# Tagged Questions

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

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### Check if ρ is an equivalence relation

Check if $$xρy \iff (x^2-y^2)(x^2y^2 - 1) = 1$$ is an equivalence relation. I know that for it to be an equivalence relation, a relation must have these properties: reflexivity, symmetry and ...
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### When is $SO(m,n)$ simple as a Lie group? What are the Zariski and Euclidean components?

Let $SO(m,n)=\operatorname{SO}(m,n)(\mathbb{R})$ denote the real $(m+n) \times (m+n)$ matrices, with determinant $1$, which preserve the quadratic form $x_1 + \cdots + x_m - x_{m+1} \cdots - x_{m+n}$ ...
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### Let $(G, *)$ be a group and let $\{g,h\}$ be a subset of $G$. Prove that $(g*h)^{-1}=h^{-1}*g^{-1 }$.

Let $(G, *)$ be a group and let $\{g,h\}$ be a subset of $G$. Prove that $(g*h)^{-1}=h^{-1}*g^{-1}$. I know that I should show that $X*Y=Y*X=e$. But I don't know how to calculate it.
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### Independence of parabolic subgroup in parabolic induction and restriction?

Suppose $G$ is a complex algebraic group, $L$ a proper Levi subgroup, and $\lambda$ an irreducible character of the subgroup $L^F$ of $F$-stable points in $L$, which is contained in $F$-stable ...
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### Analog of Hopf algebra structure for field of fractions

Let $G$ be a linear algebraic group. Then there is an additional structure on $k[G]$ called structure of Hopf algebra. Question: Is there an extra structure on field of fractions $k(G)$?
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### formula for the number of chambers incident to an $m$-simplex in a Bruhat-Tits building.

Let $\Delta$ be the Bruhat-Tits building for $PGL_n(K)$ with $K$ a non-archimedean local field with residue class field of cardinality $q$, $n\geq 2$. Suppose $m\leq n-2$, is there a formula for the ...
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### Cuspidal and Supercuspidal representation

Let $G$ be an algebraic group over a field $F$, and let $(\pi,V)$ be a smooth $G$-representation over an algebraic closed field $k$. Then $\pi$ is called a CUSPIDAL representation if $r(V)=0$ for any ...
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### Quotient of algebraic group by subgroup with trivial character group

Let $G$ be a linear algebraic group and $H$ be a closed subgroup. Suppose that all homomorphisms of algebraic groups $H\to\mathbb{G}_m$ are trivial. How to prove that $G/H$ is quasi-affine?
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### Common eigenvalue for a unipotent group of $GL_n (K)$ in positive characteristic

If we set $G$ a unipotent sub-group of $GL_n (K)$ with $car(K)>0$ ($\forall g\in G\quad g=1+n$ where $n$ is nilpotent), we wish to prove that $G$ is conjugate to a sub-group of $T$, the group of ...
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### Centralizer $C_G(T)$ of a maximal torus $T$ is in every Borel subgroup containing $T$?

I read that if $T$ is a maximal torus of a connected algebraic group $G$, then $C_G(T)$ is in every Borel subgroup containing $T$. I know $C_G(T)=N_G(T)^\circ$, the connected component of the ...
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### Bijective homomosphism of algebraic groups is isomorphism (characteristic 0)

Let $X$ and $Y$ be affine algebraic groups over an algebraically closed field $k$ of characteristic $0$, and $\phi: X \to Y$ be a group homomorphism, which is also a morphism of varieties. I would ...
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### Is Springer wrong here? Principal $F$-open sets

Let $A$ be an affine $k$-algebra with associated ring spaced $(X, \mathcal O_X)$, $F$ a subfield of $k$, and $A_0$ an $F$-structure on $A$. From Springer, Linear Algebraic Groups: A closed subset ...
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### Why is a parabolic subgroup $P$ connected?

A parabolic subgroup of a connected algebraic group is one which contains a Borel subgroup. I'm trying to understand why parabolic subgroups are connected. Let $P$ be a parabolic subgroup ...
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### Why is $\operatorname{Hom}_{\textrm{AG}}(k^\times,k^\times)\cong\mathbb{Z}$?

Suppose $k$ is an algebraically closed field, and $k^\times$ its multiplicative group. I read that $\operatorname{Hom}_{\textrm{AG}}(k^\times,k^\times)\cong\mathbb{Z}$, where the left consists of ...
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### “$F$-structures can be described in algebraic terms”

Let $(X, \mathcal O_X)$ be an affine variety (ringed space which is isomorphic to a closed subset of $k^n$). An $F$-structure on $(X, \mathcal O_X)$ is defined (Springer, Linear Algebraic Groups) to ...
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### Do these principal open sets really form a basis for the $F$-topology?

Let $F$ be a subfield of $k$ algebraically closed, and $k[\mathscr X]$ be the affine $k$-algebra associated with a Zariski closed set $\mathscr X \subseteq k^n$, and suppose $I(\mathscr X)$ can be ...
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### Showing the $F$-closed sets define a topology

Let $F \subseteq k$ be fields, $A$ an affine $k$-algebra, and $A_0$ an $F$-structure on $A$ (that is, an $F$-subalgebra of $A$ which is finitely generated as an $F$-algebra for which the natural $k$-...
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### Why is this tensor product of polynomial rings an isomorphism?

Let $F \subseteq k$ be fields, $k$ algebraically closed, $I$ a radical ideal of $k[X_1, ... , X_n]$. Then $I_0 := I \cap F[X_1, ... , X_n]$ is an ideal of $F[X_1, ... , X_n]$. Suppose that $I$ can ...
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### Projective general linear group over a ring

For $n$ a positive integer, I would like to define $PGL_n$ as a group scheme. One candidate is $X:=\mathrm{Proj}\big(\mathbb{Z}[x_{ij}:1\leq i,j\leq n]\big)\,\backslash \,\mathbb{V}(det)$, but I am ...
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### Free abelian subgroups of SL(3,$\mathbb{Z}$)

Does SL(3,$\mathbb{Z}$) have any free abelian subgroup of rank > 2? I want to find 3 $\times$ 3 integer matrices with determinant 1 such that the matrices are commutative, but there exists no other "...
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### How can we show that a product of affine $F$-varieties exists?

Let $\mathscr X \subseteq k^n, \mathscr Y \subseteq k^m$ be Zariski-closed sets, and let $k[\mathscr X] := k[X_1, ... , X_n]/I(\mathscr X)$ and $k[\mathscr Y] := k[Y_1, ... , Y_m]/I(\mathscr Y)$ be ...
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### What are $F$-structures all about?

This is a pretty open ended question. I'm reading Springer's book on algebraic groups and am very confused about these "$F$-structures." If $k$ is an algebraically closed field, and $A$ is an affine ...
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### Two-dimensional unipotent algebraic groups

How to prove that two-dimensional unipotent algebraic group over the field of characteristic 0 is commutative?
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### How can we continuously deform a height 1 formal group law into a height 2 formal group law?

A Quick Review: The complex elliptic curve $\mathbb{C}/(\mathbb{Z} + \tau \mathbb{Z})$ may be rewritten using the exponential, $\text{exp(}{2 \pi i \tau}) =: q$ as $\mathbb{C}^\times/q^\mathbb{Z}$ . ...
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### Generalization of Maschke theorem

For finite groups there is an isomorphism $$k[G]\cong \bigoplus\limits_{V -irrep}\mathrm{End}(V)$$ compatible with the group action. Can this fact be generalized to the case of linear algebraic groups?...
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### Kernel of homomorphism of algebraic groups

Let $\varphi:G\to H$ be a homomorphism of affine algebraic groups, and $\varphi^*:k[H]\to k[G]$ the corresponding homomorphism of coordinates rings. Let $I_H\subset k[H]$ be the augmentation ideal, i....
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### Examples of non-split algebraic groups.

An algebraic group over a field $K$ is called a split algebraic group if it has a Borel subgroup that has a composition series such that all the composition factors are isomorphic to either the ...
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### Different definitions of projective matrix groups, with one giving an algebraic group but not the other

Recently my professor told me that the usual definition of PSL(2,$\mathbb{R}$) = SL(2,$\mathbb{R}$)/{$\pm$I} does not give an algebraic group, but the following definition does: PSL(2,$\mathbb{R}$) =...