Fusion in compact Lie groups

D. J. Benson Department of Mathematics University of Georgia Athens GA 30602, USA djb@byrd.math.uga.edu

1 Introduction

This paper is intended as a small step in a program to understand how much of the usual machinery of finite group theory extends to compact Lie groups. We prove an analog for compact Lie groups of theorems of Alperin [2] and Goldschmidt [3] on fusion in finite groups. Our main theorem states that fusion in a compact Lie group is local, in the following precise sense.

Theorem 1.1 If A and B are nonempty subsets of a Sylow p-toral subgroup P of a compact Lie group G and B = A^{g} for some g ∈ G, then there exist elements x_{1}, …, x_{n} of G and Sylow p-toral subgroups Q_{1}, …, Q_{n} of G and an element y ∈ N_{G}(P) such that the following conditions hold. We write H_{i} for P ∩ Q_{i}.

(1) A is contained in H_{1}, and A^{x_{1}···x_{i}} is contained in H_{i+1} for 1 ≤ i ≤ n − 1,

(2) g = x_{1} ⋯ x_{n}y,

(3) x_{i} is a p-element of N_{G}(H_{i}) for 1 ≤ i ≤ n,

(4) H_{i} is a tame intersection with respect to P and Q_{i} for 1 ≤ i ≤ n,

(5) for each i with 1 ≤ i ≤ n, either x_{i} ∈ C_{G}(H_{i}) or the following hold:

(i) C_{P} (H_{i}) ≤ H_{i},

(ii) H_{i} is a Sylow p-toral subgroup of O_{p′,p}(N_{G}(H_{i})), and

(iii) either Q_{i} = P or N_{G}(H_{i})/H_{i} is a finite p-isolated group.

The definitions and basic properties of p-elements and Sylow p-toral subgroups of compact Lie groups may be found in Section 3. The definitions of tame intersection and of finite p-isolated groups may be found in Section 5.

There are several modifications needed in the proofs of Alperin and Goldschmidt. The first problem is that the ascending chain condition does not hold for subgroups of a Sylow p-toral subgroup. This means that an induction on the order in the finite case has to be replaced by the fact (proved in Section 3) that a given Sylow p-toral subgroup has only a finite number of distinct intersections with its conjugates. The second problem is that one needs to do something in order to reduce down to the case where N_{G}(H)/H is finite. The relevant property of compact Lie groups needed here is expressed in Lemma 5.2. Intuitively, this says that compact Lie groups can’t have continuous families of outer automorphisms. The third modification is that for a compact Lie group, it is not obvious (but it is true, and proved in Section 4) that any two Sylow p-toral subgroups are conjugate by a product of p-elements.

2 Intersections of tori

We begin by recalling from Adams [1] some basics on compact Lie groups and their Lie algebras. Let G be a compact Lie group with maximal torus T, and let L(G) be the Lie algebra of G; namely the tangent space at the identity element of G. Let exp : L(G) → G be the exponential map, whose image is the identity component G_{0} of G. Then exp : L(T) → T is surjective, with discrete kernel (the “integer lattice”) Λ ≅ ^{r}, and exp identifies T with L(T)/Λ ≅ ^{r}/ ^{r}, where r is the Lie rank of G.

Conjugation in G induces the adjoint action of G on L(G). Restricting to T, L(G) splits as a direct sum

L(G) = L(T) ⊕ ⨁_{i=1}^{s} V_{i}

where s is the number of positive roots in the corresponding root system, and each V_{i} is a two dimensional representation of T given by

t ↦ ( \begin{matrix}cos 2πθ_{i}(t) & − sin 2πθ_{i}(t) \\ sin 2πθ_{i}(t) & cos 2πθ_{i}(t) \end{matrix} ) .

Here, θ_{i} : T → / is induced by a linear form θ̂_{i} : L(T) → with θ̂_{i}(Λ) ⊆ . The θ_{i} are nontrivial and distinct. For each θ_{i} we set U_{i} = {t ∈ T | θ_{i}(t) = 0}, so that U_{i} is a closed subgroup of T of codimension one.

Lemma 2.1 If T and T^{g} are maximal tori in G then (i) L(C_{G}(T ∩ T^{g})) = L(T) ⊕ ⨁_{i∈I} V_{i}, where

I = {i | θ_{i}(T ∩ T^{g}) = 0} = {i | T ∩ T^{g} ≤ U_{i}}.

(ii) T ∩ T^{g} = Z(C_{G}(T ∩ T^{g})_{0})_{0}.

Proof (i) The right hand side is the set of fixed points of T ∩ T^{g} on L(G). Since T ∩T^{g} acts trivially by conjugation on C_{G}(T ∩T^{g}), it acts trivially on L(C_{G}(T ∩T^{g})), and so the left hand side is contained in the right hand side. Conversely, if x is in the right hand side then T ∩ T^{g} acts trivially on x, and hence on the one parameter subgroup H = {exp(λx), λ ∈ }. So H ≤ C_{G}(T ∩ T^{g}) and x ∈ L(C_{G}(T ∩ T^{g})). So the right hand side is contained in the left hand side.

(ii) Since T ≤ C_{G}(T ∩ T^{g})_{0} we have Z(C_{G}(T ∩ T^{g})_{0}) ≤ C_{G}(T), and so

Z(C_{G}(T ∩ T^{g})_{0})_{0} ≤ C_{G}(T)_{0} = T.

Similarly Z(C_{G}(T ∩ T^{g})_{0})_{0} is contained in T^{g}, and hence in T ∩ T^{g}. The reverse inclusion is obvious. □

Proposition 2.2 If T ≠ T^{g} are maximal tori in G then T ∩ T^{g} is equal to the intersection of the subgroups U_{i} which contain it.

Proof Write U for the intersection of the subgroups U_{i} containing T ∩ T^{g}. Then clearly T ∩ T^{g} ≤ U. Conversely, by part (i) of the lemma, U acts trivially on L(C_{G}(T ∩ T^{g})), and hence commutes with the subgroup

expL(C_{G}(T ∩ T^{g})) = C_{G}(T ∩ T^{g})_{0}.

Since U ≤ T ≤ C_{G}(T ∩ T^{g})_{0}, it follows that U is contained in Z(C_{G}(T ∩ T^{g})_{0})_{0}, which is equal to T ∩ T^{g} by part (ii) of the lemma. □

3 Sylow p-toral subgroups

We begin by recalling the analogues of the Sylow theorems for compact Lie groups.

Let T act on the left coset space G/T by left translation. The fixed points are exactly N_{G}(T)/T, and so the Euler characteristic χ(G/T) is equal to χ(N_{G}(T)/T) = |N_{G}(T) : T |. Since the Euler characteristic is multiplicative, it follows that

χ(G/N_{G}(T)) = 1.

A compact Lie group P is said to be p-toral if P_{0} is a torus and P/P_{0} is a p-group. A Sylow p-toral subgroup is defined to be the inverse image in N_{G}(T) of a Sylow psubgroup of N_{G}(T)/T. If P is a Sylow p-toral subgroup then χ(G/P) = |N_{G}(T) : P | is coprime to p.

If P′ is another closed p-toral subgroup then χ((G/P)^{P′}) is coprime to p. It follows that (G/P)^{P′} is non-empty, and so P′ is conjugate to a closed subgroup of P.

If P′ is a proper closed subgroup of P then χ(P/P′) is divisible by p, and hence so is χ(N_{P} (P′)/P′) = χ((P/P′)^{P′}). It follows that P′ is a proper closed subgroup of N_{P} (P′).

Finally, we write O_{p}(G) for the largest normal p-toral closed subgroup of G, which is equal to the inverse image in G of the largest normal finite p-subgroup of G/Z(G_{0}). Similarly, we write O_{p′}(G) for the inverse image in G of the largest normal finite p′-subgroup of G/Z(G_{0}), and O_{p′,p}(G) for the inverse image in G of the largest normal finite p-subgroup of G/O_{p′}(G). It is easy to see that these definitions make sense.

Theorem 3.1 Let P be a Sylow p-toral subgroup of G. Then there are only a finite number of subgroups of P of the form P ∩ P^{g}, as g runs over the elements of G.

Proof By definition, the identity component P_{0} is a maximal torus T of G. By Proposition 2.2, there are only a finite number of possibilities for T ∩ T^{g}. We have T ∩ T^{g} = (P ∩ P^{g})_{0}, and by the second isomorphism theorem

|P ∩ P^{g} : T ∩ T^{g}| = |P ∩ P^{g} : T ∩ P^{g}||T ∩ P^{g} : T ∩ T^{g}|
= |(P ∩ P^{g})T : T ||(T ∩ P^{g})T^{g} : T^{g}| ≤ |P : T |^{2}

It follows that P ∩ P^{g} is contained is the subgroup of N_{P} (T ∩ T^{g}) consisting of those elements whose |P : T |^{2}th power lies in T ∩ T^{g}. This subgroup contains T ∩ T^{g} with finite index, and so there are only a finite number of possibilities. □

The way we shall use this is as follows. We put a lexicographic partial order on the subgroups of P of the form P ∩ P^{g} first by the Lie rank and then by the index of the identity component. There is only a finite number of possible values for this ordered pair of numbers. If one such subgroup strictly contains a conjugate of another then it is strictly greater in the lexicographic order.

4 A strengthened conjugacy theorem

We shall need a strengthened form of the conjugacy statement for Sylow p-toral subgroups. We begin with a lemma.

Lemma 4.1 The compact Lie group SU(2) is generated by elements of order p (or 4 if p = 2).

Proof It suffices to show that SO(3) is generated by elements of order p. If g is an element of order p in SO(3), choose an element x ∈ L(SO(3)) not fixed by g. Look at the one parameter family of elements [g, exp(λx)], λ ∈ . For all values of λ, this is a product of two elements of order p. For small values of λ, we obtain in this way rotations through all small enough angles. Multiplying conjugates of these together therefore gives us all elements of SO(3). □

Lemma 4.2 If G is a perfect connected compact Lie group, then G is generated by elements of order p (or 4 if p = 2).

Proof Any perfect connected compact Lie group G is generated by the images of homomorphisms SU(2) → G, so this follows from the previous lemma. □

Lemma 4.3 If G is a connected compact Lie group, then any two maximal tori are conjugate by a product of elements of order p (or 4 if p = 2).

Proof Since G = [G, G]Z(G) and [G, G] is perfect, this follows from the previous lemma. □

Lemma 4.4 If G is a compact Lie group with the property that G_{0} = T is a torus and G/T ≅ /n, then there is an element g ∈ G of order n whose image in G/T is a generator.

Proof Let y be an arbitrary preimage in G of a generator of G/T. Let u = y^{n} ∈ T. Then u ∈ C_{T} (y), and so u is contained in a one parameter family exp(λx), λ ∈ for some x ∈ L(C_{T} (y)). If u = exp(λ_{0}x) then we take g = y exp((−λ_{0}/n)x). Then g^{n} = y^{n}exp(−λ_{0}x) = y^{n}u^{−1} = 1, as required. □

We say that an element g ∈ G is a p-element if g^{p^{a}} = 1 for some a ≥ 0.

Theorem 4.5 If G is a compact Lie group and P and Q are Sylow p-toral subgroups, then P is conjugate to Q by a product of p-elements.

Proof Let T = P_{0}, so that T is a maximal torus in G. By Lemma 4.3, we may assume without loss of generality that also T = Q_{0}. Then P/T and Q/T are Sylow p-subgroups of N_{G}(T)/T, so they are conjugate in N_{G}(T)/T by a product of pelements. Any p-element in N_{G}(T)/T lifts to a p-element in N_{G}(T) by Lemma 4.4, so we are done. □

5 The fusion theorem

We now prove Theorem 1.1. We begin by recalling the appropriate definitions.

Definition 5.1 If Q and R are Sylow p-toral subgroups of G, the Q ∩ R is said to be a tame intersection if N_{Q}(Q ∩ R) and N_{R}(Q ∩ R) are Sylow p-toral subgroups of N_{G}(Q ∩ R).

This definition is taken to depend on Q and R, not just on Q∩R. Note also that since C_{G}(Q ∩ R) ⊴ N_{G}(Q ∩ R) it follows that for a tame intersection, C_{Q}(Q ∩ R) and C_{R}(Q ∩ R) are Sylow p-toral subgroups of C_{G}(Q ∩ R).

We fix a Sylow p-toral subgroup P of G, and write F for the collection of all pairs (H, x) where H = P ∩ Q is a tame intersection with respect to P and Q, x is a p-element of N_{G}(H), and either x is contained in C_{G}(H) or the following conditions are satisfied:

(i) C_{P} (H) ≤ H,

(ii) H is a Sylow p-toral subgroup of O_{p′,p}N_{G}(H), and

(iii) either Q = P or N_{G}(H)/H is a finite p-isolated group. Recall from Goldschmidt [3] that a finite group is said to be p-isolated if there is more than one equivalence class of Sylow p-subgroups under the equivalence relation generated by the relation of having a nontrivial intersection.

The relevance of the finiteness in condition (iii) is explained by the following lemma.

Lemma 5.2 If H is a closed subgroup of a compact Lie group G then N_{G}(H)_{0} ≤ H_{0}C_{G}(H)_{0}. In particular, if C_{G}(H)_{0} ≤ H then N_{G}(H)_{0} ≤ H, which implies that N_{G}(H)/H is finite.

Proof Since H_{0} is a closed connected normal subgroup of the connected compact Lie group N_{G}(H)_{0}, we may find another closed connected normal subgroup L of N_{G}(H)_{0} with N_{G}(H)_{0} = H_{0}L and [H_{0}, L] = 1. Then L normalizes H and centralizes H_{0}, so it acts as a group of automorphisms on H/H_{0}. Since L is connected, this means it must centralize H, and so N_{G}(H)_{0} ≤ H_{0}C_{G}(H). Since H_{0}C_{G}(H)/H_{0}C_{G}(H)_{0} is finite, the lemma follows. □

Definition 5.3 If Q and R are Sylow p-toral subgroups of G, we write R ∼_{F} Q (via x) if Q = R or there are Sylow p-toral subgroups Q_{1}, …, Q_{n} and elements x_{1}, …, x_{n} of G such that the following conditions hold. We write H_{i} for P ∩ Q_{i}.

(1) x = x_{1} ⋯ x_{n} and R^{x} = Q,

(2) (H_{i}, x_{i}) is in F for 1 ≤ i ≤ n,

(3) P ∩ R ≤ H_{1} and (P ∩ R)^{x_{1}···x_{i}} ≤ H_{i+1} for 1 ≤ i ≤ n − 1.

Theorem 5.4 If Q is a Sylow p-toral subgroup of G then Q ∼_{F} P.

Proof The proof of this is a modification of the proof of Lemma 2.5 of Alperin [2]. Therefore we restrict ourselves to noting the changes that need to be made in Alperin’s proof. Throughout, Sylow p-subgroups should be replaced by Sylow ptoral subgroups, and ∼ should be replaced by ∼_{F}. As remarked in Section 5 of [2], in Lemma 2.1 (transitivity of ∼_{F}) we must divide into two cases; the first is the case where either S = R or R = Q, and the second is where S ≠ R and R ≠ Q. The first of these cases is trivial and the second proceeds as given. Lemma 2.2 carries over verbatim. In the statement and proof of Lemma 2.3, we must replace |P ∩ S| > |P ∩ Q| by the statement that P ∩ S properly contains a G-conjugate of P ∩ Q. Lemma 2.4 is sufficiently different to warrant giving the proof here. This is based on Theorem 3.3 of Goldschmidt [3].

Lemma 5.5 If Q is a Sylow p-toral subgroup of G, P ∩ Q is a tame intersection, and S ∼_{F} P for all Sylow p-toral subgroups S with the property that P ∩ S properly contains a G-conjugate of P ∩ Q, then Q ∼_{F} P.

Proof We first remark that by Theorem 4.5, any two Sylow p-toral subgroups of a compact Lie group are conjugate by a product of p-elements. Let H = P ∩ Q. By transitivity of ∼_{F}, it suffices to show that Q ∼_{F} R for some R such that P ∩ R properly contains a conjugate of H.

Suppose that H does not satisfy condition (i). Since H is a tame intersection of P and Q, C_{P} (H) and C_{Q}(H) are Sylow p-toral subgroups of C_{G}(H). So C_{Q}(H)^{x} = C_{P} (H) for some product of p-elements x ∈ C_{G}(H). Thus P ∩ Q^{x} ≥ HC_{P} (H) > H and Q ∼_{F} Q^{x} (via x). So we may assume from now on that H satisfies condition (i). Since C_{P} (H) is a Sylow p-toral subgroup of C_{G}(H), it follows that C_{G}(H)_{0} ≤ H. By Lemma 5.2, this means that the quotient N_{G}(H)/H is finite.

Suppose that H does not satisfy condition (ii). Then Q ∩ O_{p′,p}N_{G}(H) and P ∩ O_{p′,p}N_{G}(H) are Sylow p-toral subgroups of O_{p′,p}N_{G}(H) properly containing H. They are conjugate by a product of p-elements x ∈ O_{p′,p}(H). In fact, we can take x to be in O_{p′}N_{G}(H) ≤ C_{G}(H) (it is no longer necessarily a product of p-elements there, but this does not matter). Then P ∩ Q^{x} ≥ P ∩ O_{p′,p}N_{G}(H) > H and Q ∼_{F} Q^{x}. So we may assume that H satisfies condition (ii).

Suppose that H does not satisfy condition (iii). Then H is a proper subgroup of P and N_{G}(H)/H is a finite group which is not p-isolated. So there exist Sylow p-toral subgroups R_{1} = N_{P} (H), R_{2}, …, R_{n} = N_{Q}(H) of N_{G}(H) such that R_{i} ∩ R_{i+1} > H for 1 ≤ i ≤ n − 1. Let Q_{1} = P, Q_{2}, …, Q_{n} = Q be Sylow p-toral subgroups of G such that R_{i} ≤ Q_{i} for 1 ≤ i ≤ n. We prove by induction on i that Q_{i} ∼_{F} P for 1 ≤ i ≤ n. This is clear for i = 1. Suppose that Q_{i} ∼_{F} P via x. If P ∩ Q_{i+1} properly contains H then by the given condition on H, Q_{i+1} ∼_{F} P, so we may suppose that P ∩ Q_{i+1} = H, which implies that P ∩ Q_{i} ≥ P ∩ Q_{i+1}. Since Q_{i}^{x} = P, P ∩ Q_{i+1}^{x} = Q_{i}^{x} ∩ Q_{i+1}^{x} is conjugate to Q_{i} ∩ Q_{i+1}, which properly contains H, and so by the given condition on H, Q_{i+1}^{x} ∼_{F} P. By (the modified version of) Lemma 2.2 of [2], Q_{i+1} ∼_{F} P. Thus Q = Q_{n} ∼_{F} P. So we may assume that H satisfies condition (iii).

We have now shown that H satisfies conditions (i), (ii) and (iii), so that H ∈ F. Since H is a tame intersection of P and Q, we have N_{Q}(H)^{x} = N_{P} (H) for some product of p-elements x ∈ N_{G}(H). Thus P ∩ Q^{x} ≥ N_{P} (H) > H and Q ∼_{F} Q^{x} (via x), so we are done. □

Returning to the proof of Theorem 5.4, it remains to modify the proof of Lemma 2.5 of [2]. This may be done by replacing the induction on |P ∩ Q| by the use of Theorem 3.1. □

Finally, the proof of Theorem 1.1 is given by modifying the proof of the main theorem in Section 3 of [2] by replacing Sylow p-subgroups by Sylow p-toral subgroups and ∼ by ∼_{F} throughout.

Acknowledgement I would like to thank Jon Alperin, Richard Lyons and Clarence Wilkerson for conversations relevant to this work.

References

[1] J. F. Adams. Lectures on Lie groups. Chicago, 1969.

[2] J. L. Alperin. Sylow intersections and fusion. J. Algebra 6 (1967), 222–241.

[3] D. Goldschmidt. A conjugation family for finite groups. J. Algebra 16 (1970), 138–142.