Algebraic inclusions of Moufang polygons

Abstract

An inclusion of a Moufang polygon into another is called algebraic if the algebraic structures which describe them can be chosen in such a way that the one is a substructure of the other. We show that an inclusion of Moufang n-gons is always algebraic if n ∈ {3, 6, 8}, but that this is not always true when n = 4. We classify the algebraic inclusions of Moufang quadrangles in the case where none of the root groups is 2-torsion, which corresponds to the fact that the characteristic of the underlying (skew) field is different from 2. Finally, we show that all full and ideal inclusions of Moufang quadrangles without 2-torsion root groups are algebraic.

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Algebraic inclusions of Moufang polygons
Tom De Medts∗
September 2, 2003
Abstract
An inclusion of a Moufang polygon into another is called algebraic if
the algebraic structures which describe them can be chosen in such a way
that the one is a substructure of the other. We show that an inclusion of
Moufang n-gons is always algebraic if n ∈ {3,6,8}, but that this is not
always true when n = 4. We classify the algebraic inclusions of Moufang
quadrangles in the case where none of the root groups is 2-torsion, which
corresponds to the fact that the characteristic of the underlying (skew)
field is different from 2. Finally, we show that all full and ideal inclusions
of Moufang quadrangles without 2-torsion root groups are algebraic.
MSC-2000 : primary: 51E12, secondary: 08A30, 20E42, 51E24
1 Introduction
A generalized polygon is a rank-2 incidence geometry the incidence graph of
which has diameter n and girth 2n for some n ≥ 3 (and is then called a gener-
alized n-gon). A generalized polygon is in fact the same as a rank-2 spherical
building, and there is a vast literature on these objects. In many circumstances,
one is interested in subpolygons of a given generalized polygon, for various rea-
sons. To mention a few, they are used in characterizations of certain of these
polygons, they can be used to discover or describe other interesting structures
(such as spreads or ovoids), or they can be used in inductive arguments, for
example to study embeddings of generalized polygons in projective spaces or
other higher rank buildings.
A bit surprising, not too much has been written down on the study of sub-
polygons by itself. In the finite case, there are some results involving the order
of the polygons; see, for example, [10, section 1.8]. The case of the classical
compact connected polygons has been dealt with in [11].
In this paper, we will be interested in the case of the generalized polygons
satisfying the Moufang condition. Although this condition looks rather restric-
tive, it is satisfied quite often, and in particular, all classical polygons belong to
this class. Moreover, the Moufang polygons have been classified in [9] — but
there is no hope to be able to classify all generalized polygons, since there exist
free constructions, and even the finite case is still wide open. Two small pieces
∗The author is a Research Assistant of the Fund for Scientific Research - Flanders (Belgium)
(F.W.O.-Vlaanderen).
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of the study of the inclusion of Moufang polygons have already been done be-
fore, namely the inclusion of Moufang octagons [5] and a study of the Moufang
polygons which do not have full or ideal subpolygons [10, section 5.9]. It is also
noteworthy that Moufang polygons (and spherical buildings of arbitrary rank in
general) play an important role in algebraic group theory and related subjects,
so our results might have consequences on the existence of subgroups of those
groups.
We will start by recalling some definitions and notations, and prepare the
setup for our algebraic approach to the problem. Then we will be dealing with
the cases of Moufang triangles, hexagons and octagons, which can be completely
settled by taking a closer look at the proof of the classification of Moufang
polygons in [9]. It goes without saying that we will have to rely heavily on
this book. The case of Moufang quadrangles is significantly harder, and it turns
out that the inclusion of Moufang polygons does not always translate nicely into
the inclusion of the corresponding algebraic structures. However, in many cases,
it does, and in particular in the case that the characteristic of the underlying
(skew) field is not 2, we show that all inclusions are either algebraic or dual
(see below for the exact definitions of these expressions). In the section which
follows, we then classify all algebraic inclusions, with the only restriction that
we do not allow the characteristic to be equal to 2 — a case which seems to
be much harder (although many of our results can be extended to this case as
well). In the last section, we describe all full and ideal subquadrangles of a given
Moufang quadrangle, and we show that our list is complete.
2Preliminaries
We start with some definitions.
Definition 2.1. A generalized n-gon is a connected bipartite graph with diameter
n and girth 2n. A generalized polygon is a generalized n-gon for some finite
n ≥ 2. A generalized polygon Γ is called thick if |Γx| ≥ 3 for all vertices x of Γ.
A circuit of Γ of length 2n is called an apartment of Γ. A path of length n in Γ
is called a root or a half-apartment of Γ.
Definition 2.2. If α = (v0,...,vn) is a root of a generalized n-gon Γ, then the
group of all automorphisms of Γ which fix all the vertices of Γv1∪···∪Γvn−1is
called a root group of Γ (corresponding to the root α) and is denoted by Uα. If
Uαacts regularly on the set of apartments through α, then α is called a Moufang
root. If all roots of Γ are Moufang roots, then Γ is called a Moufang n-gon.
From now on, we assume that Γ is a thick Moufang n-gon for some n ≥ 3,
and we will fix an (arbitrary) apartment Σ which we label by the integers modulo
2n such that i+1 ∈ Γiand i+2 ?= i for all i. We define Ui:= U(i,i+1,...,i+n)for
all i, and we set U[i,j]= ?Ui,Ui+1,...,Uj? for all i ≤ j < i + n and U[i,i−1]= 1
for all i.
The definitions 2.3–2.6 were introduced in [9]. We present them in a different
but equivalent form.
Definition 2.3. LetˆU[1,n]be a group generated by non-trivial subgroupsˆU1,...,ˆUn
for some n ≥ 3. The (n + 1)-tuple (ˆU[1,n],ˆU1,...,ˆUn) is called a root group
sequence if there exists a Moufang n-gon Γ and a labeled apartment Σ =
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(0,...,2n − 1) in Γ such that there exists an isomorphism fromˆU[1,n]to U[1,n]
mappingˆUito Uifor all i ∈ {1,...,n}. We will denote this root group sequence
by Θ(Γ,Σ). The number n will be called the length of the root group sequence.
Definition 2.4. If Θ = (U[1,n],U1,...,Un) is a root group sequence, then (U[1,n],Un,...,U1)
is also a root group sequence. It is called the opposite of Θ and is denoted by
Θop.
Definition 2.5. Consider two root group sequences Θ = (U[1,n],U1,...,Un) and
Θ?= (U?
U[1,n]to U?
from Θ to Θ?is an isomorphism from Θ to Θ?op.
Definition 2.6. Let Θ = (U[1,n],U1,...,Un) be a root group sequence. For
each i ∈ {1,...,n}, let U?
denote the subgroup of U[1,n]generated by U?
(U?
sequence of Θ.
Recently, the classification of Moufang polygons has been completed by J.
Tits and R. Weiss in [9]. The following theorem is essential.
[1,n],U?
[1,n]mapping Ui to U?
1,...,U?
n). An isomorphism from Θ to Θ?is an isomorphism from
ifor all i ∈ {1,...,n}. An anti-isomorphism
ibe a non-trivial subgroup of Ui, and let U?
1,...,U?
n) is again a root group sequence, then Θ?will be called a sub-
[1,n]
n. If the n-tuple Θ?=
[1,n],U?
1,...,U?
Theorem 2.7. Let Γ be an arbitrary Moufang n-gon. Then:
(i) n ∈ {3,4,6,8}.
(ii) Let Σ = (0,...,2n − 1) be an arbitrary apartment of Γ. Then up to
isomorphism, Γ is uniquely determined by the isomorphism class of its root
group sequence Θ(Γ,Σ) = (U[1,n],U1,...,Un). We denote this Moufang n-
gon by by Γ(Θ).
(iii) If Θ1and Θ2are two root group sequences such that Γ(Θ1)∼= Γ(Θ2), then
Θ1and Θ2are isomorphic or anti-isomorphic.
Proof.(i) See [9, (17.1)].
(ii) See [9, (7.6) and (7.7)].
(iii) Suppose that Θ1 = Θ(Γ1,Σ1) and Θ2 = Θ(Γ2,Σ2) for some Moufang
n-gons Γ1 and Γ2 and some apartments Σ1= (0,...,2n − 1) and Σ2=
(0?,...,(2n−1)?) of Γ1and Γ2, respectively. It follows from (ii) that Γ1∼=
Γ(Θ1) and Γ2∼= Γ(Θ2), and hence Γ1∼= Γ2. Let φ be an isomorphism from
Γ1to Γ2, then φ maps Σ1to some apartment φ(Σ1) = (φ(0),...,φ(2n−1))
of Γ2. By [9, (4.12)], there exists an automorphism ψ of Γ2which maps
φ(Σ1) to Σ2and maps the edge (φ(n),φ(n +1)) to the edge (n?,(n+1)?).
So ψ ◦ φ maps Σ1to Σ2, and either it maps i to i for all i, in which case
Θ1and Θ2are isomorphic, or it maps i to 2n + 1 − i for all i, in which
case Θ1and Θ2are anti-isomorphic.
The following theorem defines the fundamental µ-maps, which play a very
important role in the whole theory of Moufang polygons, in particular for the
Moufang sets defined by opposite root groups in a Moufang polygon.
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Theorem 2.8. For each i and each ai ∈ U∗
µ(ai) ∈ U∗
element µ(ai) fixes i and i + n and reflects Σ, and Uµ(ai)
ai∈ U∗
Proof. See [9, (6.1)].
i, there exist a unique element
i+naiU∗
i+nsuch that (i − 1)µ(ai)= i + 1 and (i + 1)µ(ai)= i − 1. This
j
= U2i+n−j for each
iand each j.
By the following theorem, the study of subpolygons of Moufang polygons is
equivalent to the study of subsequences of root group sequences. Nevertheless,
we will still use the polygons, since the µ-maps which we get from Theorem 2.8
will turn out to be very useful.
Theorem 2.9.
Γ2. Then Γ1 is also a Moufang n-gon. If α is an arbitrary root of Γ1,
with corresponding root groups U(1)
α
subgroup of U(2)
In particular, let Σ be an arbitrary labeled apartment of Γ1, then Θ1:=
Θ(Γ1,Σ) is a subsequence of Θ2:= Θ(Γ2,Σ).
(i) Let Γ2be a Moufang n-gon and let Γ1be a sub-n-gon of
of Γ1and U(2)
α
of Γ2, then U(1)
α
is a
α .
(ii) Let Θ2be a root group sequence and let Θ1be a subsequence of Θ2. Then
Γ(Θ1) is isomorphic to a subpolygon of Γ(Θ2).
Proof. (i) The fact that Γ1is again Moufang is well known; see, for example,
[10, Lemma 5.2.2].
Consider an arbitrary root α of Γ1, and its corresponding root groups U(1)
of Γ1and U(2)
α
of Γ2. Let Σaand Σbbe two apartments of Γ1through α.
Then there is a unique element φ of U(2)
the subgraph ∆ := Γ1∩ φ(Γ1). Since Γ1and φ(Γ1) have the apartment
Σbin common, their intersection ∆ is again a generalized n-gon (see, for
example, [10, Proposition 1.8.4]). Since φ is a root elation, it fixes at least
one pencil and at least one point row of Γ1. It follows (see, for example,
[10, Proposition 1.8.1]) that ∆ is a full and ideal subpolygon of both Γ1
and φ(Γ1), and hence (see, for example, [10, Proposition 1.8.2]) ∆, Γ1and
φ(Γ1) coincide. We conclude that φ stabilizes Γ1, and hence its restriction
to Γ1must be the unique element of U(1)
holds for every pair of apartments Σaand Σb of Γ1through α, we have
shown that every element of U(1)
α
is the restriction of a unique element of
U(2)
α
to Γ1. Hence U(1)
α
is a subgroup of U(2)
α
α
mapping Σato Σb. Now consider
α
mapping Σato Σb. Since this
α , for all roots α of Γ1.
(ii) It follows readily from the construction in [9, (7.1) and (7.2)] that the
vertex set X1of Γ(Θ1) can be canonically identified with a subset of the
vertex set X2of Γ(Θ2), and that any two elements x,y ∈ X1which are
adjacent in Γ(Θ1) are also adjacent in Γ(Θ2). It follows that any two
elements x,y ∈ X1which have distance i in Γ(Θ1) also have distance i in
Γ(Θ2), for all i ∈ {0,...,n}; in particular, two elements x,y ∈ X1which
are non-adjacent in Γ(Θ1) are also non-adjacent in Γ(Θ2). We conclude
that Γ(Θ1) is isomorphic to a subpolygon of Γ(Θ2).
Theorem 2.10. If Θ = (U[1,n],U1,...,Un) is a root group sequence, then
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(i) [Ui,Uj] ≤ U[i+1,j−1]for all 1 ≤ i < j ≤ n;
(ii) The product map from U1× ··· × Unto U[1,n]is bijective.
Proof. See [9, (5.5) and (5.6)].
Definition 2.11. Let ai∈ Uiand aj ∈ Uj with i + 2 ≤ j < i + n. For each k
such that i < k < j, we set [ai,aj]k= ak, where akis the unique element of Uk
appearing in the factorization of [ai,aj] ∈ U[i+1,j−1].
Lemma 2.12. Let Γ2 be a Moufang n-gon and let Γ1 be a sub-n-gon of Γ2;
let Σ be an arbitrary labeled apartment of Γ1. Then the µ(1)-maps defined by
Theorem 2.8 with respect to Γ1are the restriction of the µ(2)-maps defined with
respect to Γ2, to the root groups of Γ1.
Proof. Note that, by Theorem 2.9.(i), the root groups of Γ1are indeed subgroups
of the root groups of Γ2, so the statement of this lemma makes sense.
But this same fact implies that every element µ(1)(ai) ∈ (U(1)
with ai∈ (U(1)
of the µ(2)-maps in Theorem 2.8, this element has to be equal to µ(2)(ai).
i+n)∗·ai·(U(1)
i+n)∗
i
)∗is also an element of (U(2)
i+n)∗·ai·(U(2)
i+n)∗, and by the uniqueness
For each possible value of n, we will now use the appropriate algebraic struc-
ture to describe an arbitrary Moufang n-gon, and we will redo certain steps of
the classification of Moufang n-gons, but for both the Moufang n-gon and its
sub-n-gon simultaneously, and make some appropriate choices during the proof.
3Subtriangles of Moufang triangles
We start with the study of all possible subtriangles of a given Moufang triangle.
This is the easiest case, since Moufang triangles have a very simple description,
as was already shown in 1933 (but in a slightly different form; see [1] or [3]) by
R. Moufang (see [7]):
Definition 3.1. Let (A,+,·) be an arbitrary alternative division ring, and let
U1, U2 and U3 be three groups parametrized by (A,+) via some (group) iso-
morphisms x1, x2 and x3. Let U+ be the group generated by U1, U2 and U3
with respect to the commutator relations
[U1,U2] = [U2,U3] = 1 ,
[x1(s),x3(t)] = x2(s · t) ,
for all s,t ∈ A. Then Θ = (U+,U1,U2,U3) is a root group sequence of length 3;
it is unique up to isomorphism (i.e., it does not depend on the choice of x1, x2
and x3), and will be denoted by ΘT(A). We also say that Θ is parametrized by
A via the isomorphisms x1, x2and x3.
Theorem 3.2. Let Γ be an arbitrary Moufang triangle. Then there exists an
alternative division ring (A,+,·) such that Γ∼= Γ(ΘT(A)).
Proof. See [9, (17.2)].
Theorem 3.3. Let Γ1and Γ2be two Moufang triangles. Then Γ1is isomorphic
to a subtriangle of Γ2 if and only if there exists an alternative division ring˜A
and a subring A of˜A such that Γ1∼= Γ(ΘT(A)) and Γ2∼= Γ(ΘT(˜A)).
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