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Tom De Medts

Publications and source records attributed to Tom De Medts.

33 records · Page 2Linked to original sources
arXiv2017

Structurable algebras of skew-dimension one and hermitian cubic norm structures

We study structurable algebras of skew-dimension one. We present two different equivalent constructions for such algebras: one in terms of non-linear isotopies of cubic norm structures, and one in terms of hermitian cubic norm structures. After this work was essentially finished, we became aware of the fact that both descriptions already occur in (somewhat hidden places in) the literature. Nevertheless, we prove some facts that had not been noticed before: (1) We show that every form of a matrix structurable algebra can be described by our constructions; (2) We give explicit formulas for the norm $ν$; (3) We make a precise connection with the Cayley-Dickson process for structurable algebras.

math.RA↗
arXiv2017

Tits Endomorphisms and Buildings of Type $F_4$

The fixed point building of a polarity of a Moufang quadrangle of type $F_4$ is a Moufang set, as is the fixed point building of a semi-linear automorphism of order $2$ of a Moufang octagon that stabilizes at least two panels of one type but none of the other. We show that these two classes of Moufang sets are, in fact, the same, that each member of this class can be constructed as the fixed point building of a group of order $4$ acting on a building of type $F_4$ and that the group generated by all the root groups of any one of these Moufang sets is simple.

math.GR↗
arXiv2017

Local Moufang sets and PSL_2 over a local ring

We introduce local Moufang sets as a generalization of Moufang sets. We present a method to construct local Moufang sets from only one root group and one permutation. We use this to describe $\mathsf{PSL}_2$ over a local ring as a local Moufang set, and give necessary and sufficient conditions for a local Moufang set to be of this form.

math.GR↗
arXiv2016

Local Moufang sets and local Jordan pairs

In this paper, we extend the theory of special local Moufang sets. We construct a local Moufang set from every local Jordan pair, and we show that every local Moufang set satisfying certain (natural) conditions gives rise to a local Jordan pair. We also explore the connections between these two constructions.

math.GR↗
arXiv2016

Moufang sets and structurable division algebras

A Moufang set is essentially a doubly transitive permutation group such that each point stabilizer contains a normal subgroup which is regular on the remaining vertices; these regular normal subgroups are called the root groups, and they are assumed to be conjugate and to generate the whole group. It has been known for some time that every Jordan division algebra gives rise to a Moufang set with abelian root groups. We extend this result by showing that every structurable division algebra gives rise to a Moufang set, and conversely, we show that every Moufang set arising from a simple linear algebraic group of relative rank one over an arbitrary field k of characteristic different from 2 and 3 arises from a structurable division algebra. We also obtain explicit formulas for the root groups, the τ-map and the Hua maps of these Moufang sets. This is particularly useful for the Moufang sets arising from exceptional linear algebraic groups.

math.GR↗
arXiv2015

Jordan algebras and 3-transposition groups

An idempotent in a Jordan algebra induces a Peirce decomposition of the algebra into subspaces whose pairwise multiplication satisfies certain fusion rules $Φ(\frac{1}{2})$. On the other hand, $3$-transposition groups $(G,D)$ can be algebraically characterised as Matsuo algebras $M_α(G,D)$ with idempotents satisfying the fusion rules $Φ(α)$ for some $α$. We classify the Jordan algebras $J$ which are isomorphic to a Matsuo algebra $M_{1/2}(G,D)$, in which case $(G,D)$ is a subgroup of the (algebraic) automorphism group of $J$; the only possibilities are $G = \operatorname{Sym}(n)$ and $G = 3^2:2$. Along the way, we also obtain results about Jordan algebras associated to root systems.

math.RA↗
arXiv2014

Moufang sets of mixed type F4

Moufang sets were introduced by Jacques Tits in order to understand isotropic linear algebraic groups of relative rank one, but the notion is more general. We describe a new class of Moufang sets, arising from so-called mixed groups of type F4 in characteristic 2, obtained as the fixed point set under a suitable involution.

math.GR↗
arXiv2013

A new construction of Moufang quadrangles of type E6, E7 and E8

In the classification of Moufang polygons by J. Tits and R. Weiss, the most intricate case is by far the case of the exceptional Moufang quadrangles of type E_6, E_7 and E_8, and in fact, the construction that they present is ad-hoc and lacking a deeper explanation. We will show how tensor products of two composition algebras can be used to construct these Moufang quadrangles in characteristic different from 2. As a byproduct, we will obtain a method to construct any Moufang quadrangle in characteristic different from two from a module for a Jordan algebra.

math.RA↗
arXiv2012

Trees, contraction groups, and Moufang sets

We study closed subgroups $G$ of the automorphism group of a locally finite tree $T$ acting doubly transitively on the boundary. We show that if the stabiliser of some end is metabelian, then there is a local field $k$ such that $\mathrm{PSL}_2(k) \leq G \leq \mathrm{PGL}_2(k)$. We also show that the contraction group of some hyperbolic element is closed and torsion-free if and only if $G$ is (virtually) a rank one simple $p$-adic analytic group for some prime $p$. A key point is that if some contraction group is closed, then $G$ is boundary-Moufang, meaning that the boundary $\partial T$ is a Moufang set. We collect basic results on Moufang sets arising at infinity of locally finite trees, and provide a complete classification in case the root groups are torsion-free.

math.GR↗
arXiv2012

Moufang sets arising from polarities of Moufang planes over octonion division algebras

For every octonion division algebra O, there exists a projective plane which is parametrized by O; these planes are related to rank two forms of linear algebraic groups of absolute type E6. We study all possible polarities of such octonion planes having absolute points, and their corresponding Moufang set. It turns out that there are four different types of polarities, giving rise to (1) Moufang sets of type F4, (2) Moufang sets of type 2E6, (3) hermitian Moufang sets of type C4, and (4) projective Moufang sets over a 5-dimensional subspace of an octonion division algebra. Case (3) only occurs over fields of characteristic different from two, whereas case (4) only occurs over fields of characteristic equal to two. The Moufang sets of type 2E6 that we obtain in case (2) are exactly those corresponding to linear algebraic groups of type 2E6,1^29; the explicit description of those Moufang sets was not yet known.

math.GR↗
arXiv2012

Exceptional Moufang quadrangles and structurable algebras

In 2000, J. Tits and R. Weiss classified all Moufang spherical buildings of rank two, also known as Moufang polygons. The hardest case in the classification consists of the Moufang quadrangles. They fall into different families, each of which can be described by an appropriate algebraic structure. For the exceptional quadrangles, this description is intricate and involves many different maps that are defined ad hoc and lack a proper explanation. In this paper, we relate these algebraic structures to two other classes of algebraic structures that had already been studied before, namely to Freudenthal triple systems and to structurable algebras. We show that these structures give new insight in the understanding of the corresponding Moufang quadrangles.

math.RA↗
arXiv2011

Simple locally compact groups acting on trees and their germs of automorphisms

Automorphism groups of locally finite trees provide a large class of examples of simple totally disconnected locally compact groups. It is desirable to understand the connections between the global and local structure of such a group. Topologically, the local structure is given by the commensurability class of a vertex stabiliser; on the other hand, the action on the tree suggests that the local structure should correspond to the local action of a stabiliser of a vertex on its neighbours. We study the interplay between these different aspects for the special class of groups satisfying Tits' independence property. We show that such a group has few open subgroups if and only if it acts locally primitively. Moreover, we show that it always admits many germs of automorphisms which do not extend to automorphisms, from which we deduce a negative answer to a question by George Willis. Finally, under suitable assumptions, we compute the full group of germs of automorphisms; in some specific cases, these turn out to be simple and compactly generated, thereby providing a new infinite family of examples which generalise Neretin's group of spheromorphisms. Our methods describe more generally the abstract commensurator group for a large family of self-replicating profinite branch groups.

math.GR↗
arXiv2009

Iwasawa decompositions of split Kac-Moody groups

We characterize all fields F for which a group with an F-locally split root group datum admits an Iwasawa decomposition. This class of groups in particular includes the split semisimple algebraic groups and the split Kac-Moody groups.

math.GR↗
arXiv2009

The norm of a Ree group

We give an explicit construction of the Ree groups of type $G_2$ as groups acting on mixed Moufang hexagons together with detailed proofs of the basic properties of these groups contained in the two fundamental papers of Tits on this subject. We also give a short proof that the norm of a Ree group is anisotropic.

math.GR↗