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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1answer
22 views

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 ...
1
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1answer
29 views

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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2answers
38 views

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.
1
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1answer
31 views

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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0answers
18 views

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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0answers
9 views

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 ...
3
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1answer
42 views

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 ...
2
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0answers
31 views

What is $\operatorname{ad} x$ in the Mackey Formula?

Suppose $P=L\ltimes U$ and $Q=M\ltimes V$ are $F$-stable parabolic subgroups of an algebraic group $G$, and $L$ and $M$ are $F$-stable Levi complements. Let $S(L,M)=\{x\in G:L\cap\ ^xM\textrm{ ...
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0answers
27 views

reference request: existence of a complete model of a G-variety

Sumihiro has a result in this paper: http://projecteuclid.org/download/pdf_1/euclid.kjm/1250523277, to the effect that a variety $X$ with an action by a linear algebra group has a complete model (as a ...
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0answers
16 views

paper of Ihara, why must eigenvalues of an element of a torsion-free cocompact lattice be in the ground field?

In "Discrete Subgroups of PL(2,$k_{\mathcal{P}}$)" which appears on pp. 272-278 of "Algebraic Groups and Discontinuous Subgroups: Proceedings of Symposia in Pure Mathematics, Volume IX", Ihara makes ...
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1answer
111 views

If a complex Lie group has the structure of an algebraic group, is this structure unique?

If $G$ and $H$ are algebraic groups over $\mathbb{C}$, and $f : G \rightarrow H$ is an isomorphism of complex Lie groups (i.e. a biholomorphic group isomorphism), then must $f$ be algebraic? If not, ...
4
votes
1answer
212 views

Orbits of $SL(3, \mathbb{C})/B$

Let $B= \Bigg\{\begin{bmatrix} * & *&* \\ 0 & *&*\\ 0&0&* \end{bmatrix} \Bigg\}< SL(3,\mathbb C)$. What is $SL(3,\mathbb C)/B$? Do we use these facts: Borel fixed ...
4
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5answers
360 views

Is this group of matrices cyclic? [closed]

Is the group $H$ consisting of matrices of the form $ \left( {\begin{array}{cc} 1 & n \\ 0 & 1 \\ \end{array} } \right) $ cyclic, where $n \in \mathbb{Z}$? If not, how would ...
2
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0answers
21 views

Quotient of algebraic groups $SO_n/SO_{n-1}$

It is quite an elementary question but I need a little bit help to elaborate it. Let $\text{char}\,k\neq2$. I want to show that the quotient of algebraic groups $SO_n/SO_{n-1}$ is isomorphic to the ...
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1answer
36 views

Finite solvable subgroup in $GL_2(\mathbb{C})$ which is not contained in Borel subgroup [closed]

What is the example of finite solvable subgroup in $GL_2(\mathbb{C})$ which is not contained in any Borel subgroup?
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0answers
24 views

If $Q\subset P$ are parabolic subgroups, why is $R_u(P)\subset R_u(Q)$?

Suppose $Q\subset P$ are parabolic subgroups of some group $G$, and denote $R_u(Q)$ and $R_u(P)$ to be their unipotent radicals. Why is taking the unipotent radical inclusion reversing? I know $R_u(P)...
2
votes
0answers
49 views

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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0answers
17 views

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 ...
1
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1answer
40 views

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 ...
2
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2answers
57 views

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 ...
2
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0answers
34 views

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 ...
2
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1answer
42 views

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 ...
0
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0answers
23 views

Image of adjoint representation is Zariski closed?

For Lie groups $G$ recall the adjoint representation $$\operatorname{Ad}_G:G \to \operatorname{GL}(\mathfrak{g})$$ given by $$\operatorname{Ad}_G(g)X=gXg^{-1}.$$ Question: Is the image of $\...
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1answer
31 views

Is the commutator subgroup $[B,B]$ contained in the unipotent radical $R_u(B)$?

Suppose $B$ is a connected, solvable algebraic group, and $T$ a maximal torus of $B$. I read that if $S$ is a subtorus, then $N_B(S)=C_B(S)$, the normalizer is equal to the centralizer. I know that $...
1
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1answer
34 views

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 ...
1
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2answers
43 views

“$F$-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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0answers
18 views

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 ...
1
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1answer
31 views

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 $k$-...
2
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1answer
245 views

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 ...
1
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0answers
38 views

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 ...
2
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0answers
84 views

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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0answers
29 views

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 ...
4
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1answer
87 views

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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0answers
71 views

Two-dimensional unipotent algebraic groups

How to prove that two-dimensional unipotent algebraic group over the field of characteristic 0 is commutative?
4
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1answer
73 views

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 $\mathbb{C}^\times/q^\mathbb{Z}$ . ...
2
votes
0answers
47 views

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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0answers
37 views

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....
1
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1answer
55 views

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 ...
2
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1answer
21 views

How to show that $B$ embeds to $G/U^-$ densely?

Let $G=GL_n$ and $B$ its Borel subgroup consisting of all upper triangular matrices. Let $U^-$ be the unipotent subgroup of $G$ consisting of all unipotent lower triangular matrices. How to show that $...
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1answer
31 views

Are the set of all $G$-modules the same as the set of all $U(\mathfrak{g})$-modules?

Let $G$ be an algebraic group and $\mathfrak{g}$ its Lie algebra. Are the set of all $G$-modules the same as the set of all $U(\mathfrak{g})$-modules? Thank you very much.
2
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0answers
33 views

Abelian hypersurfaces

Is there any way to tell from its defining equation when an affine hypersurface has a group law? I am aware that all such groups are matrix groups; in that case I might want to ask when a variety over ...
0
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1answer
63 views

How to compute an integral?

I am reading the lecture notes. I am trying to understand the prove of Lemma 0.0.1.1 on page 4. From line 3 to line 4 in the proof of Lemma 0.0.1.1., how to prove that $$ \int_{F^{n-1}} \hat{1}_{\...
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1answer
50 views

Why is this a tori

In her notes http://www.math.toronto.edu/fiona/courses/algp.pdf on page 383, Example 4.2 Fiona claims that the group $$ T = \left\lbrace \pmatrix{ a & b \\ -b & a } \bigg|\, a,b \in \mathbb{...
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0answers
48 views

How to realise $\mathrm{PGL}_2$ as a closed subgroup of some $\mathrm{GL}_n$ explicitly?

Let $k$ be an algebraically closed field, then it is well-known that any affine algebraic group $G$ over $k$ can be viewed as a closed subgroup of $\mathrm{GL}_n$ for some $n$. In the special case ...
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0answers
32 views

Relation between Iwasawa and Cartan decompositions

Given a real semi-simple Lie group, one have the two decompositions - $$G=KAN \text{ - Iwasawa Decomposition} $$ and $$G=KA^{+}K \text{ - Cartan Decomposition} $$ I'm interested in an explicit ...
6
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0answers
119 views

Weil Pairing on Linear Algebraic Groups

I've been studying the Weil pairing on elliptic curves recently and discovered that it has a generalisation to an abelian variety $A$ with its dual $A^{\vee}$, which then becomes a pairing on an ...
2
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1answer
90 views

Can $\mathrm{PGL}_2$ be viewed as an affine algebraic group?

I was just wondering whether or not it is possible to view $\mathrm{PGL}_2$ as an affine algebraic group.
4
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1answer
54 views

Understanding $G_2$ inside Spin(7)? (EDIT: problem solved)

This is a rather embarrassing question, so please let me know of any duplication and I will happily remove it. I am seeking to understand the $\mathbb Q$-split form of the algebraic group $G_2$, and ...
1
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1answer
54 views

Algebraic groups with no small subgroups

I have read in many textbooks proofs that any Lie group has no small subgroups, that is, there is an open neighborhood of the unity element containing no nontrivial subgroups. In particular, $GL_n(\...
2
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0answers
44 views

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,$\mathbb{R}$) =...