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Questions tagged [algebraic-groups]

Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.

2107 questions
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cohomology theory for algebraic groups

Is there a cohomology theory for algebraic groups which captures the variety structure and restricts to the ordinary group cohomology under certain conditions.
sim
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17
votes
1 answer

Is $GL_n(\mathbb{Q}_p)=GL_n(\mathbb{Z}_p)GL_n(\mathbb{Q})$?

Is $GL_n(\mathbb{Q}_p)=GL_n(\mathbb{Z}_p)GL_n(\mathbb{Q})$? Generally, let $R$ be a discrete valuation ring and $K$ its fraction field. Let $\widehat{R}$ be the completion and $\widehat{K}$ the fraction field of $\widehat{R}$. Is…
wuzx
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Diagonalizable subgroups of a connected linear algebraic group

Let $G$ be a connected linear algebraic group over an algebraically closed field $k$ of characteristic 0. Let $D\subset G$ be a closed diagonalizable subgroup of $G$ (a subgroup of multiplicative type). Is it true that $D$ is contained in some…
Mikhail Borovoi
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14
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Analog of Peter-Weyl theorem for $G[[t]]$

Let $G$ be a reductive group over ${\mathbb C}$ and let $G[[t]]$ denote the corresponding group over the formal power series ring ${\mathbb C}[[t]]$. This is a group scheme, so one can speak about its ring of functions (by definition this is the…
Alexander Braverman
  • 6,967
14
votes
3 answers

Iwasawa Decomposition

Does anyone know where I can find a proof of the Iwasawa decomposition for reductive groups? I know that there are a couple of related results that are called the Iwasawa decomposition, but I am interested in the following statement: Let G be a…
Dinakar Muthiah
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13
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3 answers

Density question in algebraic group

Suppose G is an algebraic group defined over F, the algebraic closure of F is K. Consider the Zariski topology on G(K), is G(F) Zariski dense in G(K)?
Bin Xu
  • 131
13
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2 answers

Is GL2( R ) - > PGL2( R ) surjective?

Consider $GL_2$ as the affine group scheme with coordinate ring ${\mathbb Z}[x_1,x_2,x_3,x_4,y]/(\det\left(\begin{array}{cc}x_1& x_2\\ x_3& x_4\end{array}\right)y-1)$. The group scheme $PGL_2$ is then given by the subring $S$ of $GL_1$-invariants,…
user1688
13
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2 answers

Simply connected simple algebraic groups

Before asking the question I should say that I don't know much about algebraic groups and I'm not sure if the question has the right level for MO. If not, please let me know and I will delete the question that emerged when I was trying to understand…
Demin Hu
  • 569
12
votes
2 answers

Examples of non-split algebraic groups

I am interested in knowing various examples of non-split (added hypothesis reductive) reductive linear algebraic groups. In particular, I would like to collect the following examples in my counter-example toolbox. Given an integer $n>2$, an…
Abhishek Parab
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11
votes
4 answers

Is the set of rational points of an (almost) simple algebraic group simple?

Let $G$ be an almost simple algebraic group defined over a field $K$. Then we know that, for $H = G/Z(G)$, the set of rational points $H(\overline{K})$ is a simple group (when considered with the group operation inherited from $G$). Must the set of…
H A Helfgott
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10
votes
4 answers

Connected components of the orthogonal group O(2n) in characteristic 2.

I am looking for a reference for the following fact: The orthogonal group $O_{2n}$ over an algebraically closed field of characteristic 2 has exactly two connected components. To be more precise, let $O_q$ denote the orthogonal group of the…
Mikhail Borovoi
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10
votes
1 answer

Realizations and pinnings (épinglages) of reductive groups

Let $G$ be a reductive group over an (say, algebraically closed) field $k$. Springer (in his book on algebraic groups) calls for a chosen maximal torus $T$ in $G$ a family $(u_\alpha) _{\alpha \in \Phi(G,T)}$ of immersions $u _\alpha:\mathbf{G}_a…
user717
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9
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2 answers

Difference between $\mathfrak{g}/\!/G$ and $G/\!/G$

I am studying a GIT quotient and I have a question that may be very silly. Let $G$ be a connected reductive group and $\mathfrak{g}$ its Lie algebra. Then are there some differences between $\mathfrak{g}/\!/G$ and $G/\!/G$? I started this question…
lafes
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7
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1 answer

Class number of PGL_2

Hello. Let $K = F_q(x)$ be the rational function field and let $G = \textbf{PGL}_{2,K}$. For any finite and non empty set $S$ of valuations of $K$, we refer to the subgroup of the adelic group $G(\Bbb{A})$: $$ G(A_S) = \prod_{p \in S} G_p(K_p)…
Rony Bitan
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7
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anisotropic and elliptic tori in GL(n)

Let $F$ be a commutative field and $n\geqslant 2$ be an integer. It is well known that the maximal anisotropic mod center tori in $G={\rm GL}(n,F)$ are of the form $T = {\rm Res}_{E/F}\; {\mathbb G}_m$,for some degree $n$ separable field extension…
Paul Broussous
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