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We investigate commutator operations on compatible uniformities of an algebra. We present a commutator operation for compatible uniformities of an algebra in a congruence-modular variety which extends the commutator on congruences, and…
In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.
We discuss basic topological properties of unitary dual spaces of nilpotent Lie groups, using some ideas from operator algebras and their noncommutative dimension theory. The general results are illustrated by many examples.
We describe those unipotent representations of a finite group of Lie type which are defined over the rational numbers.
In this paper, we study modular categories whose Galois group actions on their simple objects are transitive. We show that such modular categories admit unique factorization into prime transitive factors. The representations of…
In this paper we study holomorphic actions of the complex multiplicative group on complex manifolds around a singular (fixed) point. We prove linearization results for the germ of action and also for the whole action under some conditions…
Liouville property of actions of discrete groups can be reformulated in terms of existence co-F$\o$lner sets. Since every action of amenable group is Liouville, the property can be served as an approach for proving non-amenability. The…
Let $\mathcal{C}$ be a conjugacy class of involutions in a group $G$. We study the graph $\Gamma(\mathcal{C})$ whose vertices are elements of $\mathcal{C}$ with $g,h\in\mathcal{C}$ connected by an edge if and only if $gh\in\mathcal{C}$. For…
Assume that two algebraic varieties of finite type over the complex numbers are related by a morphism whose fibers are precisely the orbits for the action of a unipotent group. We show that the two varieties have the same topological Euler…
By an additive action on an algebraic variety $X$ we mean a regular effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. In this paper, we give a uniqueness criterion for…
This paper addresses various questions about pairs of similarity classes of matrices which contain commuting elements. In the case of matrices over finite fields, we show that the problem of determining such pairs reduces to a question…
Let U be the unipotent radical of a Borel subgroup of a connected reductive algebraic group G, which is defined over an algebraically closed field k. In this paper, we extend work by Goodwin-R\"ohrle concerning the commuting variety of…
In this paper, the changes of representations of a group are used in order to describe its action as algebraic Galois group of an univariate polynomial on the roots of factors of any Lagrange resolvent. By this way, the Galois group of…
Originally motivated by algebraic invariant theory, we present an algorithm to enumerate integer vectors modulo the action of a permutation group. This problem generalizes the generation of unlabeled graph up to an isomorphism. In this…
The main result of the paper is the complete classification of the compact connected Lie groups acting coisotropically on complex Grassmannians. This is used to determine the polar actions on the same manifolds.
We give a criterion of existence of a unipotent group action on the affine cone over a projective variety or, more generally, on the affine quasicone over a variety which is projective over another affine variety.
The main purpose of this work is to develop the basic notions of the Lie theory for commutative algebras. We introduce a class of $\mathbbZ_2$-graded commutative but not associative algebras that we call ``Lie antialgebras''. These algebras…
We give a new, geometric proof of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers, as well as its natural nilpotent analogue. As a part of our…
We present a generalized version of classical geometric invariant theory \`a la Mumford where we consider an affine algebraic group $G$ acting on a specific affine algebraic variety $X$. We define the notions of linearly reductive and of…
We study the invariant algebraic D-modules on an affine variety under the action of an algebraic group.For linear algebraic groups with the multiplication action by themselves, such D-modules correspond to representations of their Lie…