SK_{1} AND LIE ALGEBRAS

PETER SCHNEIDER AND OTMAR VENJAKOB

Abstract. We investigate the vanishing of the group SK_{1}(Λ(G)) for the Iwasawa algebra Λ(G) of a pro-p p-adic Lie group G (with p ≠ 2). We reduce this vanishing to a linear algebra problem for Lie algebras over arbitrary rings, which we solve for Chevalley orders in split reductive Lie algebras.

Introduction

Throughout the paper we fix a prime numer p ≠ 2, and we always let G be a pro-p p-adic Lie group with Lie algebra Lie_{ℚ_{p}}(G). Its Iwasawa algebra is defined to be Λ(G) := \underleftarrow{lim}_{N} ℤ_{p}[G/N] where N run over all open normal subgroups of G. Correspondingly we have the larger algebra Λ^{∞}(G) := \underleftarrow{lim}_{N} ℚ_{p}[G/N]. The inclusion Λ(G) ⊆ Λ^{∞}(G) induces a homomorphism of algebraic K-groups, and we define

SK_{1}(Λ(G)) := ker (K_{1}(Λ(G)) ⟶ K_{1}(Λ^{∞}(G))) .

This generalizes the well known definition of SK_{1}(ℤ[G]) for a finite p-group G as, for example, in [Oli]. The problem which we address in this paper is the vanishing of SK_{1}(Λ(G)). In the finite group case this does not seem to have a simple answer beyond abelian groups. In contrast our main result Thm. 7.1 says that any principal congruence subgroup G of a split reductive algebraic group over ℚ_{p} (at least if p ≥ 5) satisfies SK_{1}(Λ(G)) = 0. Such congruence subgroups all have the property of being uniform pro-p-groups. Although we will heavily make use of this fact, we also will see in section 5 that there are uniform groups with nonvanishing SK_{1}.

Our interest in this problem comes from noncommutative Iwasawa theory where G is a Galois group of some (infinite) extension of number fields. Cohomological invariants of arithmetic objects often are related to classes in K_{1}(Λ(G)). On the other hand the group K_{1}(Λ^{∞}(G)) can be identified with the group of Galois equivariant \overline{ℤ}_{p}^{×}-valued functions on the set of isomorphism classes of \overline{ℚ}_{p}-irreducible Artin representations of G (cf. [SV] §3). Important such functions are given by L-values. One of the basic questions of Iwasawa theory is to which extent L-values determine global cohomological invariants. This obviously is related to the injectivity of the above map whose kernel was defined to be SK_{1}(Λ(G)). In particular, the vanishing of SK_{1}(Λ(G)) is a necessary condition for p-adic L-functions to be uniquely defined (cf. [Kak]).

For a general uniform pro-p-group G Lazard ([Laz]) has constructed a corresponding Lie algebra L(G) over ℤ_{p} which is a lattice in the ℚ_{p}-Lie algebra of G in the sense of p-adic Lie groups. Generalizing a homological description of SK_{1} of a finite group by Oliver in [Oli] and using another important result by Lazard, that the cohomology algebra H^{∗}(G, 𝔽_{p}) of any uniform G is the exterior algebra on H^{1}(G, 𝔽_{p}), as well as the commutator calculus in G (cf. [GS]) we translate our problem in Thm. 4.5 to a purely Lie theoretic property of L(G).

Date: July 26, 2011.

1991 Mathematics Subject Classification. 19B28, 17B20.

Key words and phrases. uniform group, Iwasawa algebra, SK_{1}, Lie algebra .

In fact this Lie theoretic property makes sense for any Lie algebra L over any ring R which is finitely generated free as an R-module. It is the question whether the kernel of the Lie bracket viewed as a linear map ⋀^{2} L ⟶ L is generated by indecomposable elements in the kernel. To our knowledge the only situation where this has been studied in the literature is the case of semisimple complex Lie algebras. Here Kostant ([Ko]) showed by representation theoretic methods that the question always has a positive answer. We will in fact show that the answer is positive for any Chevalley order of a split reductive Lie algebra base changed to any ring in which 2 and 3 are invertible. In such a general setting representation theoretic tools do not apply anymore. We use instead the combinatorics of the root subspaces. In this sense our approach is much more elementary than the one by Kostant. We also show that the assumption that the ambient Lie algebra is reductive is not necessary. The answer also is positive for the nilpotent radical of any Borel subalgebra in the above Chevalley orders.

By a different line of reasoning which uses the Lazard isomorphism for continuous group cohomology and the classical Whitehead lemmata about the cohomology of semisimple Lie algebras we show in Thm. 2.1 that SK_{1}(Λ(G)) is finite for any pro-p p-adic Lie group G whose Lie algebra divided the Lie algebra of the center of G is semisimple.

General notation: Let tors M and M[p], for an abelian group M, denote the torsion subgroup and the subgroup of elements killed by p, respectively. If M is an abelian pro-p group then M^{∨} := Hom^{cont}(M, ℚ_{p}/ℤ_{p}) denotes its Pontrjagin dual consisting of all continuous group homomorphisms from M into ℚ_{p}/ℤ_{p}.

1. Preliminaries

In this section we give a description of SK_{1}(Λ(G)) in terms of group cohomology. This is a straightforward consequence of results in [Oli] in the finite group case.

Let H be any finite p-group. We have SK_{1}(ℤ_{p}[H]) = ker (K_{1}(ℤ_{p}[H]) ⟶ K_{1}(ℚ_{p}[H])). According to [Oli] Prop. 8.4 and Thm. 8.6 there is an exact sequence

⊕_{A⊆H}H_{2}(A, ℤ) ⟶ H_{2}(H, ℤ) ⟶ SK_{1}(ℤ_{p}[H]) ⟶ 0

where A runs over all abelian subgroups of H. All three terms in this sequence are finite abelian p-groups. Since ℚ_{p}/ℤ_{p} is an injective abelian group we have, by the universal coefficient theorem,

H^{2}(H, ℚ_{p}/ℤ_{p}) = Hom(H_{2}(H, ℤ), ℚ_{p}/ℤ_{p})

and hence the dual exact sequence

(1)  0 ⟶ SK_{1}(ℤ_{p}[H])^{∨} ⟶ H^{2}(H, ℚ_{p}/ℤ_{p}) ⟶^{res} ∏_{A⊆H} H^{2}(A, ℚ_{p}/ℤ_{p}) .

We now let H run over the factor groups G/N of G by open normal subgroups N and pass to the projective limit in (1). Since

SK_{1}(Λ(G)) = \underleftarrow{lim}_{N} SK_{1}(ℤ_{p}[G/N])

by [SV] Cor. 3.2 we obtain an isomorphism

SK_{1}(Λ(G))^{∨} = ker (H^{2}(G, ℚ_{p}/ℤ_{p}) ⟶ \underrightarrow{lim}_{N} ∏_{A⊆G/N} H^{2}(A, ℚ_{p}/ℤ_{p})) .

Lemma 1.1. We have

ker (H^{2}(G, ℚ_{p}/ℤ_{p}) ⟶ \underrightarrow{lim}_{N} ∏_{A⊆G/N} H^{2}(A, ℚ_{p}/ℤ_{p}))

= ker (H^{2}(G, ℚ_{p}/ℤ_{p}) ⟶^{res} ∏_{A⊆G} H^{2}(A, ℚ_{p}/ℤ_{p}))

where on the right hand side A runs over all closed abelian subgroups of G.

Proof. Let c ∈ H^{2}(G, ℚ_{p}/ℤ_{p}) be any fixed cohomology class. First suppose that c does not lie in the right hand kernel. We then find a closed abelian subgroup A ⊆ G such that c|A ≠ 0. Whenever N ⊆ G is an open normal subgroup such that c = inf(c′) lifts to a class c′ ∈ H^{2}(G/N, ℚ_{p}/ℤ_{p}) we must have c′|(AN/N) ≠ 0. Hence c does not lie in the left hand kernel either.

Vice versa, let us now assume that c does not lie in the left hand kernel. We find an open normal subgroup N_{1} ⊆ G such that c = inf(c_{1}) lifts to some c_{1} ∈ H^{2}(G/N_{1}, ℚ_{p}/ℤ_{p}). We now choose a decreasing sequence N_{1} ⊇ N_{2} ⊇ … of open normal subgroups N_{j} ⊆ G such that ⋂_{j} N_{j} = 1, and we put c_{j} := inf(c_{1}) ∈ H^{2}(G/N_{j}, ℚ_{p}/ℤ_{p}). Our assumption on c guarantees that the set 𝔄_{j} of all abelian subgroups A ⊆ G/N_{j} such that c_{j}|A ≠ 0 is nonempty for any j ≥ 1. In fact, the 𝔄_{j} form a projective system (w.r.t. sending A ⊆ G/N_{j+1} to its image in G/N_{j}) of finite sets. Hence their projective limit is nonempty. For any element (A_{j})_{j} ∈ \underleftarrow{lim} 𝔄_{j} its projective limit A := \underleftarrow{lim} A_{j} is a closed abelian subgroup in G such that c|A ≠ 0. This shows that c does not lie in the right hand kernel. □

Corollary 1.2. We have

SK_{1}(Λ(G))^{∨} = ker (H^{2}(G, ℚ_{p}/ℤ_{p}) ⟶^{res} ∏_{A⊆G} H^{2}(A, ℚ_{p}/ℤ_{p})) .

In order to use this criterion we consider the connecting homomorphism

δ : H^{1}(G, ℚ_{p}/ℤ_{p}) ⟶ H^{2}(G, 𝔽_{p})

in the cohomology sequence for multiplication by p on ℚ_{p}/ℤ_{p}.

Lemma 1.3. Suppose that G is torsion free; then SK_{1}(Λ(G)) vanishes if and only if the sequence

0 ⟶ H^{1}(G, ℚ_{p}/ℤ_{p})/p ⟶^{δ} H^{2}(G, 𝔽_{p}) ⟶^{res} ∏_{A⊆G} H^{2}(A, 𝔽_{p})

is exact.

Proof. Any closed abelian subgroup A ⊆ G is of the form A ≅ ℤ_{p}^{s} for some s ∈ ℕ. It follows that H^{1}(A, ℚ_{p}/ℤ_{p}) ≅ (ℚ_{p}/ℤ_{p})^{s} is divisible and further that H^{2}(A, 𝔽_{p}) = H^{2}(A, ℚ_{p}/ℤ_{p})[p].

We therefore obtain the exact commutative diagram:

\xymatrix{0 \ar[d] \\ H^{1}(G, ℚ_{p}/ℤ_{p})/p \ar[d]_{δ} \\ H^{2}(G, 𝔽_{p}) \ar[d] \ar[r] & ∏_{A} H^{2}(A, 𝔽_{p}) \ar[d]^{≅} \\ H^{2}(G, ℚ_{p}/ℤ_{p})[p] \ar[d] \ar[r] & ∏_{A} H^{2}(A, ℚ_{p}/ℤ_{p})[p] \\ 0}

By Cor. 1.2 the group SK_{1}(Λ(G)) vanishes if and only if the lower horizontal map in this diagram is injective. □

2. Finiteness

We abbreviate 𝔤 := Lie_{ℚ_{p}}(G). Let Z denote the center of G and 𝔷 := Lie_{ℚ_{p}}(Z) its Lie algebra. According to [B-LL] III§9 Ex. 4 we have

𝔷 = {𝔵 ∈ 𝔤 : Ad(g)(𝔵) = 𝔵 for any g ∈ G}.

[B-LL] III§9.3 Prop. 7 then implies that 𝔷 is contained in the center of 𝔤.

Theorem 2.1. Suppose that 𝔷 is the center of 𝔤 and that the Lie algebra 𝔤/𝔷 of G/Z is semisimple; then SK_{1}(Λ(G)) is finite.

Proof. In a first step we assume that 𝔤 is semisimple. Then

H^{1}(𝔤, ℚ_{p}) = H^{2}(𝔤, ℚ_{p}) = 0

by Whitehead’s lemmata (cf. [Wei] 7.8.10 and 7.8.12). The Lazard isomorphism for continuous group cohomology ([Laz] V.2.4.10) says that

H_{c}^{∗}(G′, ℚ_{p}) ≅ H^{∗}(𝔤, ℚ_{p})

for any sufficiently small open subgroup G′ ⊆ G. Furthermore, we have (cf. [NSW] 2.7.12)

dim_{ℚ_{p}} H_{c}^{∗}(G′, ℚ_{p}) = corank_{ℤ_{p}}H^{∗}(G′, ℚ_{p}/ℤ_{p}) .

Altogether it follows that the groups H^{1}(G′, ℚ_{p}/ℤ_{p}) and H^{2}(G′, ℚ_{p}/ℤ_{p}) are finite. A straightforward argument with the Hochschild-Serre spectral sequence then implies that also the groups H^{1}(G, ℚ_{p}/ℤ_{p}) and H^{2}(G, ℚ_{p}/ℤ_{p}) are finite. Under the assumption that 𝔤 is semisimple we therefore have established the stronger assertion that already the middle term in the identity of Cor. 1.2 is finite.

Now let 𝔤 be arbitrary as in the assertion. By our first step we know that the groups H^{1}(G/Z, ℚ_{p}/ℤ_{p}) and H^{2}(G/Z, ℚ_{p}/ℤ_{p}) are finite. The Hochschild-Serre spectral sequence gives rise to an embedding

0 ⟶ \frac{ker(H^{2}(G, ℚ_{p}/ℤ_{p}) → H^{2}(Z, ℚ_{p}/ℤ_{p}))}{im(H^{2}(G/Z, ℚ_{p}/ℤ_{p}) → H^{2}(G, ℚ_{p}/ℤ_{p}))} ⟶ H^{1}(G/Z, H^{1}(Z, ℚ_{p}/ℤ_{p}))

The denominator of the left term is finite. Since Z is a finitely generated ℤ_{p}-module its cohomology H^{1}(Z, ℚ_{p}/ℤ_{p}) is a cofinitely generated abelian group which, moreover, is trivial as a G/Z-module. Hence the right hand term is finite as well. We conclude that M := ker(H^{2}(G, ℚ_{p}/ℤ_{p}) → H^{2}(Z, ℚ_{p}/ℤ_{p})) is finite. Using Cor. 1.2 we now consider the exact diagram:

\xymatrix{ &  & 0 \ar[d] \\  &  & SK_{1}(Λ(G))^{∨} \ar[d] \\ 0 \ar[r] & M \ar[r] & H^{2}(G, ℚ_{p}/ℤ_{p}) \ar[d] \ar[r] & H^{2}(Z, ℚ_{p}/ℤ_{p}) \\  &  & ∏_{A} H^{2}(A, ℚ_{p}/ℤ_{p}) \ar@{-->}[ur]}

Clearly there is the oblique arrow which makes the diagram commutative. The diagram therefore induces a surjection

Hom(M, ℚ_{p}/ℤ_{p}) ⟶ SK_{1}(Λ(G)) ⟶ 0

and shows in this way that SK_{1}(Λ(G)) is finite. □

In the light of Cor. 1.2 the following fact is of some interest. We assume that

G is saturated rationally equi-p-valued

in the sense of [Laz] . In this situation Lazard has constructed in [Laz] IV.3.3.6 a ℤ_{p}-Lie subalgebra L := L(G) ⊆ 𝔤 which is a ℤ_{p}-lattice. Furthermore, he also has shown in [Laz] V.2.5.7.1, that the cohomology algebra

H^{∗}(G, 𝔽_{p}) = ⋀ H^{1}(G, 𝔽_{p})

is the exterior algebra over H^{1}(G, 𝔽_{p}). Both of these results will be of fundamental importance for everything which follows in this paper.

Proposition 2.2. Suppose that G satisfies the above condition and that 𝔤 is semisimple, and put d := dim G = dim_{ℚ_{p}} 𝔤; we then have dim_{𝔽_{p}} H^{2}(G, ℚ_{p}/ℤ_{p})[p] = \binom{d}{2} − d.

Proof. Under our assumption on G the Lazard isomorphism was sharpened in [HKN]Thm. 3.1.1 to an integral isomorphism

H_{c}^{∗}(G, ℤ_{p}) ⊗_{ℤ_{p}} O ≅ H^{∗}(L, ℤ_{p}) ⊗_{ℤ_{p}} O

where O is the ring of integers in an appropriate finite extension K of ℚ_{p}. Using Whitehead’s lemma we in particular obtain

H_{c}^{i}(G, ℤ_{p}) ⊗_{ℤ_{p}} K = H^{i}(L, ℤ_{p}) ⊗_{ℤ_{p}} K = H^{i}(𝔤, K) = 0

for i = 1, 2. It follows that H_{c}^{2}(G, ℤ_{p}) must be finite and that, since H_{c}^{1}(G, ℤ_{p}) by construction is torsion free, we even have H_{c}^{1}(G, ℤ_{p}) = 0. The cohomology sequence for multiplication by p on ℤ_{p} therefore gives

H^{1}(G, 𝔽_{p}) = H_{c}^{2}(G, ℤ_{p})[p] ≅ H_{c}^{2}(G, ℤ_{p})/p

as well as the exact sequence

0 ⟶ H_{c}^{2}(G, ℤ_{p})/p ⟶ H^{2}(G, 𝔽_{p}) ⟶ H_{c}^{3}(G, ℤ_{p})[p] = H^{2}(G, ℚ_{p}/ℤ_{p})[p] ⟶ 0 .

It follows that

dim_{𝔽_{p}} H^{2}(G, ℚ_{p}/ℤ_{p})[p] = dim_{𝔽_{p}} H^{2}(G, 𝔽_{p}) − dim_{𝔽_{p}} H^{1}(G, 𝔽_{p}) .

By our assumption on the group G the p-th powers G^{p} form a normal subgroup such that G/G^{p} is an 𝔽_{p}-vector space of dimension d. Hence dim_{𝔽_{p}} H^{1}(G, 𝔽_{p}) = d. But H^{2}(G, 𝔽_{p}) = ⋀^{2} H^{1}(G, 𝔽_{p}). Hence dim_{𝔽_{p}} H^{2}(G, 𝔽_{p}) = \binom{d}{2}. □

Corollary 2.3. For any open subgroup G ⊆ SL_{2}(ℤ_{p}) which satisfies the above condition we have SK_{1}(Λ(G)) = 0.

Proof. In this case we have d = 3 so that H^{2}(G, ℚ_{p}/ℤ_{p}) = 0 by Prop. 2.2. □

From now on we always will assume that our group G is uniform in the sense of [DDMS]. In Lazard’s language this is equivalent to G being saturated integrally p-valued.

3. The connecting homomorphism

Our goal in this section is to explicitly compute the connecting homomorphism δ in Lemma 1.3 (under our standing assumption that G is uniform, of course).

As recalled before the proof of Prop. 2.2 we have

H^{2}(G, 𝔽_{p}) = ⋀^{2} H^{1}(G, 𝔽_{p}) = ⋀^{2} Hom(G/G^{p}, 𝔽_{p}) = Hom(⋀^{2}(G/G^{p}), 𝔽_{p}) .

we also have noted already that in a uniform group the subset G^{p} := {g^{p} : g ∈ G} in fact is an open normal subgroup. The quotient G/G^{p}, of course, is an 𝔽_{p}-vector space. For simpler reasons (cf. [Ser] p. 117) the corresponding fact

H^{2}(A, 𝔽_{p}) = ⋀^{2} H^{1}(A, 𝔽_{p}) = ⋀^{2} Hom(A/A^{p}, 𝔽_{p}) = Hom(⋀^{2}(A/A^{p}), 𝔽_{p})

holds true for any A ≅ ℤ_{p}^{s}. In the following we write V := G/G^{p} and we introduce in ⋀^{2} V the 𝔽_{p}-vector subspace ⋀ V generated by all elements of the form aG^{p} ∧ bG^{p} where a and b are contained in the same closed abelian subgroup of G (so that [a, b] = 1). Then

ker (H^{2}(G, 𝔽_{p}) ⟶ ∏_{A} H^{2}(A, 𝔽_{p})) = Hom(⋀^{2} V / ⋀ V, 𝔽_{p}) .

The connecting homomorphism δ, by Pontrjagin duality, gives rise to a homomorphism

δ^{∨} : ⋀^{2} V ⟶ G/[G, G] =: G^{ab} .

We obtain the complex

(2)  0 ⟶ ⋀ V ⟶^{⊆} ⋀^{2} V ⟶^{δ^{∨}} G^{ab}[p] ⟶ 0 ,

which is exact except possibly in the middle, together with the following reformulation of Lemma 1.3.

Lemma 3.1. We have SK_{1}(Λ(G)) = 0 if and only if the complex (2) is exact.

On the other hand it is easy to see that

∂  : V ∧ V ⟶ (G^{p}/[G^{p}, G])[p]
gG^{p} ∧ hG^{p} ⟼ [g, h] mod [G^{p}, G]

is a well defined 𝔽_{p}-linear map. In oder to compare these two maps we claim that

φ :  G^{ab} ⟶ G^{p}/[G^{p}, G]
g[G, G] ⟼ g^{p}[G^{p}, G]

is well defined and multiplicative. This will be a consequence of the Hall-Petrescu formula (cf. [B-LL] II §5 Ex. 9). First of all, 9.d) implies that

[g, h]^{p} ≡ [g^{p}, h] ≡ 1 mod [G^{p}, G]

for any g, h ∈ G. Using 9.a) we then conclude that

(gh)^{p} ≡ g^{p}h^{p}(h^{−1}[h^{−1}, g^{−1}]h)^{\binom{p}{2}} mod [G^{p}, G]
≡ g^{p}h^{p}(h^{−1}[h^{−1}, g^{−1}]^{p}h)^{\frac{p−1}{2}} mod [G^{p}, G]
≡ g^{p}h^{p} mod [G^{p}, G]

for any g, h ∈ G. In particular, if h = [h_{1}, h_{2}] is any commutator, then

(gh)^{p} ≡ g^{p}[h_{1}, h_{2}]^{p} ≡ g^{p} mod [G^{p}, G] .

Lemma 3.2. φ is an isomorphism.

Proof. The surjectivity is obvious. For the injectivity we first of all note that according to [GS] Thm. 3.4 our group G is a torsion free PF-group. By loc. cit. Prop. 2.1(3)-(5) the commutator subgroup [G, G] therefore is a PF-embedded subgroup which satisfies [G, G]^{p} = [G^{p}, G] = [G^{p}, G]. Let now g[G, G] be any element in the kernel of φ. This means that g ∈ [G^{p}, G] = [G, G]^{p}. Then loc. cit. Prop. 2.2 implies that g ∈ [G, G]. □

Proposition 3.3. φ ∘ δ^{∨} = ∂.

Proof. We will in fact establish the dual identity δ ∘ φ^{∨} = ∂^{∨}.

Step 1: We compute the connecting homomorphism δ : Hom^{cont}(G, ℚ_{p}/ℤ_{p}) ⟶ H^{2}(G, 𝔽_{p}), where here and in the following the superscript “cont” indicates the subgroup of continuous homomorphisms. Let f ∈ Hom^{cont}(G, ℚ_{p}/ℤ_{p}) and let f̃ : G ⟶ ℚ_{p}/ℤ_{p} be any continuous map (of sets) such that pf̃ = f. Then δ(f) = [df̃] is the class of the 2-cocycle df̃(g, h) := −f̃(gh) + f̃(g) + f̃(h) on G. In order to make a suitable choice for f̃ we fix an ordered basis (g_{1}, …, g_{d}) of G. For any p-adic integer c ∈ ℤ_{p} we let 0 ≤ c̄ ≤ p − 1 denote the unique integer such that c ≡ c̄ mod p. If g = g_{1}^{c_{1}} … g_{d}^{c_{d}} ∈ G is any element we find a unique element π(g) ∈ G such that

g = gc̄_{1} ; 1 … gc̄_{d} ; dπ(g)^{p}

(cf. [DDMS] §4.2). We fix elements f_{1}, …, f_{d} ∈ ℚ_{p}/ℤ_{p} such that pf_{i} = f(g_{i}), and we choose

f̃(g) := ∑_{i=1}^{d} c̄_{i}f_{i} + f(π(g)) .

8 PETER SCHNEIDER AND OTMAR VENJAKOB

Let now g = g_{1}^{a_{1}} … g_{d}^{a_{d}} and h = g_{1}^{b_{1}} … g_{d}^{b_{d}} be any two elements in G. Then

f̃(g) = ∑_{i=1}^{d} ā_{i}f_{i} + f(π(g)) and f̃(h) = ∑_{i=1}^{d} b̄_{i}f_{i} + f(π(h)) .

On the other hand setting

p_{i} := \begin{cases}p & if ā_{i} + b̄_{i} ≥ p, \\ 0 & otherwise \end{cases}

we have

gh = gā_{1} ; 1 … gā_{d} ; dπ(g)^{p}gb̄_{1} ; 1 … gb̄_{d} ; dπ(h)^{p}

≡ g_{1}^{ā_{1}+b̄_{1}} … g_{d}^{ā_{d}+b̄_{d}}π(g)^{p}π(h)^{p} ∏_{i>j}[g_{i}^{−ā_{i}}, g_{j}^{−b̄_{j}}] mod [G^{p}, G]

≡ g_{1}^{ā_{1}+b̄_{1}−p_{1}} … g_{d}^{ā_{d}+b̄_{d}−p_{d}}π(g)^{p}π(h)^{p}( ∏_{ā_{j}+b̄_{j}≥p} g_{j}^{p}) ∏_{i>j}[g_{i}, g_{j}]^{ā_{i}b̄_{j}} mod [G^{p}, G] .

Since [G^{p}, G] = [G, G]^{p} this means that

gh = g_{1}^{ā_{1}+b̄_{1}−p_{1}} … g_{d}^{ā_{d}+b̄_{d}−p_{d}}π(g)^{p}π(h)^{p}( ∏_{ā_{j}+b̄_{j}≥p} g_{j}^{p}) ∏_{i>j}[g_{i}, g_{j}]^{ā_{i}b̄_{j}}γ_{1}^{p}

for some γ_{1} ∈ [G, G]. The element π(g)^{p}π(h)^{p}(∏_{ā_{j}+b̄_{j}≥p} g_{j}^{p}) ∏_{i>j}[g_{i}, g_{j}]^{ā_{i}b̄_{j}}γ_{1}^{p} lies in G^{p} and hence is equal to y^{p} for some y ∈ G. Since then

φ(π(g)π(h)( ∏_{ā_{j}+b̄_{j}≥p} g_{j}) ∏_{i>j}([g_{i}, g_{j}]^{\frac{1}{p}})^{ā_{i}b̄_{j}}γ_{1}) = φ(y)

Lemma 3.2 implies that

π(g)π(h)( ∏_{ā_{j}+b̄_{j}≥p} g_{j}) ∏_{i>j}([g_{i}, g_{j}]^{\frac{1}{p}})^{ā_{i}b̄_{j}}γ_{1} = yγ_{2}

for some γ_{2} ∈ [G, G]. We obtain

π(g)^{p}π(h)^{p}( ∏_{ā_{j}+b̄_{j}≥p} g_{j}^{p}) ∏_{i>j}[g_{i}, g_{j}]^{ā_{i}b̄_{j}}γ_{1}^{p} = (π(g)π(h)( ∏_{ā_{j}+b̄_{j}≥p} g_{j}) ∏_{i>j}([g_{i}, g_{j}]^{\frac{1}{p}})^{ā_{i}b̄_{j}}γ_{3})^{p}

with γ_{3} := γ_{1}γ_{2}^{−1} ∈ [G, G] and hence

gh = g_{1}^{ā_{1}+b̄_{1}−p_{1}} … g_{d}^{ā_{d}+b̄_{d}−p_{d}}(π(g)π(h)( ∏_{ā_{j}+b̄_{j}≥p} g_{j}) ∏_{i>j}([g_{i}, g_{j}]^{\frac{1}{p}})^{ā_{i}b̄_{j}}γ_{3})^{p} .

This shows that

f̃(gh) = ∑_{i}(ā_{i} + b̄_{i} − p_{i})f_{i} + f(π(g)π(h)( ∏_{ā_{j}+b̄_{j}≥p} g_{j}) ∏_{i>j}([g_{i}, g_{j}]^{\frac{1}{p}})^{ā_{i}b̄_{j}}γ_{3})

= ∑_{i}(ā_{i} + b̄_{i} − p_{i})f_{i} + f(π(g)) + f(π(h)) + ∑_{ā_{j}+b̄_{j}≥p} f(g_{j}) + ∑_{i>j} ā_{i}b̄_{j}f([g_{i}, g_{j}]^{\frac{1}{p}})

= ∑_{i}(ā_{i} + b̄_{i})f_{i} + f(π(g)) + f(π(h)) + ∑_{i>j} ā_{i}b̄_{j}f([g_{i}, g_{j}]^{\frac{1}{p}})

and consequently that

df̃(g, h) = − ∑_{i>j} ā_{i}b̄_{j}f([g_{i}, g_{j}]^{\frac{1}{p}}) .

Step 2: Let f ∈ (G^{p}/[G^{p}, G])^{∨} = Hom_{G}^{cont}(G^{p}, ℚ_{p}/ℤ_{p}) be a continuous G-equivariant (for the adjoint action of G on G^{p}) homomorphism. We obtain from Step 1 that δ ∘ φ^{∨}(f) is the class of the 2-cocycle

(3)  (g, h) = (g_{1}^{a_{1}} … g_{d}^{a_{d}}, g_{1}^{b_{1}} … g_{d}^{b_{d}}) ⟼ − ∑_{i>j} ā_{i}b̄_{j}f([g_{i}, g_{j}]) .

Step 3: We compute the map

∂^{∨} : Hom_{G}^{cont}(G^{p}, ℚ_{p}/ℤ_{p}) ⟶ (⋀^{2} V)^{∨} = ⋀^{2} V^{∨} = ⋀^{2} H^{1}(G, 𝔽_{p}) = H^{2}(G, 𝔽_{p}) .

We emphasize that the identification ⋀^{2} H^{1}(G, 𝔽_{p}) = H^{2}(G, 𝔽_{p}) is given by the cup-product. The elements e_{1} := g_{1}G^{p}, …, e_{d} := g_{d}G^{p} form an 𝔽_{p}-basis of V; let e_{1}^{∨}, …, e_{d}^{∨} denote the dual basis. Let f ∈ Hom_{G}^{cont}(G^{p}, ℚ_{p}/ℤ_{p}). Then ∂^{∨}(f) is given, in (⋀^{2} V)^{∨}, by e_{i} ∧e_{j} ⟼ f([g_{i}, g_{j}]). We observe that under the identification (⋀^{2} V)^{∨} = ⋀^{2} V^{∨} the basis dual to e_{i} ∧e_{j} (for i > j) corresponds to e_{i}^{∨} ∧ e_{j}^{∨}. Hence

∂^{∨}(f) = ∑_{i>j} f([g_{i}, g_{j}])e_{i}^{∨} ∧ e_{j}^{∨}

in ⋀^{2} V^{∨}. Finally applying the cup-product shows that in H^{2}(G, 𝔽_{p}) the class ∂^{∨}(f) is given by the 2-cocycle

(g, h) = (g_{1}^{a_{1}} … g_{d}^{a_{d}}, g_{1}^{b_{1}} … g_{d}^{b_{d}}) ⟼ − ∑_{i>j} f([g_{i}, g_{j}])e_{i}^{∨}(gG^{p})e_{j}^{∨}(hG^{p}) = − ∑_{i>j} ā_{i}b̄_{j}f([g_{i}, g_{j}]) .

This coincides with (3) and therefore establishes our assertion. □

Corollary 3.4. We have SK_{1}(Λ(G)) = 0 if and only if ⋀ V = ker ∂.

4. A Lie criterion

We recall that L := L(G) denotes the ℤ_{p}-Lie algebra of the uniform group G in the sense of Lazard (cf. [DDMS] §4.5). As sets G and L coincide. To avoid confusion we write, for any two elements g, h ∈ G = L, in the following [g, h]_{G} and [g, h]_{L} for the commutators in the group G and in the Lie algebra L, respectively. The Lie algebra structure is given in terms of the group structure as follows:

– g + h = lim_{n→∞}(g^{p^{n}}h^{p^{n}})^{p^{−n}};

– in particular, xg = g^{x} for any x ∈ ℤ_{p};
– [g, h]_{L} = lim_{n→∞}[g^{p^{n}}, h^{p^{n}}]_{G}^{p^{−2n}}.

According to [GS] Prop. 2.1(5) we have

(4)  [G^{p}, G]_{G} = [G, G]_{G}^{p} = [G, G^{p}]_{G}

and hence [G, [G, G]_{G}]_{G} ⊆ [G, G^{p}]_{G} ⊆ [G, G]_{G}^{p}, which says that [G, G]_{G} is powerfully embedded in G and therefore, in particular, is uniform. By [GS] Prop. 2.1(3) and Thm. B(1) we have

L([G, G]_{G}) = [L, L]_{L}  and  L([G^{p}, G]_{G}) = p[L, L]_{L} .

Remark 4.1. i. The map G/G^{p} ⟶^{id} L/pL is an isomorphism of 𝔽_{p}-vector spaces. ii. The map G^{p}/[G^{p}, G]_{G} ⟶^{id} pL/[pL, L]_{L} is an isomorphism of ℤ_{p}-modules.

Proof. i. See [DDMS] Cor. 4.15. ii. Using Lemma 3.2 it suffices to show that the map G/[G, G]_{G} ⟶^{id} L/[L, L]_{L} is an isomorphism of ℤ_{p}-modules. This means that we have to check that

gh ≡ g + h mod [G, G]_{G}

holds true for any g, h ∈ G. This will follow from part ii. of the following lemma. □

Lemma 4.2. i. (gh)^{p^{n}} ≡ g^{p^{n}}h^{p^{n}} mod [G, G]_{G}^{p^{n}} for all n ≥ 0.

ii. gh ≡ (g^{p^{n}}h^{p^{n}})^{p^{−n}} mod [G, G]_{G} for all n ≥ 0.

Proof. i. According to the Hall-Petrescu formula [DDMS] App. A, for any group F and x, y ∈ F one has

(xy)^{p} = x^{p}y^{p}c\binom{p}{2}_{2} ⋯ c\binom{p}{j}_{j} ⋯ c_{p−1}^{p}c_{p}

for some c_{j} ∈ γ_{j}(F), where γ_{j+1}(F) := [γ_{j}(F), F] with γ_{1}(F) := F denotes the lower central series. We apply this formula to the group G^{p^{i}}, for any i ≥ 0. Using

[G^{p^{i}}, G^{p^{j}}]_{G} = [G, G]_{G}^{p^{i+j}} ⊆ G^{p^{i+j+1}}

by [GS] Prop. 2.1(3)-(5) as well as (by induction)

γ_{j}(G^{p^{i}}) ⊆ [G, G]_{G}^{p^{ij+j−2}} for i ≥ 0, j ≥ 2

we obtain

(5)  (gh)^{p} ≡ g^{p}h^{p} mod [G, G]_{G}^{p^{2i}}  for g, h ∈ G^{p^{i}} and i ≥ 0.

We now prove the assertion via induction. The case n = 0 is trivial and the case n = 1 was shown before Lemma 3.2. For n ≥ 1 we have, by the induction hypothesis, that

(gh)^{p^{n+1}} = (g^{p^{n}}h^{p^{n}}y^{p^{n}})^{p}
≡ (g^{p^{n}}h^{p^{n}})^{p}y^{p^{n+1}} mod [G, G]_{G}^{p^{2n}}
≡ g^{p^{n+1}}h^{p^{n+1}} mod [G, G]_{G}^{p^{n+1}},

for some y ∈ [G, G]_{G} where for the two congruences we used (5).

ii. By [GS] Prop. 2.2 the element (gh)^{−1}(g^{p^{n}}h^{p^{n}})^{p^{−n}} belongs to [G, G]_{G} if and only if ((gh)^{−1}(g^{p^{n}}h^{p^{n}})^{p^{−n}})^{p^{n}} belongs to [G, G]_{G}^{p^{n}}. But by i. we do have

((gh)^{−1}(g^{p^{n}}h^{p^{n}})^{p^{−n}})^{p^{n}} ≡ (gh)^{−p^{n}}g^{p^{n}}h^{p^{n}} ≡ 1 mod [G, G]_{G}^{p^{n}} .
□

This means that source and target of the two maps

(6) G/G^{p} ∧ G/G^{p} ⟶^{[ , ]_{G}} G^{p}/[G^{p}, G]  and  L/pL ∧ L/pL ⟶^{[ , ]_{L}} pL/p[L, L]_{L}

coincide. The next lemma implies that, in fact, the two maps coincide.

Lemma 4.3. [g, h]_{G} ≡ [g^{p^{n}}, h^{p^{n}}]_{G}^{p^{−2n}} mod [G^{p}, G]_{G} for any n ≥ 0 and any g, h ∈ G.

Proof. The Hall-Petrescu identity says (cf. [B-LL] II §5 Ex. 9d)) that

[g^{p^{n}}, h]_{G} = [g, h]_{G}^{p^{n}}v_{3}(g, h)^{\binom{p^{n}}{2}} … v_{p^{n}}(g, h)^{p^{n}}v_{p^{n}+1}(g, h)

where the v_{j}(g, h) are universal products of iterated commutators in g and h of length j. In particular,

[g^{p^{n}}, h^{p^{n}}]_{G} = [g, h^{p^{n}}]_{G}^{p^{n}}v_{3}(g, h^{p^{n}})^{\binom{p^{n}}{2}} … v_{p^{n}}(g, h^{p^{n}})^{p^{n}}v_{p^{n}+1}(g, h^{p^{n}}) .

From [G, G]_{G} ⊆ G^{p} and formula (4) we conclude inductively that

v_{j}(g, h) ∈ [G, G]_{G}^{p^{j−2}}  and  v_{j}(g, h^{p^{n}}) ∈ [G, G]_{G}^{p^{n+j−2}} .

It is well known that the p-adic valuation of (j − 1)! satisfies

v_{p}((j − 1)!) ≤ \frac{j − 2}{p − 1} .

Since p > 2 we obtain v_{p}((j − 1)!) ≤ j − 3 for j ≥ 3 and hence

v_{j}(g, h)^{\binom{p^{n}}{j−1}} ∈ [G, G]_{G}^{p^{n+1}}  and  v_{j}(g, h^{p^{n}})^{\binom{p^{n}}{j−1}} ∈ [G, G]_{G}^{p^{2n+1}} .

It follows that

[g^{p^{n}}, h]_{G} ∈ [g, h]_{G}^{p^{n}}[G, G]_{G}^{p^{n+1}}  and  [g^{p^{n}}, h^{p^{n}}]_{G} ∈ [g, h^{p^{n}}]_{G}^{p^{n}}[G, G]_{G}^{p^{2n+1}} .

By passing to inverses we also have [g, h^{p^{n}}]_{G} ∈ [g, h]_{G}^{p^{n}}[G, G]_{G}^{p^{n+1}}. Inserting this into the right hand formula gives

[g^{p^{n}}, h^{p^{n}}]_{G} ∈ ([g, h]_{G}^{p^{n}}[G, G]_{G}^{p^{n+1}})^{p^{n}}[G, G]_{G}^{p^{2n+1}} .

But ([g, h]_{G}^{p^{n}}[G, G]_{G}^{p^{n+1}})^{p^{n}} ⊆ [g, h]_{G}^{p^{2n}}[G, G]_{G}^{p^{2n+1}}, since [G, G]_{G} is uniform, and hence

[g^{p^{n}}, h^{p^{n}}]_{G} ≡ [g, h]_{G}^{p^{2n}} mod [G, G]_{G}^{p^{2n+1}} .

On the other hand, (4) also implies that [g^{p^{n}}, h^{p^{n}}]_{G} ∈ [G, G]_{G}^{p^{2n}} which means, again since [G, G]_{G} is uniform, that [g^{p^{n}}, h^{p^{n}}]_{G} = y^{p^{2n}} for some y ∈ [G, G]_{G}. We see that

y^{p^{2n}} ≡ [g, h]_{G}^{p^{2n}} mod [G, G]_{G}^{p^{2n+1}} ,

which implies, a third time since [G, G]_{G} is uniform, that

y ≡ [g, h]_{G} mod [G, G]_{G}^{p} .
□

In the previous section we had introduced the 𝔽_{p}-vector subspace ⋀(G/G^{p}) ⊆ ⋀^{2}(G/G^{p}) which is generated by all elements of the form gG^{p} ∧ hG^{p} where [g, h]_{G} = 1.

Lemma 4.4. For any g, h ∈ G we have [g, h]_{G} = 1 if and only if [g, h]_{L} = 0.

Proof. If [g, h]_{G} = 1 then [g, h]_{L} = 0 is immediate from the definition of the Lie bracket. Let us suppose that [g, h]_{L} = 0. Then A := {x ∈ L : x^{m} ∈ ℤ_{p}g + ℤ_{p}h for some m ≥ 1} is an abelian Lie subalgebra of L in the sense of [DDMS] Scholium to Thm. 9.10. Hence loc. cit. says that A at the same time is a uniform subgroup of G which, moreover, by [DDMS] Cor. 7.16(i) is commutative. We conclude that [g, h]_{G} = 1 holds true. □

We now introduce in ⋀^{2} L the ℤ_{p}-submodule ⋀ L generated by all elements of the form x ∧ y where [x, y]_{L} = 0. The above lemma implies that

(7)  ⋀(G/G^{p}) = (⋀ L + p ⋀^{2} L)/p ⋀^{2} L .

We therefore may reformulate Cor. 3.4 in the following form thereby obtaining our second main result.

Theorem 4.5. For uniform G we have SK_{1}(Λ(G)) = 0 if and only if

ker(⋀^{2} L ⟶^{[ , ]_{L}} L) ⊆ ⋀ L .

Proof. The identity (7) implies that SK_{1}(Λ(G)) = 0 if and only if

⋀ L + p ⋀^{2} L = ker(⋀^{2} L ⟶^{[ , ]_{L}} pL/p[L, L]_{L}) .

But the map [ , ]_{L} : ⋀^{2} L ⟶ [L, L]_{L} is surjective. Hence

ker(⋀^{2} L ⟶^{[ , ]_{L}} pL/p[L, L]_{L}) = ker(⋀^{2} L ⟶^{[ , ]_{L}} L) + p ⋀^{2} L .

It follows that SK_{1}(Λ(G)) = 0 if and only if

⋀ L + p ⋀^{2} L = ker(⋀^{2} L ⟶^{[ , ]_{L}} L) + p ⋀^{2} L .

For trivial reasons we have ⋀ L ⊆ ker(⋀^{2} L ⟶^{[ , ]_{L}} L) so that the equivalent condition reduces to

ker(⋀^{2} L ⟶^{[ , ]_{L}} L) ⊆ ⋀ L + p ⋀^{2} L .

But this latter inclusion implies, by an easy inductive argument, that

ker(⋀^{2} L ⟶^{[ , ]_{L}} L) ⊆ ⋀ L + p^{j} ⋀^{2} L for any j ≥ 1

and hence that

ker(⋀^{2} L ⟶^{[ , ]_{L}} L) ⊆ ⋀ L + ⋂_{j} p^{j} ⋀^{2} L = ⋀ L .

The reverse direction is trivial. □

We observe that the criterion in Thm. 4.5 behaves well under products.

Remark 4.6. Let L_{i} be ℤ_{p}-Lie algebras which satisfy the condition ⋀ L_{i} = ker[ , ]_{i}, and let L_{0} := L_{1} ⊕ L_{2} be the product Lie algebra with bracket [x_{1} + x_{2}, y_{1} + y_{2}]_{0} := [x_{1}, y_{1}]_{1} + [x_{2}, y_{2}]_{2} for x_{i}, y_{i} ∈ L_{i}. Then also for L_{0} we have

⋀ L_{0} = ker[ , ]_{0} .

Proof. The following identities are easily checked and imply the assertion:

ker[ , ]_{0} = ker[ , ]_{1} + L_{1} ⊗_{ℤ_{p}} L_{2} + ker[ , ]_{2} ,
⋀ L_{0} = ⋀ L_{1} + L_{1} ⊗_{ℤ_{p}} L_{2} + ⋀ L_{2} .
□

5. A counterexample

In the following we will modify the Example 8.11 in [Oli] in order to show that the criterion in Thm. 4.5 is not always satisfied.

Let R be any commutative ring in which 2 is invertible. First we introduce the free Rmodule V := R^{4} with its standard basis e_{1}, …, e_{4} and W := ⋀^{2} V/R(e_{1} ∧ e_{2} + e_{3} ∧ e_{4}) (which has rank 5) together with the linear map

∂ : ⋀^{2} V ⟶^{pr} W .

Obviously ker ∂ does not contain any nonzero vector of the form a ∧ b.

Now define L′ := V ⊕ W. The bracket

[ , ] : ⋀^{2} L′ ⟶^{pr} ⋀^{2} V ⟶^{∂} W ⟶^{⊆} L′

makes L′ into a 2-step nilpotent Lie algebra over R with center Z(L′) = [L′, L′] = W and ⋀ L′ ≠ ker ∂. In fact, ker ∂/ ⋀ L′ is free of rank 1 over R.

Now let R = ℤ_{p} with p ≠ 2. Then L := pL′ is a powerful and torsion free, i.e. uniform ℤ_{p}-Lie algebra which corresponds to a unique uniform pro-p-group G (of dimension 9) by [DDMS] Thm. 9.8. We have pe_{1} ∧ pe_{2} + pe_{3} ∧ pe_{4} ∈ ker ∂ \ ⋀ L. Thm. 4.5 therefore implies that SK_{1}(Λ(G)) ≠ 0.

In terms of generators and relations a similar example can be described as follows. Let G be the pro-p-group generated by x_{1}, …, x_{4}, y_{1}, … y_{5} subject to the following relations

[x_{i}, y_{j}] = 1 for all 1 ≤ i ≤ 4, 1 ≤ j ≤ 5,
[y_{i}, y_{j}] = 1 for all 1 ≤ i, j ≤ 5

and

[x_{i}, x_{j}] = \{ \begin{matrix}y_{1}^{−p}, & if (i, j) = (1, 2); \\ y_{1}^{p}, & if (i, j) = (1, 3); \\ y_{2}^{p}, & if (i, j) = (1, 4); \\ y_{3}^{p}, & if (i, j) = (2, 3); \\ y_{4}^{p}, & if (i, j) = (2, 4); \\ y_{5}^{p}, & if (i, j) = (3, 4). \end{matrix}

Then G is obviously powerful, fits into an exact sequence

1 ⟶ ℤ_{p}^{5} ⟶ G ⟶ ℤ_{p}^{4} ⟶ 1,

whence is torsionfree and thus uniform. One can also show directly, using Oliver’s criterion or Cor. 3.4, that this group has non-trivial SK_{1}. Indeed, the same kind of calculation as above shows that x_{1}G^{p} ∧ x_{2}G^{p} + x_{3}G^{p} ∧ x_{4}G^{p} belongs to ker(∂ : ⋀^{2} G/G^{p} → G^{p}/[G^{p}, G]), but not to ⋀ V = ⋀ G/G^{p}.

6. Chevalley orders

Let F be any field of characteristic zero and let 𝔤 be an F -split reductive Lie algebra over F ([B-LL] Chap. VIII §2.1) with center 𝔷. We fix an F -split Cartan subalgebra 𝔥 ⊆ 𝔤 and let Φ denote the corresponding (reduced) root system viewed as a subset of the dual vector space 𝔥^{∗}. By [B-LL] Chap. VIII §2.2 Thm. 1 we have for any root α:

– The root space 𝔤^{α} = {x ∈ 𝔤 : [h, x] = α(h)x for any h ∈ 𝔥} is one dimensional.

– 𝔥_{α} := [𝔤^{α}, 𝔤^{−α}] is one dimensional; it contains a unique element H_{α} such that α(H_{α}) =

2. Obviously H_{−α} = −H_{α}. According to [B-LL] Chap. VIII §4.4 there always exists a Chevalley system for (𝔤, 𝔥): This is a family of nonzero vectors X_{α} ∈ 𝔤^{α} such that

[X_{α}, X_{−α}] = −H_{α}

and which satisfies one additional condition whose explicit form is not needed in the following. We introduce the coroot lattice

Q^{∨} := ∑_{α∈Φ} ℤH_{α} ⊆ 𝔥

as well as the coweight lattice

P^{∨} := {h ∈ ∑_{α∈Φ} ℚH_{α} : β(h) ∈ ℤ for any β ∈ Φ} ⊆ 𝔥

of the root system Φ. Throughout this section we fix a ℤ-lattice 𝔥_{ℤ} ⊆ 𝔥 such that

Q^{∨} ⊆ 𝔥_{ℤ} ⊆ P^{∨} ⊕ 𝔷 ,

and we put

𝔤_{ℤ} := 𝔥_{ℤ} + ∑_{α∈Φ} ℤX_{α} ⊆ 𝔤 .

By [B-LL] Chap. VIII §2.4 Prop. 8 and §12.7 the ℤ-submodule 𝔤_{ℤ} of 𝔤 in fact is a ℤ-Lie subalgebra (a so called Chevalley order of 𝔤). For any (commutative) ring R we then have the R-Lie algebra 𝔤_{R} := R ⊗_{ℤ} 𝔤_{ℤ}. We also put 𝔥_{R} := R ⊗_{ℤ} 𝔥_{ℤ}.

In the following we always assume that the integers 2 and 3 are invertible in the ring R. We view the Lie bracket as a linear map [ , ] : ⋀^{2} 𝔤_{R} ⟶ 𝔤_{R} on the exterior square and define

⋀ 𝔤_{R} := < x ∧ y ∈ ⋀^{2} 𝔤_{R} : [x, y] = 0 >_{R} .

We aim at showing that

⋀ 𝔤_{R} = ker[ , ]

holds true.

Remark. For R = ℂ and 𝔤 semisimple this identity was shown by Kostant in [Ko] Cor. 5.1. But his proof relies on representation theory and therefore cannot be made to work integrally.

Here is a list of relevant basic facts:

1. ⋀^{2} 𝔥_{R} ⊆ ⋀ 𝔤_{R} (obvious).

2. [X_{α}, X_{β}] = 0 and hence X_{α} ∧ X_{β} ∈ ⋀ 𝔤_{R} if α + β ∉ Φ ∪ {0}.

3. Each root induces a surjective linear form α : 𝔥_{R} ⟶ R (since α(H_{α}) = 2 is invertible

in R). The latter implies that 𝔥_{R} = RH_{α} ⊕ker(α). But for h ∈ ker(α) we have [h, X_{α}] = 0 and hence h ∧ X_{α} ∈ ⋀ 𝔤_{R}. It follows that

4. h ∧ X_{α} ≡ \frac{α(h)}{2}H_{α} ∧ X_{α} mod ⋀ 𝔤_{R}

for any h ∈ 𝔥_{R}. We fix a basis Δ ⊆ Φ of the root system; Φ^{+} denotes the corresponding subset of positive roots. The set {H_{α} : α ∈ Δ} is a basis of ℚ ⊗_{ℤ} Q^{∨}. We also need the Killing form

< , >: 𝔥 × 𝔥 ⟶ F .

It has the property (cf. [B-LL] Chap. VIII §2.2. Prop. 2(ii)) that

< Q^{∨}, Q^{∨} > ⊆ ℤ .

We then have

5. α(h) = \frac{2<H_{α},h>}{<H_{α},H_{α}>} for any h ∈ 𝔥. Since the Killing form on 𝔥/𝔷 is known to be nondegenerate ([B-LL] Chap. VIII §2.2 Remark 2) we obtain:

6. If β = ∑_{α∈Δ} m_{βα}α (with m_{βα} ∈ ℤ) then

H_{β} = ∑_{α∈Δ} m_{βα} \frac{< H_{β}, H_{β} >}{< H_{α}, H_{α} >}H_{α} .

All the ratios \frac{<H_{β},H_{β}>}{<H_{α},H_{α}>} belong to the set \{1, 2, \frac{1}{2}, 3, \frac{1}{3}\} ([B-LL] Chap. VI §1.4 Prop. 12(i) and Chap. VIII §2.4 formula (6)) and therefore are units in R. In particular, the R-module R ⊗_{ℤ} Q^{∨} is free with basis {H_{α} : α ∈ Δ}.

Let α, β be any two roots such that α + β ∈ Φ again is a root. We have

[X_{α}, X_{β}] = N_{αβ}X_{α+β} for some integer N_{αβ}.

It follows from [B-LL] Chap. VIII §2.4 Prop. 7 and Chap. VI §1.3 Cor. of Prop. 9 that

7. N_{αβ} ∈ ±{1, 2, 3} and, in particular, is a unit in R. The weight lattice of Φ is P := Hom(Q^{∨}, ℤ). It contains the root lattice Q := ∑_{α∈Φ} ℤα with finite index.

Lemma 6.1. Let α, β ∈ Φ be any two roots; we have:

i. The submodule Rα + Rβ of the R-module R ⊗_{ℤ} P = Hom(Q^{∨}, R) is a direct factor; ii. If α ≠ ±β then we find an element h ∈ R ⊗_{ℤ} Q^{∨} ⊆ 𝔥_{R} such that α(h) = 1 and

β(h) = 0.

Proof. i. Step 1: We assume that Φ is irreducible but not of type A_{ℓ}. In this case the order of the finite group P/Q is equal to 1, 2, 3, or 4 (cf. [B-LL] Plate II-X). Hence R ⊗_{ℤ} P = R ⊗_{ℤ} Q, and it suffices to show that the nonzero elementary divisors of ℤα + ℤβ in Q are invertible in R. Since any root is a member of some basis of the root system we may assume that α ∈ Δ. If α = ±β then ℤα + ℤβ = ℤα obviously is a direct factor of Q. We therefore suppose that α ≠ ±β. Let β = ∑_{δ∈Δ} m_{δ}δ. The nonzero elementary divisors in question are 1 and the gcd of the integers m_{δ} for δ ≠ α. A case by case inspection of the tables in [B-LL] shows that this gcd is equal to 1, 2, or 3 (the latter only happening for the type G_{2}).

Step 2: We assume that Φ is irreducible of type A_{ℓ}. We also may assume that α and β both are positive roots. The coroot lattice can be realized as

Q^{∨} = {(m_{1}, …, m_{l+1}) ∈ ℤ^{l+1} : m_{1} + … + m_{l+1} = 0}.

The positive roots are the maps

α_{ij} :  Q^{∨} ⟶ ℤ
(m_{1}, …, m_{l+1}) ⟼ m_{i} − m_{j}

for any 1 ≤ i < j ≤ l + 1. We leave it to the reader to explicitly check that the following facts about positive roots hold true, which by duality imply the assertion.

– α_{12}(Q^{∨}) = 2ℤ for l = 1.

– α(Q^{∨}) = ℤ for l ≥ 2 and any α.

– (ℤ ⊕ ℤ)/(α, β)(Q^{∨}) ≅ ℤ/3ℤ for l = 2 and any α ≠ β.

– (α, β)(Q^{∨}) ⊇ 2ℤ ⊕ 2ℤ for l ≥ 3 and any α ≠ β. Step 3: If we decompose a general Φ into its irreducible components then P is the direct

sum of the weight lattices of these irreducible components.

ii. By i. we have Hom_{R}(R ⊗_{ℤ} Q^{∨}, R) = Rα ⊕ Rβ ⊕ U and the claim follows by dualizing and identifying R ⊗_{ℤ} Q^{∨} with its double R-dual. □

Lemma 6.2. For any two roots α, β such that γ := α + β ∈ Φ we have

X_{α} ∧ X_{β} ≡ \frac{N_{αβ}}{2} H_{γ} ∧ X_{γ} mod ⋀ 𝔤_{R} .

Proof. The difference X_{α} ∧ X_{β} − \frac{N_{αβ}}{2} H_{γ} ∧ X_{γ} lies in the kernel of [ , ]. Therefore, if we find elements A, B ∈ 𝔤_{R} such that

A ∧ B ≡ X_{α} ∧ X_{β} − \frac{N_{αβ}}{2} H_{γ} ∧ X_{γ} mod ⋀ 𝔤_{R}

then neccesarily [A, B] = 0, and the assertion is proven.

By Lemma 6.1 we find an h ∈ 𝔥_{R} such that α(h) = −N_{αβ} and β(h) = 0. According to [B-LL] Chap. VI §1.3 Cor. of Prop. 9 the following cases can occur:

I. 2α + β ∉ Φ, II. γ′ := 2α + β ∈ Φ, but 3α + β ∉ Φ, or III. γ′ := 2α + β ∈ Φ, γ′′ := 3α + β ∈ Φ, but 4α + β ∉ Φ.

Case I: Here α + γ ∉ Φ and hence, by the choice of h and using 2. and 4.,

(X_{α} + h) ∧ (X_{β} + X_{γ}) ≡ X_{α} ∧ X_{β} + \frac{β(h)}{2}H_{β} ∧ X_{β} + \frac{γ(h)}{2}H_{γ} ∧ X_{γ} mod ⋀ 𝔤_{R}

= X_{α} ∧ X_{β} − \frac{N_{αβ}}{2} H_{γ} ∧ X_{γ} .

Case II: Here α + γ′ ∉ Φ. Using 7. we define

r′ := \frac{N_{αγ}}{2N_{αβ}} = −\frac{N_{αγ}}{2α(h)} ∈ R .

We then have

(X_{α} + h) ∧ (X_{β} + X_{γ} + r′X_{γ′})

≡ X_{α} ∧ X_{β} + X_{α} ∧ X_{γ} + \frac{β(h)}{2}H_{β} ∧ X_{β} + \frac{γ(h)}{2}H_{γ} ∧ X_{γ} + r′ \frac{γ′(h)}{2}H_{γ′} ∧ X_{γ′} mod ⋀ 𝔤_{R}

= X_{α} ∧ X_{β} + X_{α} ∧ X_{γ} − \frac{N_{αβ}}{2} H_{γ} ∧ X_{γ} − \frac{N_{αγ}}{2} H_{γ′} ∧ X_{γ′}

≡ X_{α} ∧ X_{β} − \frac{N_{αβ}}{2} H_{γ} ∧ X_{γ} mod ⋀ 𝔤_{R}

where the last congruence uses the case I. for the pair (α, γ). Case III: We define

r′ := \frac{N_{αγ}}{2N_{αβ}} , r′′ := r′ \frac{N_{αγ′}}{3N_{αβ}} ∈ R .

Using case I. for (α, γ′) and case II. for (α, γ) we have

(X_{α} + h) ∧ (X_{β} + X_{γ} + r′X_{γ′} + r′′X_{γ′′}) ≡ X_{α} ∧ X_{β} − \frac{N_{αβ}}{2} H_{γ} ∧ X_{γ} mod ⋀ 𝔤_{R} .

□

Using 1., 2., 4., and Lemma 6.2 we see that the factor R-module ⋀^{2} 𝔤_{R}/ ⋀ 𝔤_{R} is generated by the elements X_{α} ∧ X_{−α} and H_{α} ∧ X_{α} for α ∈ Φ. But [X_{α}, X_{−α}] ∈ 𝔥_{ℤ} and [H_{α}, X_{α}] = 2X_{α}. The elements X_{α} are linearly independent among each other as well as from 𝔥_{ℤ}. It follows that

(8)  ker[ , ] ⊆ 𝔄 + ⋀ 𝔤_{R} with 𝔄 :=< X_{α} ∧ X_{−α} : α ∈ Φ^{+} >_{R} .

We recall that the integers m_{βα}, for β ∈ Φ and α ∈ Δ, are defined by

β = ∑_{α∈Δ} m_{βα}α .

Lemma 6.3. For any β ∈ Φ^{+} we have

X_{β} ∧ X_{−β} ≡ ∑_{α∈Δ} m_{βα} \frac{< H_{β}, H_{β} >}{< H_{α}, H_{α} >}X_{α} ∧ X_{−α} mod ⋀ 𝔤_{R} .

Proof. By an inductive argument it suffices to show that

X_{γ} ∧ X_{−γ} ≡ \frac{< H_{γ}, H_{γ} >}{< H_{α}, H_{α} >}X_{α} ∧ X_{−α} + \frac{< H_{γ}, H_{γ} >}{< H_{β}, H_{β} >}X_{β} ∧ X_{−β} mod ⋀ 𝔤_{R}

holds true for any α, β ∈ Φ^{+} such that γ := α + β ∈ Φ. First we observe that by 6. the difference

X_{γ} ∧ X_{−γ} − \frac{< H_{γ}, H_{γ} >}{< H_{α}, H_{α} >}X_{α} ∧ X_{−α} − \frac{< H_{γ}, H_{γ} >}{< H_{β}, H_{β} >}X_{β} ∧ X_{−β}

indeed lies in the kernel of [ , ]. We put

N_{α,−β} := 0 if α − β ∉ Φ,

N_{α,−2β} := \begin{cases}N_{α,−β}N_{α−β,−β} & if α − β ∈ Φ, \\ 0 & otherwise, \end{cases}

N_{α,−3β} := \begin{cases}N_{α,−β}N_{α−β,−β}N_{α−2β,−β} & if α − β, α − 2β ∈ Φ, \\ 0 & otherwise \end{cases}

and similarly with α and β exchanged. Using Lemma 6.1 we choose h ∈ 𝔥_{R} such that

α(h) = −β(h) and α(h) = −\frac{N_{−α,γ}N_{−β,γ}}{N_{α,β}} .

We now define

A := X_{γ} + X_{α} + X_{β} + \frac{N_{α,−β}}{2N_{γ,−β}} X_{α−β} + \frac{N_{α,−2β}}{6N_{γ,−β}^{2}} X_{α−2β} + \frac{N_{α,−3β}}{24N_{γ,−β}^{3}} X_{α−3β}

+ \frac{N_{β,−α}}{2N_{γ,−α}} X_{β−α} + \frac{N_{β,−2α}}{6N_{γ,−α}^{2}} X_{β−2α} + \frac{N_{β,−3α}}{24N_{γ,−α}^{3}} X_{β−3α}

+ \frac{N_{α,β}^{2}}{N_{−α,γ}N_{−β,γ}} (1 − \frac{N_{β,−α}N_{β−α,−β}}{2N_{α,β}N_{−β,γ}} )h

(if the subscript ? in X_{?} is not a root then the corresponding summand is set to zero, and the product N_{β,−α}N_{β−α,−β} is understood to be equal to zero if β − α ∉ Φ) and

B := X_{−γ} + \frac{N_{−β,γ}}{N_{β,α}} X_{−α} + \frac{N_{−α,γ}}{N_{α,β}} X_{−β} + h .

A lengthy but straightforward computation shows that

A ∧ B ≡ X_{γ} ∧ X_{−γ} − \frac{< H_{γ}, H_{γ} >}{< H_{α}, H_{α} >}X_{α} ∧ X_{−α} − \frac{< H_{γ}, H_{γ} >}{< H_{β}, H_{β} >}X_{β} ∧ X_{−β} mod ⋀ 𝔤_{R}

Besides 4. and Lemma 6.2 it uses the following identities:

a) \frac{<H_{α′+β′},H_{α′+β′}>}{<H_{α′},H_{α′}>} = −\frac{N_{−β′,α′+β′}}{N_{β′,α′}} and \frac{<H_{α′+β′},H_{α′+β′}>}{<H_{β′},H_{β′}>} = −\frac{N_{−α′,α′+β′}}{N_{α′,β′}} whenever α′, β′, α′+ β′ ∈ Φ ([B-LL] Chap. VIII §2.4 Lemma 4).

b) N_{α′,β′} = N_{−α′,−β′} = −N_{β′,α′} whenever α′, β′, α′ + β′ ∈ Φ (cf. [B-LL] Chap. VIII §2.4 Prop. 7).

c) \frac{N_{β,−α}N_{β−α,−β}}{N_{α,β}N_{−β,γ}} = \frac{N_{α,−β}N_{α−β,−α}}{N_{β,α}N_{−α,γ}}.

To establish the last identity c) we may assume that α−β ∈ Φ. It then reduces to the equality

\frac{N_{β−α,−β}}{N_{α−β,−α}} = \frac{N_{−β,γ}}{N_{−α,γ}} .

Using a) and b) for α, β we obtain

\frac{N_{−β,γ}}{N_{−α,γ}} = −\frac{< H_{β}, H_{β} >}{< H_{α}, H_{α} >} .

On the other hand, using a) for α′ := −α and β′ := α − β we have

\frac{N_{β−α,−β}}{N_{α−β,−α}} = −\frac{< H_{−β}, H_{−β} >}{< H_{−α}, H_{−α} >} = −\frac{< H_{β}, H_{β} >}{< H_{α}, H_{α} >} .

□

Theorem 6.4. If 2 and 3 are invertible in R then ker[ , ] = ⋀ 𝔤_{R}.

Proof. By (8) and Lemma 6.3 it remains to consider any element u ∈ ker[ , ] of the form

u = ∑_{β∈Φ^{+}} r_{β} ∑_{α∈Δ} m_{βα} \frac{< H_{β}, H_{β} >}{< H_{α}, H_{α} >}X_{α} ∧ X_{−α}

with r_{β} ∈ R. We have

0 = ∑_{β∈Φ^{+}} r_{β} ∑_{α∈Δ} m_{βα} \frac{< H_{β}, H_{β} >}{< H_{α}, H_{α} >}[X_{α}, X_{−α}]

= − ∑_{β∈Φ^{+}} r_{β} ∑_{α∈Δ} m_{βα} \frac{< H_{β}, H_{β} >}{< H_{α}, H_{α} >}H_{α}

= − ∑_{α∈Δ} \frac{1}{< H_{α}, H_{α} >}( ∑_{β∈Φ^{+}} r_{β}m_{βα} < H_{β}, H_{β} > )H_{α}

and hence

∑_{β∈Φ^{+}} r_{β}m_{βα} < H_{β}, H_{β} > = 0 for any α ∈ Δ

by 6. But then also

u = ∑_{α∈Δ} \frac{1}{< H_{α}, H_{α} >}( ∑_{β∈Φ^{+}} r_{β}m_{βα} < H_{β}, H_{β} > )X_{α} ∧ X_{−α} = 0 .

□

Remark 6.5. Assuming again that 2 and 3 are invertible in R, let a ∈ R be any non-zero-divisor. The R-Lie subalgebra

𝔤_{R}(a) := a𝔤_{R} := {ax|x ∈ 𝔤_{R}}

of 𝔤_{R} also satisfies

ker[ , ] = ⋀ 𝔤_{R}(a) .

This follows immediately from the commutative diagram

\xymatrix{⋀^{2} 𝔤_{R} \ar[d]_{⋀^{2} a·}^{≅} \ar[r]^{[ , ]} & 𝔤_{R} \ar@{^{(}->}[d]^{a^{2}·} \\ ⋀^{2} 𝔤_{R}(a) \ar[r]^{[ , ]} & 𝔤_{R}(a)}

in which the vertical maps are induced by multiplication by a and a^{2}, respectively.

The negative example in section 5 compared to the above positive Thm. 6.4 could make the reader believe that the fundamental distinction here is between nilpotent and semisimple Lie algebras. But this not so. We consider the nilpotent ℤ-Lie algebra

𝔫_{ℤ} := ∑_{α∈Φ^{+}} ℤX_{α}

as well as its base change 𝔫_{R} := R ⊗_{ℤ} 𝔫_{ℤ}.

Lemma 6.6. Let α_{0}, α_{1}, β_{0}, β_{1}, γ ∈ Φ^{+} be any five roots such that

α_{0} + β_{0} = γ = α_{1} + β_{1} ;

then in any of the pairs α_{0} + α_{1}, α_{0} + β_{1} and β_{0} + α_{1}, β_{0} + β_{1} and α_{1} + α_{0}, α_{1} + β_{0} and β_{1} + α_{0}, β_{1} + β_{0} at least one member is not a root.

Proof. Let V be the underlying ℝ-vector space of the root system Φ. If V′ ⊆ V denotes the subspace generated by α_{0}, α_{1}, β_{0}, β_{1} then Φ′ := Φ ∩ V′ is a (reduced) root system in V′ ([B-LL] Chap. VI §1.1 Prop. 4). Obviously, Φ′ is of rank ≤ 3. Our assertion is an assertion about Φ′. Moreover, allroots occurring in this assertion lie in the same irreducible component of Φ′ as γ. We see that we may assume, without loss of generality, that Φ is irreducible of rank ≤ 3. The case A_{1} being empty we therefore need to consider the root systems A_{2}, B_{2}, G_{2} and A_{3}, B_{3}, C_{3}. The assertion is clearly invariant under the Weyl group of Φ. Hence it suffices ([B-LL] Chap. VI §1.3 Prop. 11) to treat the cases where γ is either the highest long root or the highest short root. These are

A_{2}  α_{1} + α_{2}
B_{2}  α_{1} + 2α_{2}  α_{1} + α_{2}
G_{2}  3α_{1} + 2α_{2}  2α_{1} + α_{2}
A_{3}  α_{1} + α_{2} + α_{3}
B_{3}  α_{1} + 2α_{2} + 2α_{3}  α_{1} + α_{2} + α_{3}
C_{3}  2α_{1} + 2α_{2} + α_{3}  α_{1} + 2α_{2} + α_{3}

(where the α_{i} form the basis of Φ corresponding to the Weyl chamber containing γ and are appropriately ordered). The assertion can now be checked by a case by case inspection using the tables at the end of [B-LL] Chap. VI. It is helpful to note that if γ is the highest long root then all the roots α_{0}, α_{1}, β_{0}, β_{1} neccessarily are positive. The most interesting case occurs for the highest short root in B_{3} where the possible decompositions are

α_{1} + α_{2} + α_{3} = α_{1} + (α_{2} + α_{3})
= (α_{1} + α_{2}) + α_{3}
= (α_{1} + α_{2} + 2α_{3}) + (−α_{3})
= (α_{1} + 2α_{2} + 2α_{3}) + (−α_{2} − α_{3}) .
□

Proposition 6.7. If 2 and 3 are invertible in R then ker[ , ] = ⋀ 𝔫_{R}.

Proof. Using 2. it remains to show the following. Fix any γ ∈ Φ^{+} which can be written in at least two different ways as a sum of two positive roots. Also fix one decomposition γ = α_{0} +β_{0} with α_{0}, β_{0} ∈ Φ^{+}. Given any other such decomposition of γ into α, β ∈ Φ^{+} we may, by Lemma 6.6, order the pair (α, β) in such a way that α_{0} + α, β_{0} + β ∉ Φ^{+}. Let S_{γ} denote the set of all such ordered pairs (including the pair (α_{0}, β_{0})). Then {X_{α} ∧ X_{β} : (α, β) ∈ S_{γ}} is a basis of the R-submodule [ , ]^{−1}(RX_{γ}) of ⋀^{2} 𝔫_{R}. We have to show that

ker[ , ] ∩ [ , ]^{−1}(RX_{γ}) ⊆ ⋀ 𝔫_{R}

holds true. Let ∑_{(α,β)∈S_{γ}} c_{αβ}X_{α} ∧ X_{β} ∈ ker[ , ], with c_{αβ} ∈ R, so that

∑_{(α,β)∈S_{γ}} c_{αβ}N_{αβ} = 0 .

Since the N_{αβ} are units in R by 7. we may compute

∑_{(α,β)∈S_{γ}} c_{αβ}X_{α} ∧ X_{β} = ∑_{(α,β)∈S_{γ}} c_{αβ}N_{αβ} \frac{1}{N_{αβ}} X_{α} ∧ X_{β}

= ∑_{(α_{0},β_{0})≠(α,β)∈S_{γ}} c_{αβ}N_{αβ}(\frac{1}{N_{αβ}} X_{α} ∧ X_{β} − \frac{1}{N_{α_{0}β_{0}}} X_{α_{0}} ∧ X_{β_{0}})

and

\frac{1}{N_{αβ}} X_{α} ∧ X_{β} − \frac{1}{N_{α_{0}β_{0}}} X_{α_{0}} ∧ X_{β_{0}} =

(\frac{1}{N_{αβ}} X_{α} + X_{β_{0}}) ∧ (X_{β} + \frac{1}{N_{α_{0}β_{0}}} X_{α_{0}}) − \frac{1}{N_{αβ}N_{α_{0}β_{0}}} X_{α} ∧ X_{α_{0}} + X_{β} ∧ X_{β_{0}} .

Since α + α_{0}, β + β_{0} ∉ Φ the three summands on the right hand side of the last identity lie in the kernel of [ . ] and hence in ⋀ 𝔫_{R}. □

We point out that for root systems of type A_{n} we have N_{αβ} ∈ {±1} and therefore Prop. 6.7 holds for any ring R.

7. Vanishing of SK_{1}

In this section we consider the case R = ℤ_{p} for p ≠ 2, 3. Let 𝔤 be a ℚ_{p}-split reductive Lie algebra and 𝔤_{ℤ} ⊆ 𝔤 a Chevalley order. Then, for any n ≥ 1, the ℤ_{p}-Lie algebra 𝔤_{ℤ_{p}}(p^{n}) is saturated in the sense of Lazard and even powerful in the sense of [DDMS], i.e., torsionfree as ℤ_{p}-module and satisfying [𝔤_{ℤ_{p}}(p^{n}), 𝔤_{ℤ_{p}}(p^{n})] ⊆ p𝔤_{ℤ_{p}}(p^{n}). Hence, by [DDMS] Thm. 9.10, there is a (up to unique isomorphism) unique uniform p-adic Lie group G(p^{n}) with ℤ_{p}-Lie algebra

L(G(p^{n})) = 𝔤_{ℤ_{p}}(p^{n}) .

Using Thm. 4.5 and Remark 6.5 we obtain our last main result.

Theorem 7.1. In the above setting we have

SK_{1}(Λ(G(p^{n}))) = 0 .

We briefly describe how these uniform groups arise as subgroups of ℚ_{p}-points of an algebraic group. To this end let G be a split reductive group scheme over ℤ (cf. [SGA3] Exp. XIX Def. 2.7 and Exp. XXII Def. 1.13; we recall that this includes the requirements that G is affine and smooth with connected fibers). Being smooth the group scheme G posseses a ℤ-Lie algebra 𝔤_{ℤ}. Its base change 𝔤 := ℚ_{p} ⊗_{ℤ} 𝔤_{ℤ} is the Lie algebra of the ℚ_{p}-split reductive algebraic group G_{ℚ_{p}} := ℚ_{p} ×_{ℤ} G and therefore is a ℚ_{p}-split reductive Lie algebra. At the same time 𝔤 = Lie_{ℚ_{p}}(G(ℚ_{p})) is the Lie algebra of the (noncompact) p-adic Lie group G(ℚ_{p}) of ℚ_{p}-rational points of G. According to [Jan] §II.1.1/11/12 the ℤ-Lie algebra 𝔤_{ℤ} indeed is a Chevalley order of 𝔤. We introduce the open pro-p subgroups

G(p^{n}) := ker (G(ℤ_{p}) → G(ℤ/p^{n}))

for n ≥ 1. It is shown in [HKN] Ex. 2.6.8 (note that they write L^{∗} for our L here) that we have

pℤ_{p} ⊗_{ℤ} 𝔤_{ℤ} = L(G(p)) .

The pro-p group G(p) can be identified with the standard group Ĝ(pℤ_{p}) associated with the formal group which arises as formal completion Ĝ of G along its unit section. In particular G(p) is uniform (cf. [DDMS] Thm. 8.31). More generally the proof of loc. cit. shows that the G(p)^{p^{n−1}} = Ĝ(p^{n}ℤ_{p}) = G(p^{n}) form the lower p-series of G(p). Hence, since the p-power map on the group coincides with the multiplication by p on the Lie algebra we obtain

L(G(p^{n})) = p^{n−1}L(G(p)) = p^{n}Lie(G_{ℤ_{p}})

for any n ≥ 1.

We finish this section with the following criterion.

Remark 7.2. For any uniform G such that its Lie algebra Lie_{ℚ_{p}}(G) is ℚ_{p}-split reductive we have SK_{1}(Λ(G)) = 0 if and only if ⋀^{2} L(G)/ ⋀ L(G) is torsion free.

Proof. Let 𝔤 := Lie_{ℚ_{p}}(G) and L := L(G). By Thm. 4.5 and Thm. 6.4 the vanishing of SK_{1}(Λ(G)) is equivalent to the inclusion ⋀ 𝔤 ∩ ⋀^{2} L ⊆ ⋀ L. But

(⋀ 𝔤 ∩ ⋀^{2} L)/ ⋀ L = tors(⋀^{2} L/ ⋀ L) .

□

References

[B-LL] Bourbaki N.: Lie Groups and Lie Algebras. Berlin - Heidelberg - New York: Springer 1998

[SGA3] Demazure M., Grothendieck A.: Schémas en Groupes III. Lect. Notes Math. 153, Berlin - Heidelberg - New York: Springer 1970

[DDMS] Dixon J.D., du Sautoy M.P.F., Mann A., Segal D.: Analytic Pro-p-Groups. Cambridge Univ. Press 1999

[GS] González-Sánchez J.: On p-saturable groups. J. Algebra 315, 809-823 (2007)

[HKN] Huber A., Kings G., Naumann N.: Some complements to the Lazard isomorphism. Compositio Math. 147, 235-262 (2011)

[Jan] Jantzen J. C.: Representations of algebraic groups, 2nd ed. AMS 2003

[Kak] Kakde M.: The main conjecture of Iwasawa theory for totally real fields. Preprint 2011

[Ko] Kostant B.: Eigenvalues of a Laplacian and commutative subalgebras. Topology 3, 147-159 (1965)

[Laz] Lazard M.: Groupes analytiques p-adique. Publ. Math. IHES 26, 389-603 (1965)

[NSW] Neukirch J., Schmidt A., Wingberg K.: Cohomology of number fields. 2nd ed. Berlin - Heidelberg - New York: Springer 2008

[Oli] Oliver R.: Whitehead groups of finite groups. London Math. Soc. Lect. Notes 132. Cambridge Univ. Press 1988

[SV] Schneider P., Venjakob O.: A splitting for K_{1} of completed group rings. To appear in Comment. Math. Helvetici.

[Ser] Serre J.-P-: Algèbre Locale Multiplicités. Lect. Notes Math. 11. Berlin - Heidelberg - New York: Springer 1975

[Wei] Weibel C.: An introduction to homological algebra. Cambridge Univ. Press 1994

Universität Münster, Mathematisches Institut, Einsteinstr. 62, 48291 Münster, Germany, http://www.uni-muenster.de/math/u/schneider/

   E-mail address: pschnei@math.uni-muenster.de

Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany, http://www.mathi.uni-heidelberg.de/˜ venjakob/

   E-mail address: venjakob@mathi.uni-heidelberg.de