[y, [y, x]]+ 3 Y 1 1 Λ[y, [y, x]] − Λ[x, R(y, y)] − Λ[y, R(x, y)] + 0(3) 4 2 where Λ : G −→ h is the projection on h parallel to V . +R(y,
4
Proof. The proof is based on the direct computation. Denote: 1 z = x + y + [x, y] + L(x, x, y) + M (x, y, y) 2 and 1 1Y h(x, y) = − [x, y] + [x, y] + 2R(x, y) + E(x, x, y) + F (x, y, y). 2 2
From (1.3) with the consideration of (1.1) and (1.1)’ we obtain the equation Y Y L(x, x, y) + M (x, y, y) + R(x, [x, y]) + R(y, [x, y]) + S(x, x, x) + 3S(x, y, y)+
1 1 [x, [x, y]]+ [y, [y, x]]+E(x, x, y)+F (x, y, y)+ 12 12 Y 1 1 1 S(x, x, x) + S(y, y, y) + [R(x, x), y] + [x, R(y, y)] + [x + y, [x, y]]− 2 2 4 1 [x + y, [x, y]] + [x + y, R(x, y)] + .... 4 Then by comparing term from V 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
3S(x, x, y)+S(y, y, y)+..... =
E(x, x, y) = R(x,
Y Y 1 1 [x, y]) + 3S(x, x, y) + Λ[x, [x, y]] − Λ[x, [x, y]]− 6 4 1 − Λ[R(x, x), y] − Λ[x, R(x, y)] 2
F (x, y, y) = R(y,
(1.4)
Y Y 1 1 [x, y]) + 3S(x, y, y) − Λ[y, [y, x]] + Λ[y, [y, x]]− 3 4 1 − Λ[x, R(y, y)] − Λ[y, R(x, y)] 2
(1.5)
Corollary 1.2: One can obtain:
(x × y) = x+y+
+
1Y 1Y 1Y 1Y Y [x, y]− [x, [x, y]]+ [R(x, x), y]+ [x, [x, y]]+ 2 6 2 4
Y 1Y 1Y [x, R(x, y)] + [y, [y, x]] + [x, R(y, y)]− 3 2 5
−
Y 1Y Y [y, [y, x]] + [y, R(x, y)] + o(3). 4
(1.6)
For the computation of terms of fourth order, denote
z = (1.6) + P (x, x, x, y) + Q(x, x, y, y) + U (x, y, y, y)
and for h to take terms of third order. 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)] · (− 12 Λ[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 past cases one obtain: ( 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) + 21 [x, y] + 1 [x, [x, y]]+ + 21 [x, R(y, y)]+ 21 [R(x, x), R(y, y)]+ 21 [x, S(y, y, y)]+ 21 [S(x, x, x), y)]+ 12 +
1 12 [x, [x, R(y, y)]]
−
1 48 [x, [y, [x, y]]]
1 +) 12 [y, [y, x]] +
+ ...
1 12 [y, [y, R(x, x)]]
−
1 48 [y, [x, [x, y]]]
−
· (− 12 Λ[x, y] + 2R(x.y) + E(x, x, y) + F (x, y, y) + ...) =
1Y 1Y 1Y 1Y [x, E(x, x, y)]+ [x, F (x, y, y)]+ [y, E(x, x, y)]+ [y, F (x, y, y)]− 2 2 2 2 1YY 1YY 1 Y 1 [ [x, y], [x, y]]+ [ [x, y], R(x, y)]+ [x, [x, − Λ[x, y]+2R(x, y)]] 8 2 12 2 1 Y 1 1 Y 1 + [y, [y, − Λ[x, y] + 2R(x, y)]] + [x, [y, − Λ[x, y] + 2R(x, y)]]+ 12 2 12 2 1 Y 1 1Y 1Y [y, [x, − Λ[x, y] + 2R(x, y)]] + [x, S(y, y, y)] + [S(x, x, x), y]+ 12 2 2 2 1 Y 1 Y 1 Y 1 Y [x, [x, R(y, y)]]+ [y, [y, R(x, x)]]−− [y, [x, [x, y]]]− [x, [y, [x, y]]]. 12 12 48 48
=
all the equality in the above expression are modulo h. Then the following proposition holds:
6
Proposition 1.3 P (x, x, x, y) = −
1Y 1 Y 1 [y, S(x, x, x)] + [x, [x, − Λ[x, y] + 2R(x, y)]]+ 2 12 12 1Y [x, E(x, x, y)] 2
U (x, y, y, y) =
1Y 1 Y 1 [x, S(y, y, y)] + [y, [y, − Λ[x, y] + 2R(x, y)]]+ 2 12 12 +
Q(x, x, y, y) =
+
1Y [y, F (x, y, y)] 2
(1.8)
1Y 1Y 1YY [y, E(x, x, y)] + [x, F (x, y, y)] − [ [x, y], [x, y]]+ 2 2 8
1YY 1 Y 1 [ [x, y], R(x, y)] + [x, [y, − Λ[x, y] + 2R(x, y)]]+ 2 12 2
+
+
(1.7)
1 1 Y 1 Y [y, [x, − Λ[x, y] + 2R(x, y)]] + [x, [x, R(y, y)]]+ 12 2 12
1 Y 1 Y 1 Y [y, [y, R(x, x)]] − [y, [x, [x, y]]] − [x, [y, [x, y]]] 12 48 48
(1.9)
Corollary 1.3:
(x × y) = x+y+
1Y 1Y 1Y 1Y Y [x, y]− [x, [x, y]]+ [R(x, x), y]+ [x, [x, y]]+ 2 6 2 4
+
Y 1Y 1Y 1Y Y [x, R(x, y)] + [y, [y, x]] + [x, R(y, y)] − [y, [y, x]]+ 3 2 4
+
Y [y, R(x, y)] + P (x, x, x, y) + Q(x, x, y, y) + U (x, y, y, y) + 0(4)
(1.10)
Where P (x, x, x, y), Q(x, x, y, y) and U (x, y, y, y) are from (1.7), (1.8) and (1.9)
7
2
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 [9] on the Cartesian product Q × Q we introduce the structure of a three-webs 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} the foliations of the third family (c ∈ Q). In the coordinate (x1 , x2 , ....., xn , y 1 , y 2 , ......., y n ), the indicated foliations are described by the system of differential 1-form [1, 11]. ω1i = o, ω2i = o, ω3i = ω1i + ω2i = o where ω1i = Pαi dxα , ω2i = Qiβ dy β , ∂µi ∂xα ∂µi Qiβ (x, y) = ∂y β Pαi (x, y) =
µi (x, y) = (x × y)i In the space of a 3-Web W , introduce the so called Chern canonical connection 1
2
∇ = (∇, ∇) [7, 21]. The indicated connection is described by: ωjk = Γkij ω1i + Γkjl ω2j , eβ Γkij = −Peiα Q j
∂ 2 µk ∂xα ∂y β
e β are inverse matrices for P α and Qβ respectively in terms where Peiα and Q i j j of the following structural equations: dω1k = ω1l ∧ ωlk + akij ω1i ∧ ωlj
dω2k = ω2l ∧ ωlk − akij ω2i ∧ ω2j
(2.1)
dωjk = ωji ∧ ωik + bkjlm ω1l ∧ ω2m where akij = −
bkjlm = (−
1 ∂ 2 µk eα e β eα e β (P Q − Pj Qi ) 2 ∂xα ∂y β i j
2 p ∂ 3 µk ∂ 3 µk eβ + e β )Pelα Q e γm − Γkpm ∂ µ Pelα Pe β + P Q j j j ∂xα ∂xβ ∂y γ ∂xα ∂y β ∂y γ ∂xα ∂xβ
8
+Γklp
∂ 2 µp e α e β P Q − Γkpm Γplj + Γklp Γpjm ∂y α ∂y β j m
The Chern connection in the 3-Web associated to the loop < Q, ×, e >, admits an alternative description in terms of anti-product of the loop Q by itself [14, 16]. In the set Q × Q introduce the covering loopouscular structure, by denoting for any pair X = (x, x′ ), Y = (y, y ′ ), A(u, v) L(X, A, Y ) = ((x(u\yv))/v, u\((uy ′ /v)x′ ))
(2.2).
Then the Chern connection coincide with the connection tangent to the covering loopuscular structure [16]. In particular, for any tensor field Ω(u, v), in the space of 3-web W = Q × Q " # o−1 1 ∂ n (e,e) [L(u,e) ]∗,(e,e) Ω(u, e) |u=e (2.3), ∇i Ω(u = e, v = e) = ∂ui " # o−1 2 ∂ n (e,e) ∇i Ω(u = e, v = e) = i [L(e,v) ]∗,(e,e) Ω(e, v) |v=e . ∂v The value in the point (e, e) of the 3-Web W = Q×Q to the loop < Q, ×, e > 1
the fundamental tensor field aijk , bijkl and their corresponding derivations ∇i , 2
∇i are called the structure tensors of the loop. The structure tensor of the smooth loop < Q, ×, e > defined uniquely by its construction up to isomorphism [7, 11, 12, 21]
Proposition 2.1 [1, 21] The following relations hold: 1
∇l aijk = bi[j|l|k] 2
∇l aijk = bi[jk]l
(2.4) (2.5).
For the proof of the proposition, it’s sufficient to consider the first differential expression of the system (2.1). Introduce the notation 1
cijklm = ∇m bijkl |(e,e) 2
dijklm = ∇m bijkl |(e,e) .
9
And consider the proposition (1.2). The law of composition (×) of the smooth local loop < Q, ×, e > in the coordinate x = (x) centralized at the point e, is given by:
(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 neighbourhood of the point (e, e) of the manifold Q × Q. Then in conformity with [7, 20] the basic tensor of the web can be expressed in term the of coefficient of the decomposition of the loop in the following way:
a(x, y) = −K(x, y), b(x, y, z) = −B(y, x, z)
(2.6)
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))
(2.7)
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))
(2.8)
where B(x, y, z) = 2L(x, y, z) − 2M (x, y, z) − K(x, K(y, z)) + K(K(x, y), z) (2.10) 10
3
Structure tensor 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 3.1 The following relations holds:
a(x, y) = −
b(x, y, z) = −
1Y [x, y] 2
Y 1Y 1YY [[x, y], z] + [ [x, y], z] − 2 [R(x, y), z] 2 2
(3.1)
(3.2)
Proof: The first relation follows from the proposition 1.1 and the relation (2.6). In the relation (2.10) we have: B(x, y, z) = 2L(x, y, z) − 2M (x, y, z) − K(x, K(y, z)) + K(K(x, y), z) and from the proposition 1.2 we have:
2L(x, y, z) = −
Y 1Y 1Y Y 1Y [x, [y, z]]+ [R(x, y), z]+ [x, [y, z]]− [y, [x, y]]+ 6 4 6
+
2M (x, y, z) =
Y Y 1Y Y [x, R(y, z)] + [y, R(x, z)] + [y, [x, z]] 4
Y 1Y 1Y Y 1Y [y, [z, x]] + [x, R(y, z)] − [y, [z, x]] + [z, [y, x]]+ 3 4 3 +
Y Y 1Y Y [y, R(x, z)] + [z, R(x, y)] − [z, [y, x]] 4
11
further more K(x, K(y, z)) =
1Y Y [x, [y, z]]. 4
1YY [ [x, y], z]. 4 Substituting these expressions in B(x, y, z), we obtain: K(K(x, y), z)) =
Y 1Y 1YY [[x, y], z] + [ [x, y], z] + 2 [R(x, y), z] 2 2 but from (2.6) we have b(x, y, z) = −B(y, x, z). Hence : B(x, y, z) = −
b(x, y, z) = −
Y 1Y 1YY [[x, y], z] − [ [x, y], z] − 2 [R(x, y), z]. 2 2
Let Ω 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-webs W = 1
2
Q × Q. Then Ω = Ω(u = e, v = e) and for ∇i Ω(u = e, v = e),∇i Ω(u = e, v = e) the formulae obtained in (2.3) hold. 1
Consider the computation of ∇i Ω(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 x × y = x × (u\y). u
As a result, ∇ is transported from Tu Q in Te Q by means of the inverse transforfu and the smooth mation Ru , which coincide with the structure of the tensor Ω local loop < Q, · , e > with the operation: u
x · y = u\((u × x) × y). u
So that
(3.3)
fu ∂Ω |u=e ∂ui in addition the law of composition (3.3) allow an intuitive algebraic interpretation in terms of the enveloping Lie group G. fu where H fu = u · H · u−1 , Consider the section Q′u = Q·u−1 of the coset space G/H u ∈ Q and the map: Ψu : Q −→ Q′u 1
∇i Ω(u = e, v = e) =
x 7−→ (u × x) × u−1 . 12
Denote by (∗) the law of composition in Q′u , so that: u
a∗b= u
′ Y (ab) u
where Q′
fu . The following : G −→ Q′u is the projection on Q′u parallel to H proposition hold. Proposition 3.2 The map Ψu : Q −→ Q′u is an isomorphism of the smooth loops < Q, · , e > and < Q′u , ∗, e > u u Proof: Let a = Ψu x, b = Ψu y and a ∗ b = Ψu z u where x, y, z ∈ Q. Then u
′ ′ Y Y a∗b= (ab) = ((u × x) · u−1 · (u × y) · u−1 ) u
u
u
(a ∗ b) × u · h · u−1 = (u × x)u−1 · (u × y) · u−1 . u Multiplying by u obtain:
(a ∗ b) × u · h = (u × x) × y. u
Applying the projection to the last equality, we obtain (a ∗ b) × u = (u × x) × y. u
Furthermore (a ∗ b) × u = (Ψu z) × u = (u × z) · u−1 × u = (u × x) × y. u
Then z = u\(u × x) × y and (a ∗ b) = (Ψu x) ∗ (Ψu y) = Ψu z = Ψu {u\(u × x) × y} = Ψu (x · y). u
u
u
Therefore Ψu (x · y) = (Ψu x) ∗ (Ψu y). u Hence the result. Similarly we establish that: f fv ∂Ω ∇i Ω(u = e, v = e) = |v=e ∂v i 2
13
e e correspond to the structure tensor of the local loop < Q, 1 , e > with the where Ω v composition law: 1 x y = (x × (y × v))/v. v
(3.4)
The law of composition (3.4) allows us to find an algebraic interpretation in terms of the enveloping Lie group G. Let us introduce in consideration the subgroup Hv′′ = vHv −1 where v ∈ Q. The following proposition holds: Proposition 3.3 ′′
Y 1 x y= (xy) v v
for all x, y ∈ Q where Q′′ ′′ v : G −→ Q is the projection on Q parallel to Hv . Proof: In the Lie group G we have xyQ= (x ⊥ y) × vhv −1 which is equivalent to xy · v = (x ⊥ y) × vh. Applying to the last formula we get x × (y × v) = (x ⊥ y) × v.
Therefore x ⊥ y = x × (y × v)/v.
4
2
1
Application:Computation of ∇l aijk and ∇l aijk 2
I: Computation of ∇l aijk For u ∈ Q, introduce the map Adu : G −→ G x 7−→ uxu−1 . Let u = exp ζ, where ζ ∈ Q and g ∈ H. Then 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 .
14
(4.1)
Let
Q′′
u
: G −→ Te Q be the projection on Te Q parallel to h′′u and exp h′′u = Hu′′ .
By fixing ξ, η from G, we find that Y [ξ, η] = [ξ, η] + h1 [ξ, η] =
′′ Y [ξ, η] + h2
(4.2)
(4.3)
u
where h1 ∈ h and h2 ∈ h′′u . From (4.1) we obtain that h2 has the form ˆ h2 = h1 + ˆ h(ζ) + [ζ, h1 ] + o(ζ), where h(ζ) ∈ h′′u . From (4.2) and (4.3) it follows that: ′′ Y Y ˆ [ξ, η] − h(ζ) − [ζ, h1 ] + o(ζ) [ξ, η] = [ξ, η] − h2 = u
=
Y Y [ξ, η] − [ζ, h1 ] + o(ζ).
But from (4.2),we have h1 = [ξ, η] −
Q [ξ, η]. It follows that
′′ Y Y Y Y Y [ξ, η] = [ξ, η] − [ζ, [ξ, η]] + [ζ, [ξ, η]] + o(ζ) u
Y Y YY [ξ, η] + [[ξ, η], ζ] − [ [ξ, η], ζ] + o(ζ). Q Denote by a′′u (ξ, η) = − 21 ′′u [ξ, η]. Then =
a′′u (ξ, η) = a(ξ, η) −
Finally we have:
1YY 1Y [[ξ, η]] + [ [ξ, η], ζ]. 2 2
d ′′ 1Y 1YY aexp tζ (ξ, η) |t=0 = − [[ξ, η]] + [ [ξ, η], ζ] dt 2 2 We obtain a result in conformity with proposition 2.1 and the relation (3.2) in deed, from the relation (3.2) 2
∇l aijk ξ j η k ζ l =
Y 1YY 1 [ [ξ, η], ζ] − 2 [R(ξ, η), ζ]. b(ξ, η, ζ) = − [[ξ, η], ζ] + 2 2
From which we find
1Y 1YY 1 [b(ξ, η, ζ) − b(η, ξ, ζ)] = − [[ξ, η], ζ] + [ [ξ, η], ζ] 2 2 2 15
2
so that ∇l aijk = bi[jk]l . II:
1
Computation of ∇l aijk
Let us introduce the map: Ψu : Q −→ Q′u x 7−→ (u × x)u−1 . Then dΨu |e : Te Q −→ Te Q′u . Then the following proposition holds: Proposition 4.1 The map define from the tangent space Te Q to tangent space Te Q′u is defined as follows: dΨu |e : Te Q −→ Te Q′u 1 1Y ξ 7−→ ξ + [u, ξ] + [u, ξ] + 2R(u, ξ) + o(u). 2 2 Proof. For the proof of this proposition, using the notion from section 2 and the relation (1.3) we have u × ξ = (u · ξ) · h but from the proposition 1.4 1 1 h(u, ξ) = − [u, ξ] + [u, ξ] + 2R(u, ξ) + o(u) 2 2 Thus u × ξ = (u · ξ) · h = u + ξ + and (u × ξ) × u−1 = u + ξ + =ξ+
1Y 1Y [u, ξ] + [u, ξ] + 2R(u, ξ) + o(u) 2 2
1Y 1 [u, ξ] + 2R(u, ξ) − u − [ξ, u] + o(u) 2 2
1 1Y [u, ξ] + [u, ξ] + 2R(u, ξ) + o(u) 2 2
Q ′ ′ f′ f′ Let g u : G −→ Te Q be the projection on Te Q parallel to hu where exp hu = uHu−1 . Then we obtain the equation ω + h1 = ω ′ + h′1 + [u, h′1 ] with ω ∈ Te Q, h1 ∈ h, ω ′ ∈ Te Q′ , h′1 ∈ h. For the computation of ω ′ = ω ′ (u, ω). From the proposition (4.1) we have: ω + h1 = ω e+
1Y 1 e ] + 2R(u, ω e ) + h′1 + [u, h′1 ] + o(u) [u, ω e ] + [u, ω 2 2 16
where ω e ∈ Te Q, so that: ω e+
It follows that:
1Y 1 e ] + 2R(u, ω e ) = ω′ [u, ω e ] + [u, ω 2 2 ω=ω e+
Y [u, ω e ] + [u, h′1 ]
h1 = h′1 + terms with u
from which ω e =ω−
Y [u, ω] − [u, h′1 ]
h′1 = h1 + term with u
Then substituting in ω ′ the expression from ω e we obtain that:
Y Y 1 1Y [u, h1 ] + [u, ω] + 2R(u, ω) + o(u) [u, ω] − [u, h1 ] + 2 2 Y 1Y 1 [u, ω] − [u, h1 ] + 2R(u, ω) + o(u) = ω + [u, ω] − 2 2 from which we find that ω′ = ω −
Y Y 1 1 f (ω + h1 ) = ω ′ = ω + [u, ω] − [u, ω] + 2R(u, ω) − [u, h1 ]. 2 2 u
(4.4)
Now let us compute Q a fu (ξ, η) = − 21 (dΨ)−1 g u [dΨξ , dΨη ] where ξ, η ∈ Te Q " Q Q Q 1 1 −1 g [u, ξ] + 2R(u, ξ), η + (dΨ)−1 g u [dΨξ , dΨη ] = (dΨ) u ξ + 2 [u, ξ] + 2 # 1 1 Q [u, η] + 2R(u, η) 2 [u, η] + 2 ( Q Q −1 g = (dΨ) [ξ, η] + 1 [ξ, [u, η]] + 1 [ξ, [u, η]] + 2[ξ, R(u, η)] − 1 [η, [u, ξ]] − u
2
2
)
2
Q [u, ξ]] − 2[η, R(u, ξ)] = ( Q Q Q Q Q Q −1 = (dΨ) [ξ, η]+ 21 [ξ, [u, η]]+ 21 [ξ, [u, η]]+2 [ξ, R(u, η)]− 12 [η, [u, ξ]]− Q Q Q Q 1 1 Q [η, [u, ξ]] − 2 [η, R(u, ξ)] + 21 [u, [ξ, η]] 2 ) − 2 [u, [ξ, η]] + Q Q Q Q + 2R(u, [ξ, η]) − [u, [ξ, η]] + [u, [ξ, η]] = − 21 [η,
17
Q Q Q Q =Q [ξ, η] + 21 [ξ, [u, η]] + 12 [ξ, Q [u, η]] + 2 [ξ, R(u, η)] − Q Q − 21 [η, [u, ξ]] − 2 [η, R(u, ξ)] − [u, [ξ, η]] =
Q Q Q Q =Q [ξ, η] + 21 [ξ, [η,Qu]] − 12 [ξ, [η, u]] + 2 [ξ, R(u, η)] − Q + 21 [η, [ξ, u]] − 2 [η, R(u, ξ)]
1 2
1 2
Q [η, [u, ξ]] −
Q [η, [ξ, u]] +
where
af u (ξ, η) = −
Y 1Y 1YY 1Y [ξ, η] − [[ξ, u], η] + [ [ξ, u], η] − [R(u, ξ), η]+ 2 4 4
+
Y 1Y 1YY [[η, u], ξ] − [ [η, u], ξ] + [R(u, η), ξ]. 4 4
From this last equation it follows that: 1
∇l aijk ξ j η k ζ l =
Y 1Y 1YY d a^ [[ξ, ζ], η]+ [ [ξ, ζ], η]− [R(ξ, ζ), η]+ exp tζ (ξ, η)|t=0 = − dt 4 4
+
Y 1Y 1YY [[η, ζ], ξ] − [ [η, ζ], ξ] + [R(η, ζ), ξ]. 4 4
We obtain a result in conformity with proposition 2.1 and the relation (3.2) in deed from the formulae (3.2) ( it follows: Q QQ Q 1 1 1 [[ξ, ζ], η]+ 12 [ [ξ, ζ], η]−2 [R(ξ, ζ), η]+ 2 [b(ξ, ζ, η)−b(η, ζ, ξ)] = 2 − 2 ) Q 1 Q Q 1 Q + 2 [[η, ζ], ξ] − 2 [ [η, ζ], ξ] + 2 [R(η, ζ), ξ] = Q QQ Q Q = − 41 [[ξ, ζ], η] + 41 [ [ξ, ζ], η] − [R(ξ, ζ), η] + 41 [[η, ζ], ξ]− −
Y 1YY [ [η, ζ], ξ] + [R(η, ζ), ξ]. 4
Therefore 1
∇l aijk = bi[j|j|k] 2
5
Computation of the tensor dijklm =∇m bijkl
Denote u · R(η, η) · u−1 by Ru′′ (η, η). For the computation of dijklm let us firstly compute Ru′′ (η, η). 18
The following proposition holds: Proposition 5.1 Ru′′ (η, η) = R(η, η) +
Y [u, R(η, η)] + 0(u, η 2 )
(5.1).
The proof of this proposition, is from section 1 It is clear that ξ + φ(ξ) ∈ Q and from section 4 h′′u = h1 + [u, h1 ] + 0(u) where h1 ∈ h. Furthermore η + Ru′′ (η, η) ∈ Q but Ru′′ (η, η) ∈ h′′u 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 :
=
(
η + Ru′′ (η, η) =
) Y Y ′′ ′′ ′′ ′′ ′′ (η+ [u, Ru (η, η)])+(Ru (η, η)−[u, Ru (η, η)])+([u, Ru (η, η)]− [u, Ru (η, η)])
Q put η + [u, Ru′′ (η, η)] = ξ then Q φ(ξ) = Ru′′ (η, η) − [u, Ru′′ (η, η)] + [u, Ru′′ (η, η)] − [u, Ru′′ (η, η)] = Ru′′ (η, η) − Q [u, Ru′′ (η, η)] + o(u) from the relation (1.1) 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 relation, we obtain: Y Ru′′ (η, η) = R(η, η) + [u, R(η, η)] + 0(u, η 2 ). Q′′ Let u : G −→ V = Te Q be the projection of G to V parallel to h′′u . Then we obtain the equation ξ+e h = ξe + h1 + [u, h1 ] e u) we have where ξ, ξe ∈ V and e h, h1 ∈ h for the search of ξe = ξ(ξ, Y Y ξ +e h = ξe + h1 + [u, h1 ] + ([u, h1 ] − [u, h1 ])
where
Y ξ = ξe + [u, h1 ] Y e h = h1 + [u, h1 ] − [u, h1 ] = h1 + terms with u.
From these two equalities we obtain
Hence
ξe = ξ −
Y [u, e h] + 0(u).
′′ Y Y (ξ + e h) = ξ − [u, e h]). u
19
(5.2)
We pass now to the computation of dijklm . From (3.2) its follows that b(ξ, η, ζ) = − that is why
Y 1Y 1YY [[ξ, η], ζ] + [ [ξ, η], ζ] − 2 [R(ξ, η), ζ] 2 2 ′′
b′′u (ξ, η, ζ) = −
′′
′′
Y 1YY 1Y [[ξ, η], ζ] + [ [ξ, η], ζ] − 2 [Ru′′ (ξ, η), ζ]. 2 u 2 u u u
From (5.2) it follows that
′′
−
1Y 1Y Y 1Y 1Y [[ξ, η], ζ] + [u, [[ξ, η], ζ]] − [u, [[ξ, η, ], ζ]]. [[ξ, η], ζ] = − 2 u 2 2 2 (5.3) Further more
′′
′′
′′
′′
′′
1YY 1YY 1YY Y 1YY [ [ξ, η], ζ] = [ [ξ, η], ζ]− [ [u, [ξ, η]], ζ]+ [ [u, [ξ, η]], ζ] = 2 u u 2 u 2 u 2 u =
Y 1YY 1Y [ [ξ, η], ζ] − [u, [ [ξ, η], ζ]]+ 2 2 +
1Y YY [u, [ [ξ, η], ζ]] + o(u). 2
(5.4)
Finally from (5.1) and (5.2) it follows that:
−2
′′ ′′ Y ′′ Y Y Y [Ru′′ (ξ, η), ζ] = −2 [Ru (ξ, η), ζ] − 2 [ [u, R(ξ, η), ζ] u
u
u
Y Y Y Y = −2 [R(ξ, η), ζ] + 2 [u, [R(ξ, η), ζ]] − 2 [u, [R(ξ, η), ζ]]− YY −2 [ [u, R(ξ, η), zeta] + o(4). (5.5) from (5.3), (5.4) and (5.5) it follows
d(ξ, η, ζ, τ ) =
2 ∇m bijkl |(e,e) ξ j η k ζ l τ m
20
! d = b′′ (ξ, η, ζ) |t=0 = dt exp tτ
=
Y 1Y 1Y Y 1Y 1Y YY [τ, [[ξ, η], ζ]]− [τ, [[ξ, η], ζ]]− [τ, [ [ξ, η], ζ]]+ [τ, [ [ξ, η], ζ]]− 2 2 2 2 −
Y 1YY 1YY Y [ [τ, [ξ, η]], ζ] + [ [τ, [ξ, η]], ζ] + 2 [τ, [R(ξ, η), ζ]]− 2 2 −2
Y Y YY [τ, [R(ξ, η), ζ]] − 2 [ [τ, R(ξ, η)], ζ].
(5.6)
In the theory of 3-Webs [1, 20, 22] the following relation is known: dijk[lm] = −bijkp aplm . Let us verify it: 1Y 1Y 1Y Y 1 (d(ξ, η, ζ, τ )−d(ξ, η, τ, ζ)) = [τ, [[ξ, η], ζ]]− [ζ, [[ξ, η], τ ]]− [τ, [[ξ, η], ζ]]+ 2 4 4 4 +
Y 1Y Y 1Y Y 1Y 1Y YY [ζ, [[ξ, η], τ ]]− [τ, [ [ξ, η], ζ]]+ [ζ, [ [ξ, η], τ ]]+ [τ, [ [ξ, η], ζ]]− 4 4 4 4 −
1Y YY 1YY 1YY [ζ, [ [ξ, η], τ ]] − [ [τ, [ξ, η]], ζ] + [ [ζ, [ξ, η]], τ ]+ 4 4 4
+
Y Y 1YY Y 1YY Y [ [τ, [ξ, τ ]], ζ]− [ [ζ, [ξ, η]], τ ]+ [τ, [R(ξ, η), ζ]]− [ζ, [R(ξ, η), τ ]]− 4 4
−
Y Y Y Y YY YY [τ, [R(ξ, η), ζ]]+ [ζ, [R(ξ, η), τ ]]− [ [τ, R(ξ, η), ]ζ]+ [ [ζ, R(ξ, η)], τ ] = Y 1Y 1YY [[ξ, η], [ζ, τ ]] + [ [ξ, η], [ζ, τ ]] − [R(ξ, η), [ζ, τ ]]. 4 4 In addition, considering that Y Y [ζ, τ ] = [ζ, τ ] + ([ζ, τ ] − [ζ, τ ]). =−
One obtain
Y Y 1Y 1YY 1 (d(ξ, η, ζ, τ ) − d(ξ, η, τ, ζ)) = − [[ξ, η], [ζ, τ ]] + [ [ξ, η], [ζ, τ ]]− 2 4 4 Y Y − [R(ξ, η), [ζ, τ ]] 21
From relations (3.1) and (3.2) it follows that:
b(ξ, η, a(ζ, τ )) =
Y Y Y 1 1Y 1YY b(ξ, η, [ζ, τ ]) = − [[ξ, η], [ζ, τ ]]+ [ [ξ, η], [ζ, τ ]]− 2 4 4 Y Y − [R(ξ, η), [ζ, τ ]].
Hence dijk[lm] = −bijkp aplm
6
Hexagonal loops
The analytic hexagonal 3-Webs and their corresponding loops can be charaterise by the following condition: bi(jkl) = 0 Q QQ Q where b(ξ, η, ζ) = − 12 [[ξ, η], ζ] + 21 [ [ξ, η], ζ] − 2 [R(ξ, η), ζ] i that is way, b(jkl) = 0 is equivalent to the following condition Y Y Y [R(ξ, η), ζ] + [R(η, ζ), ξ] + [R(ζ, ξ), η] = 0 (6.1) which can be written as follows: Y [R(ξ, η), ζ] = 0 σ ξηζ
where σ is the cyclic sum for ξ, η, ζ ξηζ
We have furthermore, for the hexagonal three-Webs the following relation di(jkl)m = 0. Considering (5.6) and (6.1) one obtain ( ) Y Y [, τ, [R(ξ, η), ζ]] − [ [τ, R(ξ, η)], ζ] = 0. σ ξηζ
where σ is the cyclic sum for ξ, η, ζ ξηζ
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