All Questions
2,543 questions
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$\sigma$-compactness of some locally compact Hausdorff topological groups
Is the topological group $(\mathbf{Q}_p/\mathbf{Z}_p)^{\oplus k}$, $k\ge 1$, a $\sigma$-compact topological group when endowed with its natural $p$-adic topology?
More generally, I'm looking for a ...
user480741
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1
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273
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Twisted forms of $\mathrm{SL}(2,q)$
$\DeclareMathOperator\SL{SL}$Let $q = p^r$ be a prime power. Let $H$ denote the subgroup of $\SL(2,\overline{\mathbb{F}}_q)$ consisting of matrices of the form $\begin{pmatrix}a & b\\ b^q & a^...
- 7,547
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1
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172
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Product of subgroups of $SU(8)$ algebraic set?
Consider the special unitary group SU(8) acting on $\mathbb{C}^8\stackrel{\sim}{=}(\mathbb{C}^2)^{\otimes 3}$.
In particular, I am interested in the two subgroups $G_1=\mathrm{id}_{\mathbb{C}^2}\...
- 449
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196
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If $\Lambda \cap U$ is Zariski-dense in $U$, then $\Lambda$ contains $U(k\mathbb Z)$ for some $k ≥ 1$?
If $G$ is a $\mathbb Q$-defined subgroup of $\operatorname{GL}_n(\mathbb C)$, $\Lambda$ is a subgroup of $G(\mathbb Z)$, and $U$ is a unipotent subgroup of $G(\mathbb C)$ such that $\Lambda \cap U$ is ...
- 332
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1
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287
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Element in finite number of Borel subgroups
Let G is a linear algebraic group over algebraic closed field, B is an
Borel subgroup of G. Does there exist g$\in$G which is only in a finite
numbers of conjugates of B (they are also Borel ...
- 403
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1
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237
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A non trivial example of an anti -affine algebraic group
An anti- affine group $G$ is defined to be an algebraic group with no global sections. Examples include abelian varieties and non trivial extensions of abelian varieties by torus (in characteristic $\...
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1
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121
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Functorial description of a certain subgroup scheme
We work with schemes over an arbitrary field $k$. Let $X$ be a scheme, and $G$ a group scheme acting on $X$. Let $Y\subseteq X$ be a locally closed subscheme. Consider the following functor $N$: for ...
- 95
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142
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does $Aut^0$ act trivially on the Neron-Severi group?
Let $X$ be a projective integral scheme over an algebraically closed field $k$. Does $\mathrm{Aut}^0_{X/k}(k)$ act trivially on $NS(X)$?
- 181
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241
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locally closed orbits in metric Hausdorff topology
I learned the following fact from Bruhat and Tits's paper "Homomorphismes “abstraits” de groupes algebriques simples" Section 3.18 that
Let $k$ be a local field. Suppose that a $k$-group $H$ acts $k$...
- 1,702
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261
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Concept of Facets in the structure of reductive algebraic groups
Where can I find a precise definition of Facet ? In some online notes it is stated that Facet is a maximal subset of co-characters having the same sign for every root. But shouldn't then every facet ...
- 61
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361
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Subgroups generated by opposite root groups
Suppose $\mathbf{G}$ is a connected reductive (possibly non-split!) group over a field $F$, $\mathbf{S} \leq \mathbf{G}$ a maximal split subtorus and $\mathbf{Z} \leq \mathbf{G}$ its centralizer. For ...
- 1,704
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179
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Reducible reductive Lie subalgebras of so(p,q)
Is it true that $S(O(p) \times O(q))$ is the only proper subgroup of $SO(p,q)$ of full rank acting on the natural representation $\mathbb{R}^{p+q}$ of $SO(p,q)$ that stabilizes a $p$-dimensional ...
- 601
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401
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$\Gamma$-action on maximal tori in Borel-Tits
This is about section 6.2 in Borel-Tits' Groupes réductifs where they define a certain $\Gamma$-action on maximal split tori, denoted as $_\Delta \gamma$, distinct from the "usual" one. (If I am not ...
- 2,454
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387
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weights and exceptional root systems
Let $G$ a simple simply connected group over $\mathbb{C}$ and $W$ his Weyl group.
Let $\lambda$ a minuscule or quasiminuscule weight.
For which types and for which weights do we have that:
$\forall ...
- 3,472
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257
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arithmetic group over function fields and its fundamental domain
Let $G$ be a semi-simple algebraic group defined over a global function field $K$.
Let $S$ be a finite set of places of $K$. For a place $v$ of $K$ let $K_v$ be the completion under $v$. We take $K_S=\...
- 853
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326
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A Criterion for Reductivity of Lie Subgroups
Let $G$ be a connected, simply-connected, complex, semisimple Lie group. Suppose that $H$ is a Zariski-closed subgroup of $G$ with reductive Lie algebra $\frak{h}$. Under what conditions may one ...
- 4,920
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265
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How we characterize a subgroup of finite group of Lie type with unipotent elements.
Let $G$ be a finite group of Lie type. Let $H$ be a subgroup of $G$ which contains unipotent elements. I want to find a 'nice' subgroup of $G$ that contains $H$, for example a Levi subgroup of $G$ ...
- 35
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1
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808
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On the Steinberg section
Let $\chi:G\rightarrow T/W$ the Steinberg map. I assume that G is simply connected. Then $T/W=\mathbb{A}^{r}$ and Steinberg constructed a section to this map given by
$\epsilon(a_{1},...,a_{r})=x_{\...
- 3,472
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185
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Let G be an affine connected algebraic group. When a subvariety of G with codimension one is a subgroup.
Let G be an affine connected algebraic group, and K[G] be its coordinate ring. Let Y be a subvariety of G defined as a zero set for some f in K[G]. For which f, Y is a closed subgroup of G
- 225
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692
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Lie Group Principal Embedding
I'm reading a paper on complex semi-simple algebraic group geometry at the moment, but finding the going a bit tough since I'm missing alot of the prerequisites. Firstly, the author refers to a ...
- 1,462
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296
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Computing the connected component without primary decomposition
Given an algebraically closed field $\mathbb{F}$ of characteristic $0$ and a closed subgroup $G$ of $GL_n(\mathbb{F})$. Let $\{g_1,\ldots, g_r\}$ be a Gr"obner basis for the correpsonding ideal $\...
- 53
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325
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How to make commutative algebraic groups strongly dualizable?
Let's use the notation of [A=>B] for Hom(A, B). Take a 1-dimensional algebraic torus G<...
- 15.1k
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80
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$p$-torsion related to algebraic groups
Definition $14.14$ from "Linear algebraic groups and finite groups of Lie type" by Malle and Testerman:
A prime $p$ is a torsion prime for a linear algebraic group $G$ if the fundamental ...
- 217
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202
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action of the extra-special group
I'm reading a paper which has this line:
A direct computation shows that $P\Omega_8$($\mathbb K$) has an elementary abelian subgroup $X = 2^2$ such that $C_{P\Omega_8(\mathbb K)}(X) = T_4.2^{1+4}_+$. ...
- 43
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371
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Weyl group actions on standard parabolic subgroups of classical groups [closed]
$\DeclareMathOperator\U{U}\DeclareMathOperator\GL{GL}$Let $E/F$ be a quadratic extension of local fields and $G=U(V)$ a unitary group associated to hermitian space $V$ over $E/F$. We fix a minimal ...
- 1,759
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350
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Small questions in studying Arthur 's book 'Introduction to the Trace formula'
I am reading Arthur's book "Introductionto the trace formula".
In reading the book, two small question has arised and so I would like to ask it.
Let $G$ be a connected reductive group over $\mathbb{...
- 1,759
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170
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Maps to additive group scheme
Let $\underline{\mathbb{Q}_p/\mathbb{Z}_p}$ be constant p-divisible group over $\mathbb{F}_p$. And let $\mathbb{G}_a$ be the additive group over $\mathbb{F}_p$. Let me prove
$$
Hom(\underline{\mathbb{...
- 397
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506
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On some notations and notions of a paper on smoothness of Schubert varieties by Lakshmibai and Sandhya
I am reading the paper Criterion for smoothness of Schubert varieties in $\mathrm{Sl}(n)/B$ by V Lakshmibai and B Sandhya; Proc. Indian Acad. Sci. (Math. Sci.), Vol. 100, No. 1, April 1990, pp. 45-52. ...
- 235
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185
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Relative position and change of torus
Let $G$ be a connected split reductive group over a field $k$ of characteristic $0$. Let $T$ and $T'$ be two split maximal tori of $G$ and $B \supset T, B' \supset T'$ be two Borel subgroups of $G$.
...
- 123
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148
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How to decompose an map $\phi: \mathbb{G}_m \to T$ as the product of a cocharacter $\phi'$ and a map $\phi'':\mathbb{G}_m \to T$?
Let $\mathbb{G}_m$ be the multiplicative group and $T$ a maximal torus of a semisimple group. Let $X^*(T)=\{ \phi: T \to \mathbb{G}_m \}$ be the set of characters and $X_*(T)=\{ \phi^{\vee}: \mathbb{G}...
- 6,211
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542
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Generalization of a theorem of Steinberg
Steinberg has a beautiful theorem counting $F$-stable maximal tori in a reductive group. Here's the version of the result that you can find as Theorem 3.4.1 of Carter's Finite groups of Lie type:
...
- 11.2k
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664
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Verifying that a differential is surjective
I've been reading "Weakly commensurable arithmetic groups and locally symmetric spaces" (Prasad and Rapinchuk, 2009). I'm having some trouble showing the following fact:
Let $K_v$ be a local field, $...
- 11
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1
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217
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Decomposing quasi-finite separated group schemes
Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps)
Why is $G = G_1\sqcup G_2$, where $G_1 \to U$ is finite and $G_2\to U$ ...
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1
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163
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when the derived group of the group of $k$-rational points has nonempty interior in the strong topology
Suppose that $G$ is an absolutely quasi-simple algebraic group defined over a non-archimedean local field $k$ of positive characteristic. Would there be any kind of reasonable sufficient condition for ...
- 2,125
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578
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Decomposing Semisimple Perverse Sheaves
So I asked this on maths SE because I don't truly consider it to be a research level question. This question mostly arises out of my completely limited understanding of perverse sheaves. However I do ...
- 2,902
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376
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on z-extensions
Let $G$ a group split over a local field $F$.
We call a $z$-extension a group $G'$ such that $G'_{der}$ is simply connected, $G'$ is a central extension of $G$ by a central torus $Z$.
Can we find a $...
- 3,472
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1
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666
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Conjugacy classes in Aut(G)
Let $G$ be a connected, simply-connected simple group over $\mathbb{C}$. The structure/classification of conjugacy classes of $G$ is described in many places.
Now, I'd like to know the structure/...
- 6,133
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311
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A weird action of SL_3 on a pair of lines
Let us consider the complex projective plane $P^2$ and two distinct lines $L,L'\subset P^2$. Let us moreover consider the restriction of the natural action of $SL_3$ to $L\cup L'$. Can you tell in ...
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89
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Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 6,006
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79
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Invariant theory (first fundamental theorem) for a direct sum of two fundamental representations
Let $G$ be a simple reductive group over $\mathbb C$, e.g. $G=\mathrm{SO}(V)$ is a special orthogonal group.
Let $W_1$ and $W_2$ be two irreducible representations of $G$. Assume both $W_i$ are ...
- 6,622
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73
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Is the subgroup of $R$-trivial classes of an algebraic group an algebraic subgroup?
Let $G$ be an algebraic group over a field $F$. I'm willing to assume it is linear if that changes anything to what I'm going to say, and even reductive if that helps (but I don't think it should make ...
- 272
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82
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Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
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138
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Quotients of open subsets of the semi-stable locus
This is a rewrite of a deleted question. I've decided to focus on one particular example mentioned in that question. Below a point means a closed point.
Let $U$ be the set of irreducible non-cuspidal ...
- 23.5k
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76
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Cartan decomposition over a not-necessarily-discretely-valued field
Let $K$ be a valued field, and let $R$ be the valuation ring of $K$. Let $G$ be a split reductive group over $K$ and $T$ a maximal torus of $G$. On page 107 Berkvoich's book "Spectral theory and ...
- 63
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161
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Computer computation of the first Galois cohomology of a $p$-adic torus?
Let $T\subset {\rm GL}(N,{\mathbb Q})$ be an $n$-dimensional ${\mathbb Q}$-torus
given by its Lie algebra $\mathfrak{t}\subset \mathfrak{gl}(N,{\mathbb Q})$.
I want to compute, in some sense ...
- 14.2k
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77
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Decomposition of $BwBw^{-1}B$
Let $(B,N,W,S)$ be a Tits system with $W$ a finite coxeter group.
Let $w\in W$, consider $BwBw^{-1}B$, then by Bruhat decomposition, it is a disjoint union of some $BxB$, $x\in W$.
My question: Let $X\...
- 653
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139
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Question on two types of Frobenius theorem in $p$-adic groups
Let $G$ be a $p$-adic classical group and let $P_0$ be a minimal parabolic subgroup of $G$. Let $P=MN$ be a
standard parabolic subgroup containing $P_0$. Let $\text{Ind}$ and $\text{Jac}$ be the ...
- 1,019
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0
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95
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Injection of $G(k)/Z(k)$ into $(G/Z)(k)$
In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? ...
- 131
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80
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How to know the character table of the twisted group algebra of the symmetric group $S_4$
Given the character table of its Schur cover group, is there a way to obtain the character table of twisted group algebra from that? I am particularly interested in the symmetric group $S_4$.
- 11
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0
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64
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A variation of the dual group of the adjoint group
Let $\mathbf{G}$ be connected reductive group over a $p$-adic field $F$. Denote by $\mathbf{Z}$ the center of $\mathbf{G}$, and $\mathbf{A}$ the maximal split torus of $\mathbf{Z}$ (also called the ...
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