Special Moufang sets of finite dimension
We prove that a special Moufang sets with abelian root subgroups derive from a quadratic Jordan division algebra if a certain finiteness condition is satisfied.
math.GR↗
arXiv subjects
Matthias Grüninger
Publications and source records attributed to Matthias Grüninger.
Moufang Twin Trees of prime order
We prove that the unipotent horocyclic group of a Moufang twin tree of prime order is nilpotent of class at most 2.
math.GR↗
On cubic action of a rank one group
We consider a rank one group $G = \langle A,B \rangle $ which acts cubically on a module $V$, this means $[V,A,A,A] =0$ but $[V,G,G,G] \ne 0$. We have to distinguish whether the group $A_0 :=C_A([V,A]) \cap C_A(V/C_V(A))$ is trivial or not. We show that if $A_0$ is trivial, $G$ is a rank one group associated to a quadratic Jordan division algebra. If $A_0$ is not trivial (which is always the case if $A$ is not abelian), then $A_0$ defines a subgroup $G_0$ of $G$ which acts quadratically on $V$. We will call $G_0$ the \textit{quadratic kernel} of $G$. By a result of Timmesfeld we have $G_0 \cong \SL_2(J,R)$ for a ring $R$ and a special quadratic Jordan division algebra $J \subseteq R$. We show that $J$ is either a Jordan algebra contained in a commutative field or a hermitian Jordan algebra. In the second case $G$ is the special unitary group of a pseudo-quadratic form $π$ of Witt index $1$, in the first case $G$ is the rank one group for a Freudenthal triple system. These results imply that if $(V,G)$ is a quadratic pair such that no two distinct root groups commute and $\characteristic V\ne 2,3$, then $G$ is a unitary group or an exceptional algebraic group.
math.GR↗
On the axioms defining a quadratic Jordan division algebra
Quadratic Jordan algebras are defined by identities that have to hold strictly, i.e that continue to hold in every scalar extension. In this paper we show that strictness is not required for quadratic Jordan division algebras.
math.RA↗
Multiplicative quadratic maps
In this paper we prove that a multiplicative quadratic map between a unital ring $K$ and a field $L$ is induced by a homomorphism from $K$ into $L$ or a composition algebra over $L$. Especially we show that if $K$ is a field, then every multiplicative quadratic map is the product of two field homomorphisms. Moreover, we prove a multiplicative version of Artin's Theorem showing that a product of field homomorphisms is unique up to multiplicity.
math.RA↗
Boundary Moufang trees with abelian root groups of characteristic p
We prove that Moufang sets with abelian root groups arising at infinity of a locally finite tree all come from rank one simple algebraic groups over local fields.
math.GR↗