Search results for "Algebraic group"

Etude de certaines familles de variétés algébriques munies d'une action de groupe algébrique

2021

Su certe classi di gruppi unipotenti

2005

We introduce some results characterizing unipotent algebraic groups having a chain as the lattice of connected subgroups and we discuss some consequent results.

The J-invariant, Tits algebras and Triality

2012

In the present paper we set up a connection between the indices of the Tits algebras of a simple linear algebraic group $G$ and the degree one parameters of its motivic $J$-invariant. Our main technical tool are the second Chern class map and Grothendieck's $\gamma$-filtration. As an application we recover some known results on the $J$-invariant of quadratic forms of small dimension; we describe all possible values of the $J$-invariant of an algebra with orthogonal involution up to degree 8 and give explicit examples; we establish several relations between the $J$-invariant of an algebra $A$ with orthogonal involution and the $J$-invariant of the corresponding quadratic form over the functi…

Invariant deformation theory of affine schemes with reductive group action

2015

We develop an invariant deformation theory, in a form accessible to practice, for affine schemes $W$ equipped with an action of a reductive algebraic group $G$. Given the defining equations of a $G$-invariant subscheme $X \subset W$, we device an algorithm to compute the universal deformation of $X$ in terms of generators and relations up to a given order. In many situations, our algorithm even computes an algebraization of the universal deformation. As an application, we determine new families of examples of the invariant Hilbert scheme of Alexeev and Brion, where $G$ is a classical group acting on a classical representation, and describe their singularities.

Wielandt's results for algebraic k-groups

2006

We analyze the relation between subnormality and nilpotence, the subnormal joint property, some criteria of subnormality, the norm and the Wielandt subgroup in the case of algebraic groups defined over an arbitrary field.

Stability conditions and related filtrations for $(G,h)$-constellations

2017

Given an infinite reductive algebraic group $G$, we consider $G$-equivariant coherent sheaves with prescribed multiplicities, called $(G,h)$-constellations, for which two stability notions arise. The first one is analogous to the $\theta$-stability defined for quiver representations by King and for $G$-constellations by Craw and Ishii, but depending on infinitely many parameters. The second one comes from Geometric Invariant Theory in the construction of a moduli space for $(G,h)$-constellations, and depends on some finite subset $D$ of the isomorphy classes of irreducible representations of $G$. We show that these two stability notions do not coincide, answering negatively a question raise…

Monothetic algebraic groups

2007

AbstractWe call an algebraic group monothetic if it possesses a dense cyclic subgroup. For an arbitrary field k we describe the structure of all, not necessarily affine, monothetic k-groups G and determine in which cases G has a k-rational generator.

Algebraic Groups and Lie Groups with Few Factors

2008

In the theory of locally compact topological groups, the aspects and notions from abstract group theory have conquered a meaningful place from the beginning (see New Bibliography in [44] and, e.g. [41–43]). Imposing grouptheoretical conditions on the closed connected subgroups of a topological group has always been the way to develop the theory of locally compact groups along the lines of the theory of abstract groups. Despite the fact that the class of algebraic groups has become a classical object in the mathematics of the last decades, most of the attention was concentrated on reductive algebraic groups. For an affine connected solvable algebraic group G, the theorem of Lie–Kolchin has b…

Algebraic groups as difference Galois groups of linear differential equations

2019

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is that every linear algebraic group, considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over $\mathbb{C}(x)$.

Algebraic singularities have maximal reductive automorphism groups

1989

LetX = On/ibe an analytic singularity where ṫ is an ideal of theC-algebraOnof germs of analytic functions on (Cn, 0). Letdenote the maximal ideal ofXandA= AutXits group of automorphisms. An abstract subgroupequipped with the structure of an algebraic group is calledalgebraic subgroupofAif the natural representations ofGon all “higher cotangent spaces”are rational. Letπbe the representation ofAon the first cotangent spaceandA1=π(A).