Related papers: Graphs and Metric 2-step Nilpotent Lie Algebras

H-type Lie algebras were introduced by Kaplan as a class of real Lie algebras generalizing the familiar Heisenberg Lie algebra $\mathfrak{h}^3$. The H-type property depends on a choice of inner product on the Lie algebra $\mathfrak{g}$.…

We undertake a comprehensive study of structural properties of graph products of von Neumann algebras equipped with faithful, normal states, as well as properties of the graph products relative to subalgebras coming from induced subgraphs.…

The goal of this paper is the study of algebraic relations on the Lie algebra of first integrals of the geodesic flow on nilpotent Lie groups equipped with a left-invariant metric. It is proved that the isometry algebra of the $k$-step…

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…

This paper deals with naturally reductive pseudo-Riemannian 2-step nilpotent Lie groups $(N, \la \,,\,\ra_N)$, such that $\la \,,\,\ra_N$ is invariant under a left action. The case of nondegenerate center is completely characterized. In…

There is a natural way to construct sub-Riemannian structures that depend on $n$ parameters on compact Lie groups. These structures are related to the filtrations of Lie subalgebras $\mathfrak g_0 < \mathfrak g_1 < \mathfrak g_2 < \dots <…

We classify real and complex infinite-dimensional narrow positively graded Lie algebras ${\mathfrak g}=\oplus_{i=1}^{{+}\infty}{\mathfrak g}_i$ with properties $$ [{\mathfrak g}_1, {\mathfrak g}_i]={\mathfrak g}_{i{+}1}, \; \dim{{\mathfrak…

This paper consists of two parts. First, motivated by classic results, we determine the subsets of a given nilpotent Lie algebra $\mathfrak{g}$ (respectively, of the Grassmannian of two-planes of $\mathfrak{g}$) whose sign of Ricci…

First, I construct an isomorphism between the categories of (topological) groups of nilpotency class 2 with 2-divisible center and (topological) Lie rings of nilpotency class 2 with 2-divisible center. That isomorphism allows us to…

We study almost inner derivations of $2$-step nilpotent Lie algebras of genus $2$, i.e., having a $2$-dimensional commutator ideal, using matrix pencils. In particular we determine all almost inner derivations of such algebras in terms of…

If a Lie algebra structure g on a vector space is the sum of a family of mutually compatible Lie algebra structures g_i's, we say that g is simply assembled from the g_i's. Repeating this procedure with a number of Lie algebras, themselves…

Every Lie algebra over a field $E$ gives rise to new Lie algebras over any subfield $F \subseteq E$ by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of…

We introduce nilpotent k-ary Lie algebras including analogues of Heisenberg Lie algebras and free nilpotent Lie algebras. We study homology of k-ary nilpotent Lie algebras by using a modification of Chevalley-Eilenberg complex. For some…

Anosov diffeomorphisms are an important class of dynamical systems with many peculiar properties. Ever since they were introduced in the sixties, it has been an open question which manifolds can admit such diffeomorphisms, where tori of…

Let $G$ be a simply connected algebraic group of type $B,C$ or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the dual vector space of the Lie algebra of $G$. In particular, we…

We classify the non-degenerate two-step nilpotent Lie algebras of dimension 8 over the field of real numbers, using known results over complex numbers. We write explicit structure constants for these real Lie algebras.

In math.DG/0312243 we developed a general classification scheme for metric Lie algebras, i.e. for finite-dimensional Lie algebras equipped with a non-degenerate invariant inner product. Here we determine all nilpotent Lie algebras l with…

We begin a systematic study of these spaces, initially following along the lines of Eberlein's comprehensive study of the Riemannian case. In particular, we integrate the geodesic equation, discuss the structure of the isometry group, and…

A Riemann-Lie algebra is a Lie algebra $\cal G$ such that its dual ${\cal G}^*$ carries a Riemannian metric compatible (in the sense introduced by th author in C. R. Acad. Paris, t. 333, S\'erie I, (2001) 763-768) with the canonical linear…

Let $G$ be a simple simply-connected algebraic group over an algebraically closed field $k$ of characteristic $p>0$ with $\mathfrak{g}={\rm Lie}(G)$. We discuss various properties of nilpotent orbits in $\mathfrak{g}$, which have previously…