Publicaties

We compare a number of different definitions of structure algebras and TKK constructions for Jordan (super)algebras appearing in the literature. We demonstrate that, for unital superalgebras, all the definitions of the structure algebra and the TKK constructions reduce to one of two cases. Moreover, one can be obtained as the Lie superalgebra of superderivations of the other. We also show that, for non-unital superalgebras, more definitions ...

A point-line space is an abstract geometric object that consists of a set of points and a set of lines such that on each line there are at least two points. A large class of point-line spaces with high symmetry comes along with buildings, combinatorial objects that are introduced by Jacques Tits and help to study algebraic objects with geometric methods.To formulate quantum mechanics as abstract and general as possible, the physicist Pascual ...

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 ...

We determine the Jordan-Holder decomposition multiplicities of projective and cell modules over periplectic Brauer algebras in characteristic zero. These are obtained by developing the combinatorics of certain skew Young diagrams. We also establish a useful relationship with the Kazhdan-Lusztig multiplicities of the periplectic Lie supergroup.

Hjelmslev-Moufang planes are point-line geometries related to the exceptional algebraic groups of type E6. More generally, point-line geometries related to spherical Tits-buildingsU+2014Lie incidence geometriesU+2014are the prominent examples of parapolar spaces: axiomatically defined geometries consisting of points, lines and symplecta (structures isomorphic to polar spaces). In this paper we classify the parapolar spaces with a similar ...

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 E6, E7 and E8, 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 ...

We study realisations of Lie (super)algebras in Weyl (super)algebras and connections with minimal representations. The main result is the construction of small realisations of Lie superalgebras, which we apply for two distinct purposes. Firstly it naturally introduces, and generalises, the Bessel operators for Jordan algebras in the study of minimal representations of simple Lie groups. Secondly, we work out the theoretical realisation ...

In this paper, we continue our study of the Green rings of finite dimensional pointed Hopf algebras of rank one initiated in [22], but focus on those Hopf algebras of non-nilpotent type. Let H be a finite dimensional pointed rank one Hopf algebra of non-nilpotent type. We first determine all non-isomorphic indecomposable H -modules and describe the Clebsch–Gordan formulas for them. We then study the structures of both the Green ring r(H) and ...