Let $𝒜$ be a commutative algebraic group defined over a number field $k$. We consider the following question:Let $r$ be a positive integer and let $P\in 𝒜\left(k\right)$. Suppose that for all but a finite number of primes $v$ of $k$, we have $P=r{D}_v$ for some $D_v\in 𝒜\left(k_v\right)$. Can one conclude that there exists $D\in 𝒜\left(k\right)$ such that $P=rD$?A complete answer for the case of the multiplicative group ${𝔾}_m$ is classical. We study other instances and in particular obtain an affirmative answer when $r$ is a prime and $𝒜$ is either an elliptic curve or a torus of small dimension...

Suslin’s local-global principle asserts that if a matrix over a polynomial ring vanishes modulo the independent variable and is locally elementary then it is elementary. In this article we prove Suslin’s local-global principle for principal congruence subgroups of Chevalley groups. This result is a common generalization of the result of Abe for the absolute case and Apte, Chattopadhyay and Rao for classical groups. For the absolute case the localglobal principle was recently obtained by Petrov and...

Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on noncommutative localizations, we introduce localizations compatible with coactions. Coinvariants of these localized coactions give local information about quotients. We define Zariski locally trivial quantum...

There is a well known relation between simple algebraic groups and simple singularities, cf. [5], [28]. The simple singularities appear as the generic singularity in codimension two of the unipotent variety of simple algebraic groups. Furthermore, the semi-universal deformation and the simultaneous resolution of the singularity can be constructed in terms of the algebraic group. The aim of these notes is to extend this kind of relation to loop groups and simple elliptic singularities. It is the...

Let Λ be a directed finite-dimensional algebra over a field k, and let B be an upper triangular bimodule over Λ. Then we show that the category of B-matrices mat B admits a projective generator P whose endomorphism algebra End P is quasi-hereditary. If A denotes the opposite algebra of End P, then the functor Hom(P,-) induces an equivalence between mat B and the category ℱ(Δ) of Δ-filtered A-modules. Moreover, any quasi-hereditary algebra whose category of Δ-filtered modules is equivalent to mat...

Let ${𝒪}_K$ be a discrete valuation ring of mixed characteristics $(0,p)$, with residue field $k$. Using work of Sekiguchi and Suwa, we construct some finite flat ${𝒪}_K$-models of the group scheme ${\mu }_{p^n,K}$ of $p^n$-th roots of unity, which we call Kummer group schemes. We carefully set out the general framework and algebraic properties of this construction. When $k$ is perfect and ${𝒪}_K$ is a complete totally ramified extension of the ring of Witt vectors $W\left(k\right)$, we provide a parallel study of the Breuil-Kisin modules of finite flat models...

Let A be a finitely generated associative algebra over an algebraically closed field. We characterize the finite-dimensional A-modules whose orbit closures are local hypersurfaces. The result is reduced to an analogous characterization for orbit closures of quiver representations obtained in Section 3.

With this work and its sequel, Moduli of unipotent representations II, we initiate a study of the finite dimensional algebraic representations of a unipotent group over a field of characteristic zero from the modular point of view. Let $G$ be such a group. The stack ${ℳ}_n\left(G\right)$ of all representations of dimension $n$ is badly behaved. In this first installment, we introduce a nondegeneracy condition which cuts out a substack ${ℳ}_n^{\mathrm{nd}}\left(G\right)$ which is better behaved, and, in particular, admits a coarse algebraic space, which...

We extend the methods of geometric invariant theory to actions of non–reductive groups in the case of homomorphisms between decomposable sheaves whose automorphism groups are non–reductive. Given a linearization of the natural action of the group $\mathrm{Aut}\left(E\right)\times \mathrm{Aut}\left(F\right)$ on Hom(E,F), a homomorphism is called stable if its orbit with respect to the unipotent radical is contained in the stable locus with respect to the natural reductive subgroup of the automorphism group. We encounter effective numerical conditions for...

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely...