Journal of Lie Theory Volume 11 (2001) 459–468 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

On a Special Class of Frobenius Groups admitting Planar Partitions

Peter Maier

Communicated by K. Strambach

Abstract. Among all Frobenius Lie groups having a complement isomorphic either to ℂ^{×} or to ℍ^{×} and a kernel which is a vector group those are determined that admit a planar partition into closed subgroups. Moreover, it is shown that for each of these groups the exponential function induces a bijection between the set of planar partitions of the group and the set of planar partitions of the associated Lie algebra.

1. Introduction

One of the reasons for studying groups with partitions is in doing linear incidence geometry, in particular, topological incidence geometry. If one takes a group G with a partition P, and sets G := {gH | g ∈ G, H ∈ P}, then the pair (G, G) is a linear space on which the group G acts as a group of automorphisms via left translation (the notion “linear space” simply means that for any two distinct “points” g, h ∈ G there exists exactly one “line” L ∈ G which contains both g and h). The structures obtained by this construction were first investigated by André in [1] and they are examples of point-regular geometries as considered in [11]. In dealing with stable planes (a special kind of topological linear spaces; see [6] for a definition) one is led to take G to be a Lie group and P a partition of G into closed subgroups of half dimension. Such partitions are called planar partitions. In [7] the author shows that for a Lie group G with a planar partition P the incidence structure (G, G) is a stable plane exactly if the induced partition LP := {LP | P ∈ P} of the Lie algebra LG is compact in the respective Grassmann topology.

So one has a construction method for point-regular stable planes which starts from a Lie group with planar partition, and now it remains to find Lie groups which admit such partitions. Thanks to the work of Plaumann and Strambach (cf. [8] and [9]) one already knows that such a group is either exponential (that is, the exponential function is a diffeomorphism) or is a Frobenius group whose kernel is a vector group and whose complements are isomorphic to ℂ^{×} or to ℍ^{×}.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag In the present paper the author classifies those of the aforementioned Frobenius groups which admit planar partitions.

Throughout this note, all Lie groups are assumed to be real Lie groups of finite dimension and all vector spaces over skewfields are assumed to be left vector spaces.

2. Partitions of Vector Spaces, Algebras, and Groups

Definition 2.1. By a partition of a vector space we mean a set of non-trivial subspaces which cover the whole space and which pairwise intersect trivially. Partitions of algebras and groups are defined analogously. By a partition of a Lie group we mean a partition into closed subgroups. A partition is called trivial if it consists of one set only.

Examples 2.2. (a) For any vector space the set of all 1-dimensional subspaces is a partition, and the same holds true for Lie algebras. So if one has a collection of subalgebras of a Lie algebra which pairwise intersect trivially one can extend this to a partition of the algebra by adding a suitable set of 1-dimensional subalgebras.

(b) Let F be a skewfield and V a left vector space over F. On the F -vector space F × V we define a bracket multiplication by

[(a, v), (b, w)] := a(b, w) − b(a, v).

Endowed with this multiplication F × V becomes a Lie algebra over the center Z(F) of F. We denote it by dil(V), or dil_{n}F if V = F^{n}. Algebras of this type are called dilatation algebras. As is easy to see from the definition, any F -subspace of dil(V) is a Lie subalgebra and hence, any partition of dil(V) into vector subspaces is a Lie algebra partition.

(c) Let F and V be as in (b). On the set F^{×} ×V we define a multiplication by

(a, v)(b, w) := (ab, aw + v).

Endowed with this multiplication F^{×} × V becomes a group which we denote by Dil(V), or Dil_{n}F if V = F^{n}. Groups of this type are called dilatation groups. It is easy to see that for an F -subspace U of F × V the set Ũ := ((1, 0) + U) ∩ Dil(V) is a subgroup of Dil(V) (cf. proof of Proposition 4.7). That means that for any partition 𝔓 of the Lie algebra dil(V) into F -subspaces the set P := {𝔭̃ | 𝔭 ∈ 𝔓} is a partition of the group Dil(V).

(d) If we assume V in example (c) to be finite dimensional over F and if we take F to be a locally compact connected skewfield (that is, F is isomorphic to one of the skewfields ℝ, ℂ, or ℍ), then the group Dil(V) is a Lie group and for any partition 𝔓 of the Lie algebra dil(V) into F -subspaces the set P := {𝔭̃ | 𝔭 ∈ 𝔓} is a partition of the Lie group Dil(V).

Since we are not interested in partitions of vector spaces, groups, or algebras in general but in those which give rise to geometric objects such as translation planes (see [6] for a definition) and stable planes, we now turn to so-called planar partitions. Before giving the definition we just note that any partition P of a Lie group G induces a partition LP := {LP | P ∈ P} of the associated Lie algebra LG (as is easy to deduce from the fact that the exponential function exp : LG → G is a local homeomorphism).

Definition 2.3. A partition of a finite-dimensional vector space or a Lie algebra is called planar if it consists of subspaces of half dimension. A partition of a Lie group is called planar if the induced partition of the Lie algebra is planar.

Obviously, planar partitions of vector spaces or Lie groups can only exist in even dimension, but at the present point, it is not at all clear whether in any even dimension there exists a Lie group or a vector space which admits a planar partition. Indeed, there is a result which states that any even-dimensional vector space admits a planar partition (cf. [2]), and clearly, this result covers the Lie group case, since any finite-dimensional real vector space is an abelian Lie group. So the question arises to classify all Lie groups which admit planar partitions. In dealing with this problem one is led to the following result due to Plaumann and Strambach (cf. [9]).

Theorem 2.4. (Plaumann–Strambach) Let G be a connected Lie group that admits a partition into subgroups of a fixed dimension d > 1. Then exactly one of the following holds:

(i) The exponential function is a diffeomorphism of LG onto G.

(ii) The group G is a Frobenius group whose kernel is a vector group and whose complement is isomorphic to ℂ^{×} or to ℍ^{×}.

In the remaining part of this note we are concerned with Frobenius groups mentioned in part (ii) of Theorem 2.4. Among these groups we classify exactly those which admit planar partitions. It turns out that, up to one exceptional class, all of these groups are dilatation groups over the quaternions or the complex numbers. Moreover, it is shown that for any such group the planar partitions are exactly the exponential images of the planar partitions of the corresponding Lie algebra.

3. A Result on Invariant Partitions

In the sequel we answer the following question: Consider a finite dimensional vector space V and an endomorphism φ of V . Under which conditions on φ the vector space V admits a partition into φ-invariant subspaces? The problem can also be formulated in terms of modules. This will be done in the proof of the following theorem.

Theorem 3.1. Let F be a field, V a finite dimensional F -vector space, and let φ be a non-zero endomorphism of V . Then the vector space V admits a nontrivial partition into φ-invariant subspaces if, and only if, the subring F [φ] of End_{F} V is a field, and dim_{F} F [φ] < dim_{F} V .

Proof. We set R := F [x] and define a F -algebra morphism ev_{φ} : R → End_{F} V by sending x to φ. With respect to this morphism the vector space V becomes an R-module and the φ-invariant subspaces are exactly the R-submodules of V .

Suppose first that F [φ] is a field and dim_{F} F [φ] < dim_{F} V . Then V is a F [φ]-vector space and because of dim_{F} F [φ] < dim_{F} V the set of all 1-dimensional F [φ]-subspaces is a non-trivial partition of V into R-submodules.

Now suppose that V admits a non-trivial partition P into R-submodules. As R is a principal ideal domain and V is a finitely generated R-module it is a direct sum of non-trivial cyclic submodules Rv_{1}, …, Rv_{n} (cf. [5]). We pick such a decomposition

V = Rv_{1} ⊕ ⋯ ⊕ Rv_{n}

which is maximal with respect to the number of summands, and claim that all these summands are isomorphic simple R-modules. If this is true, then ker ev_{φ} is a maximal ideal in R and F [φ] is a field, since it is isomorphic to R/ ker ev_{φ}. So it remains to prove the claim. We do this by induction on n, starting with n = 2, since a cyclic R-module only admits the trivial partition.

As P is a partition its elements cover V . Therefore we find P_{1}, P_{2} ∈ P containing v_{1} and v_{2}, respectively, and thus obtain Rv_{1} ⊆ P_{1} and Rv_{2} ⊆ P_{2}, because P_{1} and P_{2} are R-submodules of V . Since we have V = Rv_{1} ⊕ Rv_{2} and since P was supposed to be non-trivial this implies P_{1} = Rv_{1} and P_{2} = Rv_{2}. Denoting the annihilator of an element v ∈ V by Ann(v) the indecomposability of Rv_{1} and Rv_{2} implies that Ann(v_{k}) = (p_{k}^{n_{k}}) holds for suitable prime polynomials p_{k} ∈ R and natural numbers n_{k}. Now, we have to show n_{1} = n_{2} = 1 and (p_{1}) = (p_{2}). Without loss we assume n_{1} ≤ n_{2}. We distinguish three cases.

First case: (p_{1}) ≠ (p_{2}). In this case we have Ann(v_{1}) + Ann(v_{2}) = R. Taking into account that we also have Ann(v_{1} + v_{2}) = Ann(v_{1}) ∩ Ann(v_{2}) (since the sum Rv_{1} + Rv_{2} is direct) the chinese remainder theorem yields R(v_{1} + v_{2}) = Rv_{1} ⊕ Rv_{2} = V . So V is cyclic and therefore P has to be trivial, which was forbidden.

Second case: (p_{1}) = (p_{2}) and n_{2} > 1. Let w_{1} := p_{1}^{n_{1}−1}v_{1}. Since R(w_{1} + v_{2}) is cyclic it is contained in some P ∈ P. Because of {0} ≠ Rpv_{2} ⊆ R(w_{1} +v_{2})∩Rv_{2} we obtain P = P_{2}, which is impossible, since w_{1} + v_{2} ∉ P_{2}.

The remaining third case is the desired one. So we have Rv_{1} ≅ R/(p_{1}) = R/(p_{2}) ≅ Rv_{2} and the claim is proved for n = 2.

Now let n > 2 and assume the claim holds true for n − 1. We write Rv_{k} ≅_{s} Rv_{l} if Rv_{k} and Rv_{l} are isomorphic simple R-modules. Suppose there exist k, l such that Rv_{k} ≅_{s̸} Rv_{l}. Without loss let (k, l) = (1, 2). Then we have Rv_{1} ≅_{s̸} Rv_{3} or Rv_{2} ≅_{s̸} Rv_{3} and we can assume that Rv_{2} ≅_{s̸} Rv_{3} holds. Setting U := Rv_{1} ⊕ … Rv_{n−1} and W := Rv_{2} ⊕ … Rv_{n} the induction hypothesis yields that U and W only admit trivial partitions into R-modules and therefore we find P, Q ∈ P containing U and W, respectively. Since we have Rv_{2} ⊆ U ∩ W this implies P = Q and thus P = {P }, contradicting the fact that P is non-trivial.

Because an algebraically closed field admits no proper finite extension one immediately gets the following consequence of Theorem 3.1.

Corollary 3.2. Let F an algebraically closed field, V a finite dimensional F - vector space, and let φ be a non-zero endomorphism of V . Then the vector space V admits a non-trivial partition into φ-invariant subspaces if, and only if, the endomorphism φ is contained in the center of End_{F} V .

4. Application to Frobenius Lie Groups

Definition 4.1. By a Frobenius group we mean a group G that admits a semidirect decomposition G = KN where N is a normal subgroup of G, called the kernel of G, and K is a complement of N such that for any g ≠ h ∈ N we have gKg^{−1} ≠ hKh^{−1} and the set {gKg^{−1} | g ∈ N} together with N forms a partition of G. This partition is also called the natural partition of the group G. If G is supposed to be a Lie group we require K and N to be closed subgroups.

Remark 4.2. The condition in Definition 4.1 that gKg^{−1} ≠ hKh^{−1} holds for any two distinct elements g, h ∈ N implies that the action of K on N by conjugation is free, i.e., for any n ∈ N \ {1} we have {k ∈ K | knk^{−1} = n} = {1}. As we shall see in Example 4.3(b) this is sometimes sufficient for KN to be a Frobenius group. (Quite general, if a group G acts as a group of automorphisms on a group H, then we call this action free if for any element h ∈ H \ {1} the stabilizer is trivial.)

Examples 4.3. (a) Any dilatation group Dil(V) is a Frobenius group with kernel {1} × V .

(b) Example (a) is a special case of the following construction: Let V be a finite dimensional vector space over some skewfield F and let G ⊆ GL(V) be a non-trivial group that acts freely on V . Then the semidirect product G ⋉ V is a Frobenius group with kernel {1} × V . In order to see this, we have to show that for any element (g, v) ∈ (G \ {1}) × V there exists exactly one element w ∈ V such that

(g, v + (1 − g)w) = (1, w)(g, v)(1, w)^{−1} ∈ G × {0}.

By freeness of the action of G on V we obtain that 1 − g ∈ End_{F} V is injective and thus invertible. So w := −(1 − g)^{−1}v ∈ V is the unique element with the desired property.

(c) There is a generalisation of dilatation groups which one obtains by admitting F = (F, +, ·) in the definition of Dil_{n}F to be a nearfield (that is, (F, +) is an abelian group, (F^{×}, ·) is a group, a(x + y) = ax + ay and 0x = 0 hold for all a, x, y ∈ F). For any nearfield the set K(F) := {k ∈ F | (∀x, y ∈ F) (x + y)k = xk + yk} is a skewfield and F is a left vector space over K(F) (cf. [4]). If F is finite-dimensional over K(F), then Dil_{n}F is a Frobenius group with kernel {1} × F^{n}, which immediately follows from the discussion in Example (b).

(d) One can for each positive real number b define proper nearfield ℍ_{b} := (ℍ, ∘_{b}, +) by letting x ∘_{b} y := x|x|^{ib}y|x|^{−ib} for x, y ∈ ℍ. (A theorem due to Kalscheuer states that any locally compact connected nearfield which is not a skewfield is isomorphic to exactly one of the nearfields ℍ_{b} (cf. [10] 64.20).) Now, for any such nearfield ℍ_{b} the group Dil_{n}ℍ_{b} is a Frobenius Lie group (cf. [6] 3.1).

We call a linear representation λ : G → GL(V) free if the associated action of G on V is free in the sense of Remark 4.2. Throughout the remaining part of this note 𝔽 stands for the field of the complex numbers or the quaternions, furthermore, we write S_{𝔽} for the subgroup consisting of all elements of modulus 1 in the multiplicative group 𝔽^{×}.

Proposition 4.4. Let G be a Frobenius Lie group whose kernel is a vector group V and whose complements are isomorphic to S_{𝔽}. Then G is isomorphic to one of the groups S_{𝔽} ⋉ 𝔽^{n} where the action of S_{𝔽} on 𝔽^{n} is given by scalar multiplication. In particular, the kernel V is an 𝔽-vector space and for any subgroup H of G that contains a complement of G the subgroup H ∩ V is an 𝔽-subspace of V .

Proof. We have G ≅ S_{𝔽} ⋉_{λ} V with some representation λ : S_{𝔽} → GL(V). As S_{𝔽} is a compact group this representation is completely reducible whence we have a decomposition V = V_{1}⊕⋯⊕V_{n} into irreducible S_{𝔽}-modules. Now the assumption on G to be a Frobenius group implies that each of the respective subrepresentations is free. By looking at the irreducible representations of S_{ℂ} ≅ SO_{2}ℝ one gets that the only ones that are free are those which are equivalent to the natural representation on ℂ given by left multiplication.

The analogous result holds true for S_{ℍ} but it requires a little more work in order to prove it. We focus on V_{1}. By complexification we get a representation S_{ℍ} → GL_{ℂ}(V_{1} ⊗ ℂ), for which we have two possibilities: V_{1} ⊗ ℂ is an irreducible complex S_{ℍ}-module or it decomposes into two isomorphic irreducible complex submodules. In both cases the associated action of S_{ℍ} on V_{1} ⊗ ℂ is free if the action of S_{ℍ} on V_{1} is free. In the latter case even V_{1} has a complex structure and the real representation S_{ℍ} → GL(V_{1}) can be viewed as a complex representation with respect to this structure. So we have to deal with irreducible complex representations of S_{ℍ}. Since S_{ℍ} ≅ SU_{2}ℂ we are concerned with irreducible representations of SU_{2}ℂ, but these are well-known. Up to isomorphism they all can be obtained by the following construction (see [3] II.5): let ℂ[z_{1}, z_{2}] be the vector space of complex polynomials in two commuting variables and let P_{n}, n ∈ ℕ_{0}, be the subspace of homogeneous polynomials of degree n. Viewing the elements of ℂ[z_{1}, z_{2}] as functions on ℂ^{2} we can define an action of SU_{2}ℂ on ℂ[z_{1}, z_{2}] by letting gp(z) := p(g^{−1}z) where g ∈ SU_{2}ℂ, p ∈ ℂ[z_{1}, z_{2}], and z ∈ ℂ^{2}. Then P_{n} is an (n + 1)-dimensional irreducible SU_{2}ℂ-submodule. In order to see that only one of these irreducible submodules, namely P_{1}, leads to a free action, we argue as follows: clearly, the action of SU_{2}ℂ on P_{0} is not free. In order to see that this is likewise the case for P_{n} if n > 1, take p := (z ↦ z_{1}^{n}) ∈ P_{n} and g = diag(a, ā) ∈ SU_{2}ℂ where a ∈ ℂ \ {1} is an n-th root of unity. Then we have gp = p but g ≠ 1.

Since there is no 2-dimensional real representation of SU_{2}ℂ whose complexification could yield the aforementioned representation on P_{1}, we get that the action of SU_{2}ℂ on P_{1} is the only free action of SU_{2}ℂ on a real vector space. By uniqueness, this action of SU_{2}ℂ on P_{1} is equivalent to the action of S_{ℍ} on ℍ by left multiplication. So an irreducible ℍ^{×}-module decomposes into irreducible S_{ℍ}- submodules which we can identify with ℍ and on which S_{ℍ} acts via multiplication from the left.

The next result is a consequence of Result 6.3 in [8]. Nevertheless, we give here a direct proof.

Lemma 4.5. The Lie groups ℂ^{×} and ℍ^{×} only admit the trivial partition.

Proof. Let P be a partition of ℂ^{×}. If P were non-trivial then it would consist of the images of all one-parameter subgroups of ℂ^{×}. Because a one-parameter subgroup of ℂ^{×} is given by a homomorphism α_{c} : ℝ → ℂ^{×} : t ↦ e^{ct}, where c = a + bi ∈ ℂ, this would imply \{e^{d} | d ∈ ℕ\frac{2πa}{b}\} ⊆ α_{c}(ℝ) ∩ α_{1}(ℝ) for c ∈ ℂ \ ℝ, contradicting that fact that the elements of P intersect trivially.

Now let P be a partition of ℍ^{×}. Using what we have already shown, we obtain that for any h ∈ ℍ^{×} the subgroup C_{h} := hℂ^{×}h^{−1} of ℍ^{×} is contained in some P_{h} ∈ P, and this implies that P is trivial, since we have ⋂{C_{h} | h ∈ ℍ^{×}} = ℝ^{×} and ⋃{C_{h} | h ∈ ℍ^{×}} = ℍ^{×}.

Proposition 4.6. Let G be a Frobenius Lie group whose kernel is a vector group V and whose complements are isomorphic to 𝔽^{×}. Then any planar partition of G induces a partition of V into 𝔽^{×}-invariant subspaces.

Proof. Let G = 𝔽^{×} ⋉_{λ} V with some representation λ : 𝔽^{×} → GL(V) and assume that G admits a planar partition P. Since ℂ^{×} and ℍ^{×} only admit the trivial partition any element of P which is not contained in V contains a complement of G. As all complements of G are conjugate to 𝔽^{×} by an element of V , and as V is abelian, this implies that for any Q ∈ Q := {P ∈ P | P ⊈ V } the subgroup Q ∩ V is an 𝔽^{×}-invariant subspace of V . So it remains to show that any P ∈ P \ Q is also an 𝔽^{×}-invariant subspace of V . Since dim P + dim P′ = dim G > dim V holds for all P, P′ ∈ P the set P \Q contains at most one element and hence, the assertion now follows from the fact that ⋃{Q ∩ V | Q ∈ Q} is an 𝔽^{×}-invariant subset of V .

If one views the group Dil_{n}𝔽 as a group of matrices by identifying the

element (a, v) with the matrix \binom{a1_{n} v}{1}, and doing the analogue for the

Lie algebra dil_{n}𝔽, one sees that dil_{n}𝔽 is the Lie algebra of Dil_{n}𝔽 and that the exponential function is given by

exp(a, v) = \{ \begin{matrix}(e^{a}, (e^{a} − 1)a^{−1}v) & if a ≠ 0 \\ (1, v) & otherwise \end{matrix} .

Using this, easy calculations show that exp : dil_{n}𝔽 → Dil_{n}𝔽 is surjective and that

exp 𝔽x ⊆ (1, 0) + 𝔽x

holds for each x ∈ dil_{n}𝔽 (see [6] 3.1 for the details). Both facts together imply that

exp 𝔽x = ((1, 0) + 𝔽x) ∩ Dil_{n}𝔽  (1)

holds for each x ∈ dil_{n}𝔽, since the 1-dimensional 𝔽-subspaces of dil_{n}𝔽 form a partition of dil_{n}𝔽.

Proposition 4.7. If U is an 𝔽-subspace of the Lie algebra dil_{n}𝔽, then we have exp U = ((1, 0) + U) ∩ Dil_{n}𝔽 and exp U is a closed subgroup of Dil_{n}𝔽. In particular, if 𝔓 is any partition of the Lie algebra dil_{n}𝔽 into 𝔽-subspaces, then the set exp 𝔓 := {exp 𝔭 | 𝔭 ∈ 𝔓} is a partition of the Lie group Dil_{n}𝔽.

Proof. The fact that exp U = ((1, 0) + U) ∩ Dil_{n}𝔽 holds for any 𝔽-subspace U of dil_{n}𝔽 is an immediate consequence of equation (1), since any 𝔽-subspace is cover by its 1-dimensional 𝔽-subspaces. In order to see that exp U is a group, we pick (1 + a, v), (1 + b, w) ∈ exp U and compute

\begin{aligned}(1 + a, v)(1 + b, w) & = & ((1 + a)(1 + b), (1 + a)v + u) \\ & = & (1, 0) + (a, u) + (1 + a)(b, v), \end{aligned}

and

(1 + a, v)^{−1} = ((1 + a)^{−1}, −(1 + a)^{−1}v) = (1, 0) − (1 + a)^{−1}(a, v).

Since U is an 𝔽-subspace and a, b ≠ −1 both results are contained in exp U.

Theorem 4.8. Let 𝔽 ∈ {ℂ, ℍ} and let G be a Frobenius Lie group whose kernel is a vector group and whose complements are isomorphic to 𝔽^{×}. If G admits a planar partition, then exactly one of the following holds:

(i) The group G is isomorphic to one of the groups Dil_{n}𝔽.

(ii) The group G is isomorphic to one of the groups Dil_{1}ℍ_{b}. Moreover, the planar partitions of Dil_{n}𝔽 are exactly the exponential images of the planar partitions of dil_{n}𝔽, which all consist of 𝔽-subspaces. The only planar partitions of the groups Dil_{1}ℍ_{b} are the natural ones.

Proof. Let G = 𝔽^{×} ⋉_{λ} V and assume that G admits a planar partition P. Then, in view of Proposition 4.4 and Proposition 4.6, the kernel V is an 𝔽-vector space and the set P_{V} := {P ∩ V | P ∈ P} is a partition of V into 𝔽^{×}-invariant vector subspaces. Identifying 𝔽^{×} with ℝ × S_{𝔽} via the map (t, s) ↦ e^{t}s we obtain that α := λ|_{ℝ} : ℝ → GL(V) is a one-parameter group with im α ⊆ GL_{𝔽}V and α(t)P ⊆ P for any P ∈ P_{V} and any t ∈ ℝ. Hence, there exists some X ∈ End_{𝔽}V such that α(t) = exp(tX) for any t ∈ ℝ. We claim that P_{V} is an X -invariant partition, and in order to prove this, we pick a non-zero vector v ∈ V . As P_{V} is a partition of V there exists exactly one element P_{v} ∈ P_{V} that contains v. Since P_{v} is invariant under α(ℝ) it contains the image of the curve α_{v} : ℝ → V : t ↦ exp(tX)v and since it is closed in V it also contains the tangent vector Xv = α̇_{v}(0), as was to be shown. Now we set n := dim_{𝔽} V and distinguish several cases.

Let first 𝔽 = ℂ. If n = 1 then we have End_{ℂ}V = ℂid_{V} and thus X ∈ ℂid_{V} . If n > 1 then Corollary 3.2 applies and yields that X is contained in ℂid_{V} . Consequently, α is of the form t ↦ e^{ct}id_{V} for some c ∈ ℂ. In fact, we even have c ∈ ℂ \ ℝi, because of the injectivity of α. Now the map (t, s) ↦ e^{ct}s induces an automorphism of ℂ^{×}, and this automorphism again induces an isomorphism of G = ℂ^{×} ⋉_{λ} V onto Dil_{n}ℂ.

Now let 𝔽 = ℍ and n > 1. As V is a ℍ-vector space it is a vector space over any subfield K of ℍ. We regard ℂ as a subfield of ℍ and set K := {hℂh^{−1} | h ∈ ℍ^{×}}. Obviously, we have inclusions GL_{ℍ}V ⊆ GL_{K}V ⊆ GL(V) for any K ∈ K. As in case 𝔽 = ℂ we obtain α(ℝ) ⊆ K^{×}id_{V} for any K ∈ K, and hence

α(ℝ) ⊆ ⋂_{K∈K} K^{×}id_{V} = ℝ^{×}id_{V}

since ⋂ K = ℝ. Consequently, α is of the form t ↦ e^{rt}id_{V} for some r ∈ ℝ. Now, in analogy with the preceding case, we obtain that the map (t, s) ↦ e^{rt}s : ℝ × S_{ℍ} → ℍ^{×} induces an isomorphism of G = ℍ^{×} ⋉_{λ} V onto Dil_{n}ℍ.

It remains to treat the case where 𝔽 = ℍ and n = 1. Identifying V with ℍ, we obtain End_{ℍ}V = {ρ_{h} | h ∈ ℍ}, where ρ_{h} denotes multiplication from the right with h. Since the map h ↦ ρh̄ : ℍ → End_{ℍ}V is an isomorphism of ℝ-algebras the one-parameter group α : ℝ → GL_{ℍ}V is determined by a continuous injective homomorphism β : ℝ → ℍ^{×}. Such a homomorphism is of the form t ↦ e^{ht} for some h ∈ ℍ. Because of the injectivity of β, we even have h ∈ ℍ \ Pu ℍ, where Pu ℍ denotes set of pure quaternions. As a commutative real subalgebra of ℍ the algebra ℝ[e^{ht}] is isomorphic to ℝ or to ℂ and hence, after conjugation with a suitable quaternion, we can assume ℝ[e^{ht}] ⊆ ℂ. Thus we have h = a + bi ∈ ℂ \ ℝi and by rescaling we can achieve a = 1. Furthermore, we can assume b ≥ 0, if necessary by replacing the representation α by the equivalent real representation ᾱ : t ↦ κ ∘ α(t) ∘ κ, where κ : ℍ → ℍ : x ↦ x̄ denotes conjugation in ℍ. Now the action of ℍ^{×} on ℍ is given by (a, v) ↦ a|a|^{−1}v\overline{|a|^{h}} = av|a|^{−bi}. For b = 0 the group G is obviously isomorphic with Dil_{1}ℍ. If b > 0 then the map (a, v) ↦ (a|a|^{−bi}, v) from ℍ^{×} × ℍ onto itself gives an isomorphism of the group G onto Dil_{1}ℍ_{b}.

Let 𝔓 be a planar partition of dil_{n}𝔽. We show that 𝔓 consists of 𝔽subspaces. Because of the dimension formula there exists at most one element of 𝔓 which is contained in the subspace V := {0} × 𝔽^{n}, and for any element 𝔭 ∈ 𝔓 which is not contained in V we obtain that the projection of 𝔭 onto 𝔽 × {0} along V is surjective. Noting this and that [(a, 0), v] = av holds for any a ∈ 𝔽 and any v ∈ V one sees as in the proof of Proposition 4.6 that the set 𝔓_{V} := {𝔭 ∩ V | 𝔭 ∈ 𝔓} consists of 𝔽-subspaces of V . Let 𝔭 ∈ 𝔓 be an element which is not contained in V and let (1, v) ∈ 𝔭 \ V . For any a ∈ 𝔽 we find w ∈ 𝔽^{n} such that (a, w) ∈ 𝔭. Now we compute (0, w −av) = [(1, v), (a, w)] ∈ 𝔭 and obtain a(1, v) = (a, w) − (0, w − av) ∈ 𝔭, as desired. Since 𝔓 consists of 𝔽-subspaces of dil_{n}𝔽 its exponential image exp 𝔓 is a planar partition of Dil_{n}𝔽, according to Proposition 4.7.

If P is a planar partition of Dil_{n}𝔽, then LP is a planar partition of dil_{n}𝔽, and thus consists of 𝔽-subspaces. So we can apply Proposition 4.7 and obtain P = exp LP.

The rest follows from the fact that for any of the groups Dil_{1}ℍ_{b} the complements only admit the trivial partition, since they all are isomorphic with ℍ^{×}.

References

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Peter Maier Fachbereich Mathematik Technische Universität Darmstadt Schloßgartenstr. 7 D-64289 Darmstadt maier@mathematik.tu-darmstadt.de

Received June 19, 2000