On eigenvectors of nilpotent Lie algebras of linear operators

On Kirillov's lemma for nilpotent Lie algebras

We establish a sharpening of Kirillov's lemma on nilpotent Lie algebras with 1-dimensional center and use it to study the structure of 3-step nilpotent Lie algebras.

Faithful Representations of Minimal Dimension of 6-dimensional nilpotent Lie algebras

The main goal of this paper is to compute $μ(\g)$ and $μ_{nil}(\g)$ for each nilpotent Lie algebra $\g$ of dimension 6 over a field of characteristic zero $\k$. Here $μ(\g)$ and $μ_{nil}(\g)$ is the minimal dimension of a faithful representation of $\g$ and the minimal dimension of a faithful nilrepresentation of $\g$, respectively. We also give a minimal faithful representation and nilrepresentation for each nilpotent Lie algebra of dimension $\leq 6$.

Normal forms of nilpotent elements in semisimple Lie algebras

We find the normal form of nilpotent elements in semisimple Lie algebras that generalizes the Jordan normal form in $\mathfrak{sl}_N$, using the theory of cyclic elements.

Uncertainty principles on nilpotent Lie groups

Hardy's type uncertainty principle on connected nilpotent Lie groups for the Fourier transform is proved. An analogue of Hardy's theorem for Gabor transform has been established for connected and simply connected nilpotent Lie groups. Finally Beurling's theorem for Gabor transform is discussed for groups of the form $\mathbb{R}_n \times K$, where $K$ is a compact group

On index of certain nilpotent Lie algebras

We introduce the method of calculation of index of Lie algebras that are factors of the unitriangular Lie algebra with respect to ideals spanned by subsets of root vectors.

Weights of Semiregular Nilpotents in Simple Lie Algebras of D Type

We compute the weights of the adjoint action of semiregular $sl_2$-triples in simple Lie algebras of type $D_n$ using mathematical induction.

Pseudo-metric 2-step nilpotent Lie algebras

The metric approach to studying 2-step nilpotent Lie algebras by making use of non-degenerate scalar products is realised. We show that any 2-step nilpotent Lie algebra is isomorphic to its standard pseudo-metric form, that is a 2-step nilpotent Lie algebra endowed with some standard non-degenerate scalar product compatible with Lie brackets. This choice of the standard pseudo-metric form allows to study the isomorphism properties. If the elements of the centre of the standard pseudo-metric form constitute a Lie triple system of the pseudo-orthogonal Lie algebra, then the original 2-step nilpo…

Nilpotent orbits in classical Lie algebras over $\textbf{F}_{2^n}$ and the Springer correspondence

We give the number of nilpotent orbits in the Lie algebras of orthogonal groups under the adjoint action of the groups over $\tF_{2^n}$. Let $G$ be an adjoint algebraic group of type $B,C$ or $D$ defined over an algebraically closed field of characteristic 2. We construct the Springer correspondence for the nilpotent variety in the Lie algebra of $G$.

八维实非退化两步幂零李代数

Real non-degenerate two-step nilpotent Lie algebras of dimension eight

本文利用复数域结果，对实数域上八维非退化两步幂零李代数进行分类，并给出显式结构常数。

Faithful representations of infinite-dimensional nilpotent Lie algebras

For locally convex, nilpotent Lie algebras we construct faithful representations by nilpotent operators on a suitable locally convex space. In the special case of nilpotent Banach-Lie algebras we get norm continuous representations by bounded operators on Banach spaces.

Subalgebra generated by ad-locally nilpotent elements of Borcherds Generalized Kac-Moody Lie algebras

We determine the Lie subalgebra $\mathfrak{g}_{nil}$ of a Borcherds symmetrizable generalized Kac-Moody Lie algebra $\mathfrak{g}$ generated by $ad$-locally nilpotent elements and show that it is `essentially' the same as the Levi subalgebra of $\mathfrak{g}$ with its simple roots precisely the real simple roots of $\mathfrak{g}$.

Nilpotent orbits in classical Lie algebras over finite fields of characteristic 2 and the Springer correspondence

Let $G$ be an adjoint algebraic group of type $B$, $C$, or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of $G$. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is determined and the number of nilpotent orbits over finite fields is obtained.

Shannon-Like Wavelet Frames on a Class of Nilpotent Lie Groups

We prove and construct Shannon-like Parseval wavelet frames for a class of two step connected, and simply connected nilpotent Lie groups, using a mixture of representation theory, group Fourier theory, and Gabor theory. Moreover, we are able to compute an upperbound for the norm of these Parseval frame wavelets.

Unitary representations of nilpotent super Lie groups

We show that irreducible unitary representations of nilpotent super Lie groups can be obtained by induction from a distinguished class of sub super Lie groups. These sub super Lie groups are natural analogues of polarizing subgroups that appear in classical Kirillov theory. We obtain a concrete geometric parametrization of irreducible unitary representations by nonnegative definite coadjoint orbits. As an application, we prove an analytic generalization of the Stone-von Neumann theorem for Heisenberg-Clifford super Lie groups.

Unitary Representations of Lattices of Free Nilpotent Lie Groups of Step-Two

Using a theorem proved by Bekka and Driutti, we show that if $\mathfrak{f}$ is a freely generated nilpotent Lie algebra of step-two, then almost every irreducible representation of the corresponding Lie group restricted to some lattice $Γ$ is an irreducible representation of $Γ$ if the dimension of the Lie algebra is odd. However, if the dimension of the Lie algebra is even, then almost every unitary irreducible representation of the Lie group restricted to $Γ$ is reducible.

Boundedness for Pseudo-Differential Calculus on Nilpotent Lie Groups

We survey a few results on the boundedness of operators arising from the Weyl-Pedersen calculus associated with irreducible representations of nilpotent Lie groups.

Beurling's Theorem and $L^p-L^q$ Morgan's Theorem for Step Two Nilpotent Lie Groups

We prove Beurling's theorem and $L^p-L^q$ Morgan's theorem for step two nilpotent Lie groups

On the isomorphism problem for $C^*$-algebras of nilpotent Lie groups

We investigate to what extent a nilpotent Lie group is determined by its $C^*$-algebra. We prove that, within the class of exponential Lie groups, direct products of Heisenberg groups with abelian Lie groups are uniquely determined even by their unitary dual, while nilpotent Lie groups of dimension $\le 5$ are uniquely determined by the Morita equivalence class of their $C^*$-algebras. We also find that this last property is shared by the filiform Lie groups and the $6$-dimensional free two-step nilpotent Lie group.

Weyl-Pedersen calculus for some semidirect products of nilpotent Lie groups

For certain nilpotent real Lie groups constructed as semidirect products, algebras of invariant differential operators on some coadjoint orbits are used in the study of boundedness properties of the Weyl-Pedersen calculus of their corresponding unitary irreducible representations. Our main result is applicable to all unitary irreducible representations of arbitrary 3-step nilpotent Lie groups.

Nilpotent metric Lie algebras of small dimension

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 dim l'=2 which are used in this scheme. Furthermore, we classify all nilpotent metric Lie algebras of dimension at most 10.

A classification of the principal nilpotent pairs in simple Lie algebras and related problems

We give a classification of the principal and distinguished nilpotent pairs in all classical Lie algebras. As a classification of the principal pairs in the exceptional simple Lie algebras was obtained earlier (see Appendix to Ginzburg's preprint math.AG/9903059), we thus have a complete classification of principal nilpotent pairs.

A Paley-Wiener Theorem for Nilpotent Lie Groups

A version of Paley-Wiener like theorem for connected, simply connected nilpotent Lie groups is proven.

A survey on Weyl calculus for representations of nilpotent Lie groups

We survey some aspects of the pseudo-differential Weyl calculus for irreducible unitary representations of nilpotent Lie groups, ranging from the classical ideas to recently obtained results. The classical Weyl-Hörmander calculus is recovered for the Schrödinger representation of the Heisenberg group. Our discussion concerns various extensions of this classical situation to arbitrary nilpotent Lie groups and to some infinite-dimensional Lie groups that allow us to handle the magnetic pseudo-differential calculus.

On the existence of nilsolitons on 2-step nilpotent Lie groups

A 2-step nilpotent Lie algebra n is said to be of type (p,q)if dim(n)=p+q and dim([n,n])=p. By considering a class of 2-step nilpotent Lie algebras naturally attached to graphs, we prove that there exist indecomposable, 2-step nilpotent Lie groups of type (p,q) which do not admit a nilsoliton metric for every pair (p,q) such that 20 < q and q-2 < p < 1/2q^2-5/2q+10. This improves a result due to Jablonski.

Lie groups as permutation groups: Ulam's problem in the nilpotent case

Ulam asked whether every connected Lie group can be represented on a countable structure. This is known in the linear case. We establish it for the first family of non-linear groups, namely in the nilpotent case. Further context is discussed to illustrate the relevance of nilpotent groups for Ulam's problem.

Smooth vectors and Weyl-Pedersen calculus for representations of nilpotent Lie groups

We present some recent results on smooth vectors for unitary irreducible representations of nilpotent Lie groups. Applications to the Weyl-Pedersen calculus of pseudo-differential operators with symbols on the coadjoint orbits are also discussed.

Abundance of nilpotent orbits in real semisimple Lie algebras

We formulate and prove that there are "abundant" in nilpotent orbits in real semisimple Lie algebras, in the following sense. If S denotes the collection of hyperbolic elements corresponding the weighted Dynkin diagrams coming from nilpotent orbits, then S span the maximally expected space, namely, the (-1)-eigenspace of the longest Weyl group element. The result is used to the study of fundamental groups of non-Riemannian locally symmetric spaces.

Approximation of nilpotent orbits for simple Lie groups

We propose a systematic and topological study of limits $\lim_{ν\to 0^+}G_\mathbb{R}\cdot(νx)$ of continuous families of adjoint orbits for non-compact simple Lie groups. This limit is always a finite union of nilpotent orbits. We describe explicitly these nilpotent orbits in terms of Richardson orbits in the case of hyperbolic semisimple elements. We also show that one can approximate minimal nilpotent orbits or even nilpotent orbits by elliptic semisimple orbits. The special cases of $\mathrm{SL}_n(\mathbb{R})$ and $\mathrm{SU}(p,q)$ are computed in detail.

Boundedness of Relative Convolutions on Nilpotent Lie Groups

We discuss some norm estimations for integrated representations. We use the covariant transform to extend Howe's method from the Heisenberg group to general nilpotent Lie groups.

A remark on cohomology of nilpotent Lie algebras

We prove that the cohomology algebra of a conilpotent Lie coalgebra is generated in degree 1 as an A-infinity algebra. By dualizing, the same is true about cohomology algebras of finite dimensional nilpotent Lie algebras. In the process, we provide a proof of the conilpotent version of the dual Poincare-Birkhoff-Witt theorem.

On integrability of transverse Lie-Poisson structures at nilpotent elements

We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie-Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform construction of these integrable systems for an infinite family of distinguished nilpotent elements of semisimple type.

On exceptional nilpotents in semisimple Lie algebras

We classify all pairs (m,e), where m is a positive integer and e is a nilpotent element of a semisimple Lie algebra, which arise in the classification of simple rational W-algebras.

关于例外李代数中幂零轨道的第二上同调

On the second cohomology of nilpotent orbits in exceptional Lie algebras

本文研究非紧致非复例外李代数中幂零轨道的第二上同调，计算了大多数情形的维数，并对其余情形给出上界。

The second cohomology groups of nilpotent orbits in classical Lie algebras

The second de Rham cohomology groups of nilpotent orbits in non-compact real forms of classical complex simple Lie algebras are explicitly computed. Furthermore, the first de Rham cohomology groups of nilpotent orbits in non-compact classical simple Lie algebras are computed; they are proven to be zero for nilpotent orbits in all the complex simple Lie algebras. A key component in these computations is a description of the second and first cohomology groups of homogeneous spaces of general connected Lie groups which is obtained here. This description, which generalizes a previous theorem of th…

A curve of nilpotent Lie algebras which are not Einstein nilradicals

The only known examples of noncompact Einstein homogeneous spaces are standard solvmanifolds (special solvable Lie groups endowed with a left invariant metric), and according to a long standing conjecture, they might be all. The classification of Einstein solvmanifolds is equivalent to the one of Einstein nilradicals, i.e. nilpotent Lie algebras which are nilradicals of the Lie algebras of Einstein solvmanifolds. Up to now, there have been found very few examples of graded nilpotent Lie algebras that can not be Einstein nilradicals. In particular, in each dimension, there are only finitely man…