We construct Moufang sets, Moufang triangles and Moufang hexagons using inner ideals of Lie algebras obtained from structurable algebras via the Tits–Kantor–Koecher construction. The three different ...types of structurable algebras we use are, respectively: (1) structurable division algebras, (2) algebras
D
⊕
D
for some alternative division algebra
D
, equipped with the exchange involution, (3) matrix structurable algebras
M
(
J
, 1) for some cubic Jordan division algebra
J
. In each case, we also determine the root groups directly in terms of the structurable algebra.
We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an ...associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E_8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.
Modules over Axial Algebras De Medts, Tom; Van Couwenberghe, Michiel
Algebras and representation theory,
02/2020, Volume:
23, Issue:
1
Journal Article
Peer reviewed
Open access
We introduce axial representations and modules over axial algebras as new tools to study axial algebras. All known interesting examples of axial algebras fall into this setting, in particular the ...Griess algebra whose automorphism group is the Monster group. Our results become especially interesting for Matsuo algebras. We vitalize the connection between Matsuo algebras and 3-transposition groups by relating modules over Matsuo algebras with representations of 3-transposition groups. As a by-product, we define, given a Fischer space, a group that can fulfill the role of a universal 3-transposition group.
In 2000, M. Burger and S. Mozes introduced universal groups acting on trees with a prescribed local action. We generalize this concept to groups acting on right-angled buildings. When the ...right-angled building is thick and irreducible of rank at least 2 and each of the local permutation groups is transitive and generated by its point stabilizers, we show that the corresponding universal group is a simple group. When the building is locally nite, these universal groups are compactly generated totally disconnected locally compact groups, and we describe the structure of the maximal compact open subgroups of the universal groups as a limit of generalized wreath products.
We introduce the notion of city products of right-angled buildings that produces a new right-angled building out of smaller ones. More precisely, if M is a right-angled Coxeter diagram of rank n and ...Δ1,…,Δn are right-angled buildings, then we construct a new right-angled building ▪. We can recover the buildings Δ1,…,Δn as residues of Δ, but we can also construct a skeletal building of type M from Δ that captures the large-scale geometry of Δ.
We then proceed to study universal groups for city products of right-angled buildings, and we show that the universal group of Δ can be expressed in terms of the universal groups for the buildings Δ1,…,Δn and the structure of M. As an application, we show the existence of many examples of pairs of different buildings of the same type that admit (topologically) isomorphic universal groups, thereby vastly generalizing a recent example by Lara Beßmann.
We extend the theory of Matsuo algebras, which are certain non-associative algebras related to 3-transposition groups, to characteristic 2. The decompositions of our algebras are now induced by ...nilpotent elements associated to lines in the corresponding Fischer space, rather than idempotent elements associated to points. For many 3-transposition groups, this still gives rise to a Z/2Z-graded fusion law, and we provide a complete classification of when this occurs. In one particular small case, arising from the 3-transposition group Sym(4), the fusion law is even stronger, and the resulting Miyamoto group is an algebraic group Ga2⋊Gm.
We study structurable algebras of skew-dimension one. We present two different equivalent constructions for such algebras: one in terms of nonlinear 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:
We show that every form of a matrix structurable algebra can be described by our constructions;
We give explicit formulas for the norm ν;
We make a precise connection with the Cayley-Dickson process for structurable algebras.
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