Search Results

As the example of nfdc23 shows, the answer is generally no. But maybe it helps to think about this question in a somewhat wider context, where the notions of split and anisotropic tori arise: the stu …

The question is stated in a somewhat confusing way, so a little historical perspective may help. Going back to the Chevalley seminar of 1956-58, where these ideas originate, one deals mainly with co …

I'm not sure where your "statement" comes from, or why the specific type of embedding of $H$ is assumed here. But the closest relative of this situation I'm aware of goes back to work of Kostant (in …

Gary Seitz (and various collaborators over the years) have worked out lots of concrete information about maximal closed subgroups of classical groups and exceptional algebraic groups over an algebraic …

Maybe it's helpful to add some comments to what Allen has already said, since the question is formulated loosely and can be looked at from different viewpoints. 1) It would help to point out explicit …

Since the question is formulated in the language of the papers by Borel-Tits, I'd recommend $\S 22$ of Borel's GTM 126 Linear Algebraic Groups (on central isogenies) as a reasonable reference. See e …

Here is a short answer, which can be filled in further. Whether your group is of adjoint type or not probably makes little difference. Work going back to a fundamental paper of Borel and deSiebenth …

To reinforce Angelo's example, it's worthwhile to point out the broader setting for this kind of question: the study of centralizers and connectedness properties in a semisimple (or more generally red …

The answer seems to be no, according to II, 5.8 (and following material) in the Springer-Steinberg lecture notes (Lect. Notes in Math 131, 1970), though I might be overlooking something in your questi …

Centralizers of arbitrary elements in a general linear group (over an arbitrary algebraically closed field) are connected for an easy reason: the centralizer in the space of $n \times n$ matrices is …

First of all, it's probably intended that "linear algebraic groups" are semisimple or at least reductive (and connected). For example, a solvable group might have no semisimple elements except 1 (et …

The question itself falls somewhat short of being "research-level", but maybe it's useful to expand a little on the comment by anon (which the proposed "answer" by Anupam doesn't improve on). The …

For a full treatment of the foundations it's best to consult Part I of the book Representations of Algebraic Groups by J.C. Jantzen (2nd ed., AMS, 2003) even though it's not easily available online. …

To amplify Brian Conrad's semi-answer, I need a more precise definition of "simply connected" at the outset. In characteristic 0 some of the classical ways of thinking about this concept can be carr …

To amplify Pete's answer, there is a reasonable discussion in Section 18 of the second edition of Borel's Linear Algebraic Groups (Springer GTM 126). In particular, his Corollary 18.3 following a di …