Ssang Yong
10.7. Modelo de Las Cinco Fuerzas de Porter 1 Rivalidad Entre Empresas Competidoras.
The goal of this section is to present Thompson’s Theorem.
8.4 Theorem [Tho86, Thm 4.7] On a simply connected space form, the space of Killing tensors attains its maximal dimension in each degree and is generated by Killing one-forms, so every Killing tensor is completely decomposable. 8.4 ◂
To prove this, Thompson introduced special coordinates on the space formM and showed, that in the flat case, the coefficients of every Killing tensor must be poly- nomial functions in the introduced coordinates. Using this result he presented a method to recursively deconstruct a Killing tensor, that is a homogeneous polyno- mial in the coordinates, into a sum of products of Killing one-forms. Afterwards he presents a comparison argument to conclude the same result for the other cases. The proof given in [Tak83, §4] also employs a local chart. A different proof for the standard sphere was given by Delong in [Del82]. The author proved, that Killing tensors onSnare those that are Killing on Rn+1and are in involution with two certain elements. He explicitly calculates all such Killing vectors and shows that their products produce the maximal possible amount of linear independent generators. He also found the decomposition of the space of Killingp-tensors into its SO(n)-irreducible components. [Del82, Proposition in §4.3]
The proof presented here has some similarities to Delong’s approach, but takes a different route and is motivated by the article of McLenaghan, Milson, and Smirnov, which shows that the space of Killing tensors on a space formM is an irreducible SL(n +1) representation on a cone ˆM over M [MMS04]. Although the next theorem was already stated in that paper its proof relied on Thompson’s result about the total decomposability of higher-order Killing tensors. The prove given here uses
the formalism developed in the last chapter. Thompson’s theorem follows then as a corollary to this result.
8.5 Theorem ([MMS04, Theorem 3.5]) On a simply connected space form(M,д) of dimension n and constant curvature κ ≠ 0, the symmetric algebra of Killing tensorsK(M) is isomorphic to the algebra ⊕∞
p=0S{p,p}Rn+1generated by the Cartan product [FH91, p. 428] [Eas05b] of parallel two-forms on the manifold ˆM = R+× M
with metric ˆд = κdr ⊗ dr + r2π∗д. 8.5 ◂
LetMn,s,κbe a space form of dimensionn signature n − 2s for 0 ≤ s ≤ n and constant
curvatureκ ∈ {±1}. Consider the model Mn,s,κ ⊂ Rn+1, where the ambient space Rn+1carries the canonical metric of signaturen +1−2s −κ and Mn,s,κ is a connected component of the set{x ∈ Rn+1∣ ⟨x , x⟩ = κ}. [Wol11, Thm 2.4.4] Fixn,s and κ and write, until otherwise stated,Mn,s,κ= M. Let
ˆ
Mn,s,κ= ˆM = R
+× M, ˆд = κdr ⊗ dr + r 2π∗д
be the cone overM with projection π∶ ˆM → M on the second component. ˆM is isomorphic to an open subset of Rn+1equipped with the canonical flat metric of signaturen + 1 − 2s − κ. In this case, Theorem 7.17provides an isomorphism of vector spaces between the Killing tensors onM and certain parallel sections on
ˆ
M.Theorem 8.5is proved by showing that this isomorphism is actually an algebra isomorphism.
Product of Two-Forms Since the symmetric product of Killing one-forms is a Killing two-tensor, it is to suspect that there is a corresponding product on the associated parallel two-forms on the cone. Theorem 7.17implies, that it has to take values in S{2,2}Tˆ∗, the symmetrized algebraic curvature tensors on ˆT= T ⊕R, the
tangent space at a point in ˆM.
8.6 Lemma For everyp ∈ N the map
⊙∶ S{k,k}Tˆ ⊗ S{p,p}Tˆ → S{p+k,p+k}Tˆ k ⊗ K ↦ proj(p+k,p+k)k ⊗ K
is a surjective SLn+1-equivariant homomorphism. 8.6 ◂
resentation whose highest weight is the sum of the highest weights of both factors, and always occurs with multiplicity one. As a projection on a subrepresentation it
is equivariant and surjective.
8.7 Corollary ([MMS04, Proposition 2.1]) The space S{p,p}Tˆ∗ is generated for
everyp ∈ N by the p-fold ⊙-products of S{1,1}Tˆ∗= ⋀2Tˆ∗. 8.7 ◂
Proof. This is proved by induction by using the above lemma for the induction step. For the beginning of the induction choosep = 1 and k = 1. Then assume the statement to be true forp ∈ N and apply the lemma for p and k = 1 to prove the
statement forp + 1 ∈ N.
8.8 Lemma ([Eas05b, (9)]) The union of spaces S{p,p}Tˆ∗forp ∈ N forms a graded,
commutative, unital algebra. 8.8 ◂
Proof. It needs to be shown, that ⊙ is associative and commutative. The first is always true for the Cartan productU ⊙V ⊙W of irreducible representations U ,V andW . Let λ, µ and ν be their highest weights, then U ⊙V andV ⊙W are respectively the unique subrepresentations ofU ⊗ V and V ⊗ W with highest weights λ + µ and µ + ν. Thus (U ⊙V ) ⊙ W and U ⊙ (V ⊙W ) have highest weight λ + µ + ν and are isomorphic and unique inU ⊗ V ⊗W . That means that, up to rescaling, there is only one non-zero equivariant projection sending elementsu ⊗v ⊗ w to (u ⊙v) ⊙ w = u ⊙ (v ⊙ w). Commutativity of the Cartan product is shown in the same way. Up to rescaling, there is only one projection sendingu ⊗v ∈ U ⊗ U to
u ⊙v = v ⊙ u.
Proof ofTheorem 8.5 As already mentioned above,Theorem 7.17implies an isomorphism of vector spaces between symmetric Killing p-tensors on M and parallel sections of S{p,p}T∗M. The latter are generated by ⊙-products of parallelˆ
sections of⋀2T∗M = Sˆ {1,1}T∗M. This follows fromˆ Lemma 8.8and the fact that ˆM
is flat and simply connected. It is therefore left to show, that under this isomorphism of vector spaces the ⊙-product of parallel two-forms on ˆM is equivalent to the product of Killing one-forms onM.
Letω ∈ ⋀2Tˆ∗≃ S{1,1}Tˆ∗andS ∈ S{p,p}Tˆ∗. ByTheorem 7.17the tensorsω and S are
related by the equations
α = P ⌟ ω and k = Pp⌟ S.
The symmetric productα ⋅ k of the Killing fields α and k is given by a cyclic sum over all slots of the tensorα ⊗ k ∈ Γ ((T∗⊗SpT∗)M)
α ⋅ k = S1,...,p+1(P ⌟ ω) ⊗ (Pp⌟ S) .
Because the product is a Killing tensor onM, it corresponds to a parallel section ˜S
of S{p+1,p+1}T∗M which, byˆ Theorem 7.17, is given by
˜
S = ˆ∇p+1 S
1,...,p+1(P ⌟ ω) ⊗ (P p⌟ S) .
Using ˆ∇P = id, ˆ∇ω = 0, ˆ∇S = 0 and the Leibniz rule, shows that ˜S is some sym- metrization ofω ⊗S. To see that ˜S is actually the Cartan product, one can either use the Young symmetrizer and compute the product directly, or proceed as follows. The map S{1,1}Tˆ∗⊗ S{p,p}Tˆ∗ → S{p+1,p+1}Tˆ∗ which sendsω ⊗ S to ˜S, as defined
above is equivariant, since it is just a symmetrization. It is not zero, because for a non-zeroα, the fields k = αp andα ⋅ k are not vanishing Killing tensors, and therefore correspond to non-vanishing sectionsS and ˜S. By Schur’s Lemma, this map is therefore a multiple of the Cartan product.
The Flat Case and Thompson’s Operator This paragraph recasts Thompson’s proof ofTheorem 8.4in the flat case [Tho86]. As observed inCorollary 7.3, every Killing tensork ∈ Kp(M) is a polynomial of degree p in the coordinate functions of a chart.
The two-forms{ei∧ ej∣ 1 ≤ i < j ≤ n} form a basis of ⋀2T∗≃ so(n), which induce
all non-parallel Killing one-forms{ki,j∣ 1 ≤ i < j ≤ n} on Rn. Thompson defines on
sectionsk of SpT∗M the operator
Lk = ∑
1≤i,j≤nki,j⋅ (ei⌟ ∇ej − ej ⌟ ∇ei)k.
Note, effectively just giving a factor 2, it is important for a later calculation, that the sum runs over all pairs of indicesi and j.
8.9 Lemma Ifk is a Killing tensor, then for every 1 ≤ i, j ≤ n the tensor ei⌟ ∇ejk − ej ⌟ ∇eik
is also Killing. 8.9 ◂
Proof. Since covariant derivatives commute, and the fieldseiare parallel in the flat
case, one computes:
d(ei⌟ ∇ej− ej⌟ ∇eik) = ∑ l el ⋅ ∇ el(ei⌟ ∇ejk − ej⌟ ∇eik) = ∇ej(−∇eik + ei⌟ dk) − ∇ei(−∇ejk + ej⌟ dk) . 8.10 Lemma ([Tho86, (4.17)]) Letk be a Killing p-tensor that is homogeneous of degreer in the coordinates, then
Lk = −2r(p + 1)k.
8.10 ◂
Proof. The proof uses the Euler formulas from the Lemmata2.9and2.10. Note, that in coordinateski,j = xiej − xjei. Therefore,
Lk = ∑ i,j (x iej− xjei) ⋅ (e i⌟ ∇ej − ej⌟ ∇ei)k = ∑ i,j ( xiej ⋅ ei⌟ ∇ejk + xjei⋅ ej⌟ ∇eik − ej⋅ e j⌟ (xi∇eik) − ei⋅ ei⌟ (xj∇ejk)) .
Since,k is homogenous in the coordinates of degree r, one has ∑j(xj∇ejk) = rk.
Hence,− ∑i,jej ⋅ ej ⌟ (xi∇eik) = −2rpk. By that
Lk = −2rpk + 2 ∑
i,j
xiej⋅ (e
i⌟ ∇ejk) .
Using the Leibniz rule to commute(ej⋅) and (ei⌟) gives
2∑ i,j xiej ⋅ (e i⌟ ∇ejk) = −2rk + 2 ∑ i xie i⌟ dk.
Because dk = 0 for a Killing tensor, one finally has Lk = −2r(p + 1)k.
Combining the two results above, every Killing tensor, that is homogeneous in the coordinate variables of orderr > 0, decomposes as
k = −2r(p + 1) ∑1
1≤i<j≤nki,j⋅ (ei⌟ ∇ej− ej⌟ ∇ei)k.
Because the Killing operator dk = ∑iei∇eik is homogeneous in the coordinate
variables, the homogeneous partskj, in the expansion of a Killing tensork = ∑rkr
with 0 ≤ r ≤ p, are all homogeneous. [Tho86, Proposition 4.2] By the equation above, every suchkr splits into Killing tensors of lesser degree forr > 0 while k0
is parallel and is generated by the Killing one-forms{ei∣ 1 ≤ i ≤ n}. Applying this algorithm recursively decomposes every Killing tensork into products of ki,j and ei.