Questions tagged [algebraic-groups]

For questions about groups which have additional structure as algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure. These include both linear algebraic groups as well as abelian varieties. Consider using with the (group-theory) tag.

This tag is for questions about algebraic groups. There are two main types of algebraic groups: linear algebraic groups and abelian varieties. The prototypical example of a linear algebraic group is $\mathrm{GL}_n$, the group of $n\times n$ matrices. The prototypical example of an abelian variety is an elliptic curve, which is the set of solutions to an equation $y^2 = x^3 + Ax + B$.

Over a field $k$, an algebraic group consists of (i) an underlying set $G$ defined as an algebraic subset of either affine space $G \subset \mathbb{A}^n_k$ (in the case of linear algebraic groups) or projective space $G \subset \mathbb{P}^n_k$ (in the case of abelian varieties) and (ii) a group operation called multiplication, which is a polynomial function $m\colon G \times G \to G$ satisfying axioms of associativity, invertibility, and identity. The set $G$ is referred to as an algebraic variety, and it is endowed with the Zariski topology, which is defined as the coarsest topology such that all subsets $Z$, which are cut out by the vanishing of a collection of polynomials, are closed. These closed subsets $Z \subset G$ are also algebraic varieties, called sub-varieties. If $Z$ is closed under the restriction of the multiplication map, i.e. if $m(Z \times Z) \subset Z$, then $Z$ also inherits a group structure and is called an algebraic subgroup of $G$.

Note that here we have not required a variety to be irreducible, which is a condition used by some authors. We have also not required $k$ to be algebraically closed, as some authors do.

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Is there an atlas of Algebraic Groups and corresponding Coordinate rings?

I was wondering if there was a resource that listed known algebraic groups and their corresponding coordinate rings. Edit: The previous wording was terrible. Given an algebraic group $G$, with Borel subgroup $B$ we can form the Flag Variety $G/B$…
BBischof
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p-adic Lie groups vs. algebraic groups over $\mathbb{Q}_p$

I am somewhat confused about the following two concepts and the relations between them- One concept is a Lie group $G$ over the $p$-adic field. This is defined in a similar fashion to a (real) Lie group - using atlases of charts, while making sure…
the_lar
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Description of Levi factors and unipotent radicals of parabolic subgroups in classical groups

For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor of $P$. If $G = GL(V)$ with $\dim V = n$, then $$P…
spin
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What is a $p$-adic group

I saw the name $p$-adic group on a book I was reading, so I tried to find some related documents. Although I've found something on this topic, there is no definition. Would anyone please explain the definition for a $p$-adic group to me? Thanks very…
ShinyaSakai
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Why are parabolic subgroups called "parabolic" subgroups?

I used to think that things called "parabolic" must have something to do with parabolas or their defining quadratic equations. In fact, terms like parabolic coordinate, parabolic partial differential equation and so on, are indeed related to…
ShinyaSakai
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What are the Borels/parabolics of the orthogonal or symplectic groups?

Does anyone know where I can find info about the Borel subgroups and parabolic subgroups of algebraic groups? I know what they are for the general linear group and for the special linear group you can just take the determinant 1 elements of a Borel…
Guest
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How to represent the naive PGL functor?

Let $k$ be a field. Consider the functor $F : \mathrm{Alg}(k) \to \mathrm{Set}, ~ R \mapsto \mathrm{GL}_n(R) / R^*$ [You might call this $\mathrm{PGL}_n^{\text{naive}}$, since it does not coincide on the correct $\mathrm{PGL}_n = \mathrm{GL}_n /…
Martin Brandenburg
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Geometric intuition for linearizing algebraic group action

I am reading an overview of geometric invariant theory and find myself stuck when we begin linearizing the action of an algebraic group on a variety. The definition given in my notes is that given an action $\sigma: G\times X\rightarrow X$ of an…
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Is there a surjective group homomorphism $\operatorname{GL}_{n}(k) \to \operatorname{GL}_{m}(k)$ where $n > m$?

Does there exist a field $k$, two positive integers $n > m > 1$, and a surjective group homomorphism $\operatorname{GL}_{n}(k) \to \operatorname{GL}_{m}(k)$? Here $k$ can be any field, and $\operatorname{GL}_{n}(k)$ is viewed as an abstract group…
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Unitary (algebraic) groups

I am looking for references on unitary groups in the algebraic setting: that is, given a quadratic extension $E/F$, the unitary groups (if I understand correctly) are subgroups of the Weil restriction of scalars of $GL(n)$ from $E$ down to $F$,…
M Turgeon
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Transcendental Galois Theory

Is there a good reference on transcendental Galois Theory? More precisely, if $K/k$ admits a separating transcendence basis (or maybe if it is a separably generated extension) it seems to me that many of the usual theorems of Galois theory go…
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Is the universal cover of an algebraic group an algebraic group?

Here algebraic group means affine algebraic group in both instances. Also I'm mainly interested in groups over $\mathbb{C}$. In fact I'm taking $\pi_1(G)$ to mean the fundamental group of $G_{an}$, the analytification. So I guess my question only…
solbap
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References on Linear Algebraic Groups/Lie Theory

I am currently doing a course on Lie groups, Lie Algebras and Representation theory based on Brian Hall's book of the same name. We should cover upto chapter 4/5 in this book by the end of the semester. Now for this course, our lecturer has…
user38268
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Is there a group between $SO(2,\mathbb{R})$ and $SL(2,\mathbb{R})$?

Is there a non-trivial subgroup $H \subset SL(2,\mathbb{R})$ such that $H \supset SO(2,\mathbb{R})$ ? My intuition is that, since $\dim SO(2)=1$ and $\dim SL(2)=3$, there should be some group between, but I can't point out one. Note : in the complex…
user10676
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Abelian subgroups of $GL_n(\mathbb{F}_p)$

Let $p$ be a prime number, and let $k=\mathbb{F}_p$ be the field of $p$ elements. Let $G=GL_n(k)$. We know that $$|G|=\prod_{i=0}^{n-1}(p^n-p^i)=p^{\binom{n}{2}}\prod_{i=0}^{n-1}(p^{n-i}-1)$$ so that the Sylow $p$-subgroups of $G$ have order…
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