Talk:Algebraic group

Semi-Direct Product Versus Extensions

Every algebraic group G contains a linear subgroup H such that G/H is an abelian variety, but the sequence 1 -> H -> G -> G/H -> 1 is in general not split. Hence it is wrong to say that G is a semidirect product of H and G/H. Joerg Winkelmann 11:53, 1 May 2006 (UTC)

Dear Joerg,

I think this has been fixed in the current version.

DeaconJohnFairfax (talk) 23:51, 30 June 2008 (UTC)

Examples

The text says that each finite group $G$ is also algebraic. I think that needs further explaining: why is that? Why is $G$ even the set of zeros of some polynomials (over which field? The group-algebra?). And why are inversion and multiplication morphismns? —Preceding unsigned comment added by 77.189.130.148 (talk) 16:03, 24 March 2008 (UTC)

Dear Anonymous User,

I think you're right about the finite group example needing more explanation.

Regarding inversion and multiplication being given by morphisms, nobody said they were. In fact, inversion is and multiplication is not (at least not with the natural definition x -> xy where y is fixed; i.e., xz -> xyxz = xzy is not true in general.)

The article said they were given by regular functions. For matrix inversion in Gln(C), that is Cramer's rule. The variety is defined by det(X) != 0. I'm not absolutely sure how matrix multiplication is given by a regular function, or by regular functions, but I think it is just by the inner product of a row and a column giving a coordinate of the product of the two matricies. I'm pretty sure the coordinate functions on an algebraic variety are regular functions (that's what the previous statement boils down to.)

For elliptic curves, the regular functions are given in the ususal proof that the product of two rational points on the curve is rational.

So, if you replace "morphism" by "regular function" in your request, then, I agree that it is a good suggestion.

DeaconJohnFairfax (talk) 23:49, 30 June 2008 (UTC)

A truly pathetic article

Despite containing true statements, this is a truly pathetic article!

It neglects to define its subject. And any Wikipedia that does not even say what it is about is not worth having in Wikipedia.

But algebraic groups is an important subject in mathematics. I hope that someone knowledgeable about this subject can patch up this article so that it is worth keeping in Wikipedia.2601:200:C000:1A0:F59A:E52F:20D8:1279 (talk) 02:45, 7 February 2021 (UTC)

Improvement needed: Algebraic groups vs other groups in the lead

I added this edit that was reverted, given to be too technical for the lead, hopefully part of it can be reused in future versions, maybe in the lead, maybe in the body:

A crucial aspect of algebraic groups compared to analytic and continuous groups is that they 
do not need to be topological groups (which would require a product topology).
In algebraic geometry they are defined on the basis of the Zariski topology, i.e. the topology of 
closed sets which is non-Haussdorff, a 
part from the case of dimension zero. 

I assume the average reader of this page to be a batchelor in whatever STEM subjects, and I hope we can agree on this principle. That the average reader does not need to know neither algebraic geometry, nor category theory, he has some notions of neighborhoods from a calculus course, and a notion of group from a first course in algebra but not much else.

In general the statement in the article lead is also containing too much tech terms:

• "regular maps" is quite a technical concept of algebraic geometry itself and too technical for the lead: the first intution about a map I would expect it to be for example smooth, not to be locally a polynomial, i.e locally $C^{n}$ minus special points for the sake of internal consistency arguments of algebraic geometry.

• the categoric definition of algebraic groups is also too technical for the lead, it's everything but constructive, intuitive or useful for educational purposes and it has quite a lot of ramifications that needs to be made explicit.

Not coming from an algebraic geometry background, I was actually very much confused by the definition in the lead, that's why I added the exact definition from humphreys, and edited it here. One objective of a lead would at least to mention distinctive features, I would expect to find intuitively why we need algebraic groups at all and why they are different from other groups. I perfectly know that you don't need a topology to define a group but tell me then why we need an algebraic group definition which is distinct from a standard group.

I think that reverting maybe a quick and easy fix, but does not really solve the problem, i.e the quality of the page. It ends up to leave a page like this in a bad shape for much longer, and it's purely done for time management issues of the person doing the revert. — Preceding unsigned comment added by Flyredeagle (talk • contribs) 09:07, 5 May 2022 (UTC)

The technical statements currently in the lede are actual definitions of an algebraic group. They are not the most fitting for being there and should eventually be replaced but they give an idea of what the article is about (groups and algebraic geometry).

On the other hand the sentences you added are about a technical property of algebraic groups (they are not topological groups when endowed with the Zariski topology) which does not belong in the lede at all. It should probably be mentioned in a section on the relations between Lie and algebraic groups in the body of the article. jraimbau (talk) 07:57, 5 May 2022 (UTC)

I actually can agree that they do not belong to the lead, and also agree that the other tech statements are kind of formally correct in any case. I think it should be more or less this way: the major reason of algebraic group to exist with a definition based on regular maps is to have everything in algebraic geometry as local polynomial maps, and ultimately as algebraic varieties, I imagine to use then polynomial rings, by consequence comes then the different topology, and let's say somewhat by accident is a Zariski one. Probably I was trying a topology first approach that is somewhat wrong, at least from an algebraic geometry only perspective, and I imagine that things such as the motives were invented to harmonize these kind of mismatches, in fact the de Rham one is smooth. Flyredeagle (talk) 21:00, 5 May 2022 (UTC)

References