Publicaties
A dichotomy for integral group rings via higher modular groups as amalgamated products Vrije Universiteit Brussel
We show that U(ZG), the unit group of the integral group ring ZG, either satisfies Kazhdan's property (T) or is, up to commensurability, a non-trivial amalgamated product, in case G is a finite group satisfying some mild conditions. A key step in the proof is the construction of amalgamated decompositions of the elementary group E2(O), where O is an order in rational division algebra, and of certain arithmetic groups Γ. The methods for the ...
Nilpotent decomposition in integral group rings Vrije Universiteit Brussel
A finite group G is said to have the nilpotent decomposition property (ND) if for every nilpotent element α of the integral group ring Z[G] one has that αe also belong to Z[G], for every primitive central idempotent e of the rational group algebra Q[G]. Results of Hales, Passi and Wilson, Liu and Passman show that this property is fundamental in the investigations of the multiplicative Jordan decomposition of integral group rings. If G and ...
Every finite abelian group is a subgroup of the additive group of a finite simple left brace Vrije Universiteit Brussel
Left braces, introduced by Rump, have turned out to provide an important tool in the study of set-theoretic solutions of the quantum Yang–Baxter equation. In particular, they have allowed to construct several new families of solutions. A left brace (B,+,⋅) is a structure determined by two group structures on a set B: an abelian group (B,+) and a group (B,⋅), satisfying certain compatibility conditions. The main result of this paper shows that ...
Set-theoretic solutions of the Yang–Baxter equation, associated quadratic algebras and the minimality condition Vrije Universiteit Brussel
Given a finite non-degenerate set-theoretic solution (X, r) of the Yang–Baxter equation and a field K, the structure K-algebra of (X, r) is A=A(K,X,r)=K⟨X∣xy=uvwheneverr(x,y)=(u,v)⟩. Note that A= ⊕ n ≥A n is a graded algebra, where A n is the linear span of all the elements x 1⋯ x n, for x 1, ⋯ , x n∈ X. One of the known results asserts that the maximal possible value of dim (A 2) corresponds to involutive solutions and implies several deep ...
Structure of group rings and the group of units of integral group rings: an invitation Vrije Universiteit Brussel
These constructions rely on explicit constructions of units in $\Z G$ and proofs of main results make use of the description of the Wedderburn components of the rational group algebra $\Q G$. The latter relies on ...
Corrigendum and addendum to "The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang-Baxter equation" Vrije Universiteit Brussel
An abundance of simple left braces with abelian multiplicative Sylow subgroups Vrije Universiteit Brussel
non-de\-gene\-rate set-theoretic solutions of the Yang-Baxter
equation. A constructive method for producing all such finite
solutions from a description of all finite left braces has been
recently discovered. It is thus a fundamental problem to construct
and classify all simple left braces, as they can be considered as
building blocks for the general ...