Linear Algebraic Groups

On One-Sided Ideals of Rings of Linear Transformations

Linear Algebraic Groups

We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups.

Tannakian Categories, Linear Differential Algebraic Groups, and Parametrized Linear Differential Equations

Tannaka’s theorem (cf. [19]) states that a linear algebraic group is determined by its category of representations. The problem of recognizing when a category is the category of representations of a linear algebraic group (or, more generally, an affine group scheme) is attacked via the theory of neutral Tannakian categories (see [18], [9]). This theory allows one to detect the underlying presen...

Addendum to: "Infinite-dimensional versions of the primary, cyclic and Jordan decompositions", by M. Radjabalipour

In his paper mentioned in the title, which appears in the same issue of this journal, Mehdi Radjabalipour derives the cyclic decomposition of an algebraic linear transformation. A more general structure theory for linear transformations appears in Irving Kaplansky's lovely 1954 book on infinite abelian groups. We present a translation of Kaplansky's results for abelian groups into the terminolo...

On categories of merotopic, nearness, and filter algebras

We study algebraic properties of categories of Merotopic, Nearness, and Filter Algebras. We show that the category of filter torsion free abelian groups is an epireflective subcategory of the category of filter abelian groups. The forgetful functor from the category of filter rings to filter monoids is essentially algebraic and the forgetful functor from the category of filter groups to the cat...

Algebraic Groups RWTH Aachen , WS 2006 Jürgen

Algebraic groups are analogues of the classical Lie groups, such as the linear, orthogonal or symplectic groups, over arbitrary algebraically closed fields. Hence they are no longer classical manifolds, but varieties in the sense of algebraic geometry. In particular, they are used in the uniform description of the finite groups of Lie type, which encompass a substantial part of all finite simpl...