Related papers: Algebraic (Volume) Density Property for Affine Hom…

For a homogeneous space X (not necessarily principal) of a connected algebraic group G (not necessarily linear) over a number field k, we prove a theorem of strong approximation for the adelic points of X in the Brauer-Manin set. Namely,…

A Danilov-Gizatullin surface is an affine surface $V$ which is the complement of an ample section $S$ of a Hirzebruch surface. The remarkable theorem of Danilov and Gizatullin states that the isomorphism class of $V$ depends only on the…

We consider the density properties of divergence-free vector fields $ b \in L^1([0,1],\textit{BV}([0,1]^2)) $ which are ergodic/weakly mixing/strongly mixing: this means that their Regular Lagrangian Flow $X_t$ is an ergodic/weakly…

For an irreducible affine variety $X$ over an algebraically closed field of characteristic zero we define two new classes of modules over the Lie algebra of vector fields on $X$ - gauge modules and Rudakov modules, which admit a compatible…

Building on work of Baldwin and Beaudoin, assuming Martin's Axiom, we construct a zero-dimensional separable metrizable space $X$ such that $X$ is countable dense homogeneous while $X^2$ is not. It follows from results of Hru\v{s}\'ak and…

Divergence-free Lie algebras are originated from the Lie algebras of volume-preserving transformation groups. Xu constructed a certain nongraded generalization, which may not contain any toral Cartan subalgebra. In this paper, we give a…

Let $V$ be a complex finite dimensional super vector space with an action of a connected semisimple group $G$. We classify those pairs $(G,V)$ for which all homogeneous components of the super symmetric algebra of $V$ decompose…

A cubic space is a vector space equipped with a symmetric trilinear form. Using categorical Fra\"iss\'e theory, we show that there is a universal ultrahomogeneous cubic space $V$ of countable infinite dimension, which is unique up to…

We solve the problem of constructing a genus-zero full conformal field theory (a conformal field theory on genus-zero Riemann surfaces containing both chiral and antichiral parts) from representations of a simple vertex operator algebra…

It is shown by the author in [J. Lie Theory 29:4, 1045-1070, 2019] that for every connected linear complex Lie group the algebra of polynomials (regular functions) is dense in the algebra of holomorphic functions of exponential type.…

In this paper we construct a linear space that parameterizes all invariant bilinear forms on a given vertex algebra with values in a arbitrary vector space. Also we prove that every invariant bilinear form on a vertex algebra is symmetric.…

Apart from global topological problems an affine homogeneous space is locally described by its curvature, its torsion and a slightly less tangible object called its connection in a given base point. Using this description of the local…

This thesis is devoted to the study of geometric properties of affine algebraic varieties endowed with an action of an algebraic torus. It comes from three preprints which correspond to the indicated points (1), (2), (3). Let $X$ be an…

A non-degenerate toric variety $X$ is called $S$-homogeneous if the subgroup of the automorphism group $\text{Aut}(X)$ generated by root subgroups acts on $X$ transitively. We prove that maximal $S$-homogeneous toric varieties are in…

The faithful representations of real Lie algebras g with dimensions $n\leq 4$ acting on the dual space of the universal covering algebra of an ideal s of g, are calculated. With the help of left and right translations of the Lie group, we…

A 'prehomogeneous vector space' is a rational representation $\rho:G\to\mathrm{GL}(V)$ of a connected complex linear algebraic group $G$ that has a Zariski open orbit $\Omega\subset V$. Mikio Sato showed that the hypersurface components of…

Let G be a reductive algebraic group and H a closed subgroup of G. An affine embedding of the homogeneous space G/H is an affine G-variety with an open G-orbit isomorphic to G/H. We start with some basic properties of affine embeddings and…

There exist three vector fields with complete polynomial flows on $\mathbb{C}^n$, $n \geq 2$, which generate the Lie algebra generated by all algebraic vector fields on $\mathbb{C}^n$ with complete polynomial flows. In particular, the flows…

The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of…

A commutative associative algebra A with an identity over the field of real numbers which has a basis, where all elements are invertible, is considered in the work. Moreover, among matrixes consisting of the structure constants of A, there…