Related papers: Simply transitive NIL-affine actions of solvable L…

We generalize the BF theory action to the case of a general Lie crossed module $(H \to G)$, where $G$ and $H$ are non-abelian Lie groups. Our construction requires the existence of $G$-invariant non-degenerate bilinear forms on the Lie…

Let $G$ be a complex reductive algebraic group, $g$ its Lie algebra and $h$ a reductive subalgebra of $g$, $n$ a positive integer. Consider the diagonal actions $G:g^n, N_G(h):h^n$. We study a relation between the algebra $C[h^n]^{N_G(h)}$…

It is known that if the special automorphism group $\text{SAut}(X)$ of a quasiaffine variety $X$ of dimension at least $2$ acts transitively on $X$, then this action is infinitely transitive. In this paper we address the question whether…

Let $G$ be a semisimple algebraic group whose decomposition into a product of simple components does not contain simple groups of type $A$, and $P\subseteq G$ be a parabolic subgroup. Extending the results of Popov [7], we enumerate all…

A description of transitive actions of a semisimple algebraic group G on toric varieties is obtained. Every toric variety admitting such an action lies between a product of punctured affine spaces and a product of projective spaces. The…

For a semisimple real Lie group $G$ with an irreducible representation $\rho$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $\rho$ for existence of a group of affine transformations of $V$ whose linear…

Let $G$ be a finite group. There is a standard theorem on the classification of $G$-equivariant finite dimensional simple commutative, associative, and Lie algebras (i.e., simple algebras of these types in the category of representations of…

Let $X$ be an affine algebraic variety endowed with an action of complexity one of an algebraic torus $\mathbb{T}$. It is well known that homogeneous locally nilpotent derivations on the algebra of regular functions $\mathbb{K}[X]$ can be…

We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner…

We provide an algebraic characterization of transitive, finite-dimensional algebraic Lie pseudogroups (or $\mathcal{D}$-groupoids) that are algebraic integrable, that is, isogenous to the action groupoid of an algebraic group action. Our…

Consider a transitive action of a Lie group $G$ on a (real analytic) manifold $M$ of dimension $m$, and two (embedded) submanifolds $A$ and $B$ in $M$ of sufficiently large class and of dimension $k$ and $l$, respectively. We prove that,…

The equivariant movability of topological spaces with an action of a given topological group $G$ is considered. In particular, the equivariant movability of topological groups is studied. It is proved that a second countable group $G$ is…

This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results…

The adjoint action of a finite group of Lie type on its Lie algebra is studied. A simple formula is conjectured for the number of split semisimple orbits of a given genus. This conjecture is proved for type A, and partial results are…

We prove that if $G$ is a group of finite Morley rank which acts definably and generically sharply $n$-transitively on a connected abelian group $V$ of Morley rank $n$ with no involutions, then there is an algebraically closed field $F$ of…

The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if $G=N\times A$ is a connected, simply connected, nilpotent Lie group with an…

A transitive Lie algebra g of rational vector fields on a projective manifold which do not preserve any foliation determines a rational map to an algebraic homogenous space G/H which maps g to lie(G).

Let $G$ be a group with socle a simple group of Lie type defined over the finite field with $q$ elements where $q$ is a power of the prime $p$. Suppose that $G$ acts transitively upon the lines of a linear space $\mathcal{S}$. We show that…

In this paper, we study a natural class of groups that act as affine transformations of $\mathbb T^N$. We investigate whether these solvable, "abelian-by-cyclic," groups can act smoothly and nonaffinely on $\mathbb T^N$ while remaining…

Let $\mathfrak{g}$ be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows to define a generalized multiplication $f \# g = (f^{\vee} * g^{\vee})^{\wedge}$ of two…