Related papers: Geometric rigidity of simple modules for algebraic…

Let $\frak g$ be a simple finite-dimensional Lie algebra over an algebraically closed field $\mathbb F$ of characteristic 0. We denote by $\operatorname{U}(\frak g)$ the universal enveloping algebra of $\frak g$. To any nilpotent element…

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of positive characteristic. We will untwist the structure of G-modules by a newly found splitting of the Frobenius endomorphism on the algebra of…

We study the problem of determining, for a polynomial function $f$ on a vector space $V$, the linear transformations $g$ of $V$ such that $f g = f$. In case $f$ is invariant under a simple algebraic group $G$ acting irreducibly on $V$, we…

For a complex connected reductive group G, we classify the simple modules whose cone of primitive vectors admits a nontrivial G-invariant deformation. We relate this classification to that of simple Jordan algebras, and to that (due to…

In this article we describe the $G\times G$-equivariant $K$-ring of $X$, where $X$ is a regular compactification of a connected complex reductive algebraic group $G$. Furthermore, in the case when $G$ is a semisimple group of adjoint type,…

Let $k$ be a number field, $\mathbf{G}$ an algebraic group defined over $k$, and $\mathbf{G}(k)$ the group of $k$-rational points in $\mathbf{G}.$ We determine the set of functions on $\mathbf{G}(k)$ which are of positive type and…

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geq 0$ and let $V$ be an irreducible rational $G$-module with highest weight $\lambda$. When $V$ is self-dual, a basic question to ask…

We call a finite-dimensional complex Lie algebra $\mathfrak{g}$ strongly rigid if its universal enveloping algebra $\Ug$ is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation.…

Let $kG$ be the group algebra of a finite group scheme defined over a field $k$ of characteristic $p>0$. Associated to any closed subset $V$ of the projectivized prime ideal spectrum $\operatorname{Proj} \operatorname{H}^*(G,k)$ is a thick…

Let G be a reductive complex algebraic group and V a finite-dimensional G-module. From elements of the invariant algebra C[V]^G we obtain by polarization elements of C[kV]^G, where k\geq 1 and kV denotes the direct sum of k copies of V. For…

We study the structure of a metric $n$-Lie algebra $\mathcal {G}$ over the complex field $\mathbb C$. Let $\mathcal {G}= \mathcal S\oplus {\mathcal R}$ be the Levi decomposition, where $\mathcal R$ is the radical of $\mathcal {G}$ and…

A graph $G = (V,E)$ is globally rigid in $\mathbb{R}^d$ if for any generic placement $p : V \rightarrow \mathbb{R}^d$ of the vertices, the edge lengths $||p(u) - p(v)||, uv \in E$ uniquely determine $p$, up to congruence. In this paper we…

Let $k$ be a field and let $G$ be an affine algebraic group over $k$. Call a $G$-torsor weakly versal for a class of $k$-schemes $\cal C$ if it specializes to every $G$-torsor over a scheme in $\cal C$. A recent result of the first author,…

Let G be a finite group and let T(G) be the abelian group of equivalence classes of endotrivial kG-modules, where k is an algebraically closed field of characteristic p. We determine, in terms of the structure of G, the kernel of the…

Classifying endotrivial kG-modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite group G, has been a long-running quest. By deep work of Dade, Alperin, Carlson, Thevenaz, and others, it has been…

Let $k$ be a field, $\tilde{G}$ a connected reductive $k$-group, and $\Gamma$ a finite group. In a previous work, the authors defined what it means for a connected reductive $k$-group $G$ to be "parascopic" for $(\tilde{G},\Gamma)$.…

Let $p$ be a prime, let $G$ be a finite group of order divisible by $p$, and let $k$ be a field of characteristic $p$. An endotrivial $kG$-module is a finitely generated $kG$-module $M$ such that its endomorphism algebra…

In this article we define $G$-algebras, that is, graded algebras on which a reductive group $G$ acts as gradation preserving automorphisms. Starting from a finite dimensional $G$-module $V$ and the polynomial ring $\mathbb{C}[V]$, it is…

Let $V$ be a finite-dimensional vector space over a finite field, and suppose $G \leq \Gamma \mathrm{L}(V)$ is a group with a unique subnormal quasisimple subgroup $E(G)$ that is absolutely irreducible on $V$. A base for $G$ is a set of…

Let $k$ be a field of characteristic $p>0$, and let $W$ be a complete discrete valuation ring of characteristic $0$ that has $k$ as its residue field. Suppose $G$ is a finite group and $G^{\mathrm{ab},p}$ is its maximal abelian $p$-quotient…