Related papers: Sampling Theorems for Some Two-Step Nilpotent Lie …

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let $Z$ be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus…

Let $G= \exp(\g)$ be a connected, simply connected nilpotent Lie group. We show that for every $G$-invariant smooth sub-manifold $M$ of $g^*$, there exists an open relatively compact subset $\mathcal{M}$ of $M$ such that for any smooth…

We calculate the homology of three families of 2-step nilpotent Lie (super)algebras associated with the symplectic, orthogonal, and general linear groups. The symplectic case was considered by Getzler and the main motivation for this work…

Let L\subset V=\bR^{k,l} be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k,l) is 2-step nilpotent and is defined by an element \eta \in…

We study symplectic structures on nilpotent Lie algebras. Since the classification of nilpotent Lie algebras in any dimension seems to be a crazy dream, we approach this study in case of 2-step nilpotent Lie algebras (in this sub-case also,…

Let $\mathfrak{g}$ be a real semisimple Lie algebra with Iwasawa decomposition $\mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$. We show that, except for some explicit exceptional cases, every derivation of the nilpotent subalgebra…

Let $\mathfrak{d}$ be the Lie superalgebra of superderivations of the sheaf of sections of the exterior algebra of the homogeneous vector bundle $E$ over the flag variety $G/P$, where $G$ is a simple finite-dimensional complex Lie group and…

Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We collect some general results on the Poisson center of S(g), including some simple criteria regarding its polynomiality, and also on…

The space K^n of all n-dimensional { Lie} algebras has a natural non-Hausdorff topology k^n, which has characteristic limits, called transitions, A -> B, between distinct Lie algebras A and B. The entity of these transitions are the natural…

Gromov claimed, with a sketch of proof, that simply connected nilpotent Lie groups have polynomially bounded filling invariants. The literature establishes this, often with a stronger conclusion where the exponent of polynomiality is…

Let $k$ be a field of characteristic not two or three. We classify up to isomorphism all finite-dimensional Lie superalgebras $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ over $k$, where $\mathfrak{g}_0$ is a three-dimensional simple…

Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nilpotent. A primitive…

We find the irreducible decomposition of the Weil representation of the unitary group $\mathrm{U}_{2n}(A)$, where $A$ is a ramified quadratic extension of a finite, commutative, local, principal ideal ring $R$ and the nilpotency degree of…

We prove that any Novikov algebra over a field of characteristic $\neq 2$ is Lie-solvable if and only if its commutator ideal $[N,N]$ is right nilpotent. We also construct examples of infinite-dimensional Lie-solvable Novikov algebras $N$…

We consider a Leibniz algebra ${\mathfrak L} = {\mathfrak I} \oplus {\mathfrak V}$ over an arbitrary base field $\mathbb{F}$, being ${\mathfrak I}$ the ideal generated by the products $[x,x], x \in {\mathfrak L}$. This ideal has a…

From a Lie algebra $\mathfrak{g}$ satisfying $\mathcal{Z}(\mathfrak{g})=0$ and $\Lambda^2(\mathfrak{g})^\mathfrak{g}=0$ (in particular, for $\g$ semisimple) we describe explicitly all Lie bialgebra structures on extensions of the form…

Any abstract (not necessarily continuous) group automorphism of a simple, compact Lie group must be continuous due to Cartan (1930) and van der Waerden (1933). The purpose of this paper is to study a similar question in nilpotent Lie…

We study the centrally generated primitive ideals of $U(\mathfrak{n})$, where $\mathfrak{n}$ is the (locally) nilpotent radical of a (splitting) Borel subalgebra of a simple complex Lie algebra…

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…

We give a full description of the Lie algebra generated by locally nilpotent derivations (short LNDs) on smooth Danielewski surfaces $D_p$ given by $xy=p(z)$. In case $\mathrm{deg}(p)\geq 3$ it turns out to be not the whole Lie algebra…