Related papers: Longer nilpotent series for classical unipotent su…
We find the lower central series for residually nilpotent Baumslag-Solitar groups, and find the intersection of all terms of the lower central series. Also, we find non-abelian Bauslag-Solitar groups for which the lower central series has…
We give an efficient algorithm for Lang's Theorem in split connected reductive groups defined over finite fields of characteristic greater than 3. This algorithm can be used to construct many important structures in finite groups of Lie…
Let $\mathcal{N}_{\mathfrak{g}^*}$ be the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. In \cite{Lu2} Lusztig proposes a definition of a partition of…
Following Lazard, we study the $N$-series of a group $G$ and their associated graded Lie algebras. The main examples we consider are the lower central series (LCS), Stallings' rational and mod-$q$ versions, and Zassenhaus' mod-$p$ version…
Consider the Macdonald groups $G(\alpha)=\langle A,B\,|\, A^{[A,B]}=A^\alpha,\, B^{[B,A]}=B^\alpha\rangle$, $\alpha\in{\mathbf Z}$. We fill a gap in Macdonald's proof that $G(\alpha)$ is always nilpotent, and proceed to determine the order,…
We introduce a special class of powerful $p$-groups that we call powerfully nilpotent groups that are finite $p$-groups that possess a central series of a special kind. To these we can attach the notion of a powerful nilpotence class that…
Let $G$ be a connected reductive group over an algebraically closed field $\Bbbk$. Under mild restrictions on the characteristic of $\Bbbk$, we show that any $G$-module with a good filtration also has a good filtration as a module for the…
Category of pro-nilpotently extended differential graded commutative algebras is introduced. Chevalley-Eilenberg construction provides an equivalence between its certain full subcategory and the opposite to the full subcategory of strong…
The pro-isomorphic zeta function of a finitely generated nilpotent group is a Dirichlet generating series that enumerates all finite-index subgroups whose profinite completion is isomorphic to that of the ambient group. We study the…
In this paper, several theorems of Macdonald \cite{Mac1961,Mac1962} on the varieties of nilpotent groups will be generalized to the case of Lie rings. We consider three varieties of Lie rings of any characteristic associated with some…
These are expanded notes from graduate courses about Lie algebras and Chevalley groups held at the University of Stuttgart. In the 1950s Chevalley showed how linear groups over arbitrary fields could be obtained~ -- ~by a uniform procedure~…
In this paper, we focus on how we can interpret the actions of the elements in the Gelfand spectrum of a weighted Fourier algebra on connected Lie groups. They can be viewed as evaluations on specific points of the complexification of the…
We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain…
In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central…
We generalize the common notion of descending and ascending central series. The descending approach determines a naturally graded Lie ring and the ascending version determines a graded module for this ring. We also link derivations of these…
We give a unified description of twisted forms of classical reductive groups schemes. Such group schemes are constructed from algebraic objects of finite rank, excluding some exceptions of small rank. These objects, augmented odd form…
This paper investigates some properties of complex structures on Lie algebras. In particular, we focus on $\textit{nilpotent}$ $\textit{complex structures}$ that are characterized by a suitable $J$-invariant ascending or descending central…
A group is called capable if it is a central factor group. We consider the capability of certain nilpotent products of cyclic groups, and obtain a generalisation of a theorem of Baer for the small class case. The approach may also be used…
The paper is devoted to give a full classification of all finite dimensional nilpotent Lie algebras $ L $ of class $4$ such that $ \dim L^2=3. $ Moreover, we classify the capable ones.
An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical. Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular…