Titel Deelnemers "Jordan algebras and 3-transposition groups" "Tom De Medts, Felix Rehren" "On structure and TKK algebras for Jordan superalgebras" "Sigiswald Barbier, Kevin Coulembier" "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 become nonequivalent. As an application, we obtain the corresponding Lie superalgebras for all simple finite dimensional Jordan superalgebras over an algebraically closed field of characteristic zero." "Unified products for Jordan algebras. Applications" "A.L. Agore, Gigel Militaru" "Given a Jordan algebra A and a vector space V, we describe and classify all Jordan algebras containing A as a subalgebra of codimension dimk(V) in terms of a non-abelian cohomological type object JA(V,A). Any such algebra is isomorphic to a newly introduced object called unified product A♮V. The crossed/twisted product of two Jordan algebras are introduced as special cases of the unified product and the role of the subsequent problem corresponding to each such product is discussed. The non-abelian cohomology Hnab2(V,A) associated to two Jordan algebras A and V which classifies all extensions of V by A is also constructed. Several applications and examples are given: we prove that Hnab2(k,kn) is identified with the set of all matrices D∈Mn(k) satisfying 2D3−3D2+D=0, where we consider the abelian Jordan algebra structure on k and kn." "Exceptional Lie algebras, SU(3) and Jordan pairs: part 2. Zorn-type representations" "Alessio Marrani" "The factorization problem for Jordan algebras: applications" "A.L. Agore, Gigel Militaru" "We investigate the factorization problem as well as the classifying complements problem in the setting of Jordan algebras. Matched pairs of Jordan algebras and the corresponding bicrossed products are introduced. It is shown that any Jordan algebra which factorizes through two given Jordan algebras is isomorphic to a bicrossed product associated to a certain matched pair between the same two Jordan algebras. Furthermore, a new type of deformation of a Jordan algebra is proposed as the main step towards solving the classifying complements problem." "Local Moufang sets and local Jordan pairs" "Tom De Medts, Erik Rijcken" "Point-line spaces related to Jordan pairs" "Simon Huggenberger" "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 Jordan invented a non-associative algebraic structure which is now called Jordan algebra. A generalisation of Jordan algebras are the so-called Jordan pairs. In 1975, Ottmar Loos classified all Jordan pairs with finite dimension. As a matter of fact, the list of Ottmar Loos matches in a certain way a part of the list of types of buildings given by Jacques Tits. The buildings of the types that correspond to the types of the Jordan pairs provide a class of point-line spaces. This class consists of two exceptional types and an infinite number of types that occur in four series of increasing dimension. There is a natural way to enlarge these series to the cases of infinite dimension. Together with the two exceptional types this is the class of point-line spaces that we consider to be the point-line spaces related to Jordan pairs of arbitrary dimension. The present work gives a characterisation of point-line spaces that determines exactly this class and uses four rather simple axioms. Moreover, we give a full classification of the point-line spaces satisfying these axioms and prove that it is the class mentioned above." "Moufang sets and structurable division algebras" "Lien Boelaert, Tom De Medts, Anastasia Stavrova" "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 tau-map and the Hua maps of these Moufang sets. This is particularly useful for the Moufang sets arising from exceptional linear algebraic groups." "Algebraic constructions for Jacobi-Jordan algebras" "A.L. Agore, Gigel Militaru" "For a given Jacobi-Jordan algebra A and a vector space V over a field k, a non-abelian cohomological type object HA2(V,A) is constructed: it classifies all Jacobi-Jordan algebras containing A as a subalgebra of codimension equal to dimk(V). Any such algebra is isomorphic to a so-called unified product A♮V. Furthermore, we introduce the bicrossed (semi-direct, crossed, or skew crossed) product A⋈V associated to two Jacobi-Jordan algebras as a special case of the unified product. Several examples and applications are provided: the Galois group of the extension A⊆A⋈V is described as a subgroup of the semidirect product of groups GLk(V)⋊Homk(V,A) and an Artin type theorem for Jacobi-Jordan algebra is proven." "Polynomial realisations of lie (super)algebras and Bessel operators" "Sigiswald Barbier, Kevin Coulembier" "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 concretely for the exceptional Lie superalgebra D(2, 1; alpha), giving a useful hands- on realisation."