Gradings for nilpotent Lie algebras

从立方范数对到 $G_2$- 和 $F_4$- 分级的群和李代数

From cubic norm pairs to $G_2$- and $F_4$-graded groups and Lie algebras

本文提出了一种从立方范数对构造 $G_2$ 和 $F_4$ 分级李代数及群的方法，并揭示了其结构特性。

Fine gradings of complex simple Lie algebras and Finite Root Systems

A $G$-grading on a complex semisimple Lie algebra $L$, where $G$ is a finite abelian group, is called quasi-good if each homogeneous component is 1-dimensional and 0 is not in the support of the grading. Analogous to classical root systems, we define a finite root system $R$ to be some subset of a finite symplectic abelian group satisfying certain axioms. There always corresponds to $R$ a semisimple Lie algebra $L(R)$ together with a quasi-good grading on it. Thus one can construct nice basis of $L(R)$ by means of finite root systems. We classify finite maximal abelian subgroups $T$ in $\Aut(L…

The action of a compact Lie group on nilpotent Lie algebras of type {n,2}

We classify finite-dimensional nilpotent Lie algebras with $2$-dimensional central commutator ideals admitting a Lie group of automorphisms isomorphic to $SO_2(\mathbb R)$. This enables one to enlarge the class of nilpotent Lie algebras of type {n,2}.

Nilpotent Lie groups and hyperbolic automorphisms

A connected Lie group admitting an expansive automorphism is known to be nilpotent, but all nilpotent Lie groups do not admit expansive automorphism. In this article, we find sufficient conditions for a class of nilpotent Lie groups to admit expansive automorphism.

Approximate Lattices and Meyer Sets in Nilpotent Lie Groups

We show that uniform approximate lattices in nilpotent Lie groups are subsets of model sets. This extends a theorem due to Yves Meyer about quasicrystals in Euclidean spaces. To do so we study relatively dense subsets of simply connected nilpotent Lie groups and their logarithms. We then deduce a simple criterion for the existence of an approximate lattice in a given nilpotent Lie group.

The Spherical Maximal Function on the Free Two-step Nilpotent Lie Group

We consider here the free two step nilpotent Lie group, provided with the homogeneous Korányi norm; we prove the $L^p$-boundedness of the maximal function corresponding to the homogeneous unit sphere, for some $p$.

Relatively free nilpotent torsion-free groups and their Lie algebras

For a torsion free finitely generated nilpotent group G we naturally associate four finite dimensional nilpotent Lie algebras over a field of characteristic zero. We show that if G is a relatively free group of some variery of nilpotent groups then all the above Lie algebras are isomorphic. As a result, any two quasi-isometric relatively free nilpotent groups are isomorphic. Moreover let L be a relatively free nilpotent Lie algebra over Q generated by X. We give L the structure of a group by means of the Baker-Campbell-Hausdorff formula and we show that the subgroup H generated by X is relativ…

On quasi-isometric nilpotent Lie groups

In this article we provide evidence for a well-known conjecture which states that quasi-isometric simply-connected nilpotent Lie groups are isomorphic. We do so by constructing new examples which are rigid in the sense that whenever they are quasi-isometric to any other simply-connected nilpotent Lie group, the groups are actually isomorphic.

C*-algebras with norm controlled dual limits and nilpotent Lie groups

Motivated by the description of the C*-algebras of 5 dimensional nilpotent Lie groups as algebras of operator fields defined over their spectra, we introduce the family of C* -algebras with norm controlled dual limits and we show that the C* -algebras of the 5 dimensional nilpotents Lie groups belong to this class.

Some examples of quasiisometries of nilpotent Lie groups

We construct quasiisometries of nilpotent Lie groups. In particular, for any simply connected nilpotent Lie group N, we construct quasiisometries from N to itself that is not at finite distance from any map that is a composition of left translations and automorphisms. We also construct biLipschitz maps of the Heisenberg groups that send vertical lines to non-vertical curves.