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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7 views

Why does the commutator subgroup of a unipotent algebraic group have smaller dimension?

Suppose $U$ is a unipotent linear algebraic group. Is there an explanation why the commutator subgroup $[U,U]$ has strictly smaller dimension, or at least why it is a proper subgroup? This fact is ...
2
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0answers
49 views

On algebraic groups of dimension 1

I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following : Let $K$ be an algebraically closed field. ...
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1answer
20 views

Definition of schematically dense

In these notes on algebraic groups (http://www.jmilne.org/math/CourseNotes/iAG200.pdf), assume $(X, \mathcal O_X)$ is an algebraic $k$-scheme for some field $k$, and $S$ is a subset of $X(k)$, where ...
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0answers
13 views

$k$-closed, defined over $k$, and pure inseparability

Let $\Omega$ be a large algebraically closed field, $k$ a subfield of $\Omega$, $\overline{k}$ the algebraic closure of $k$ in $\Omega$, and $k^i$ the field of purely inseparable elements over $k$. ...
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0answers
31 views

Group action of linear algebraic group $G$ on itself induces a representaion of $G$ on $Lie(G)$

Let us be given a linear algebraic group $G$ over a field $K$ of characterstic zero. This group $G$ is defined as the common zeroes of a finite set of polynomials $\{f_1, \ldots ,f_r\}$ $\in K ...
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1answer
36 views

Action of an algebraic group induce a representation of its Lie algebra

Let $G$ be a linear algebraic group over a field $K$ of characterstic zero acting on a vector space $V$. Then does this action induce a representation : $$\Gamma : Lie(G) \to gl(V)$$ If yes, how ? ...
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31 views

Relation between

Can anybody please give a relationship among these objects. Varieties, schemes, moduli spaces, stacks, algebraic spaces, groupoids. I mean here, we can define an abstract Variety as a scheme with ...
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0answers
22 views

Pic of a variety of type G/P

Let $G$ be an simple algebraic group an let $P$ be a parabolic subgroup of $G$. Let $X$ be the projective, homogeneous variety $G/P$. Is it true that the following holds: Pic($X$) has rank $1$ iff ...
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2answers
30 views

Why are alternating matrices only used when $n$ is even?

Consider $\textrm{GL}_n(k)$, where $k$ is a field. If $Q \in \textrm{GL}_n(k)$, then $Q$ induces a $k$-bilinear form $B_Q: k^n \times k^n \rightarrow k$ by the formula $$B_Q(v,w) = v^t Q w$$ $Q$ is ...
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1answer
34 views

On Simple Algebraic Groups

I was skimming a paper and got stuck in the middle. As you see in the underlined parts, the authors first assumed that $\mathcal{G}$ is a simple algebraic group. Then $\mathcal{G}$ is defined to be ...
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4answers
75 views

affine variety definition

I had this very elementary question which baffles me. Most introductions to the topic define an affine variety as a subset of affine space that is the zero-locus of a set of polynomials. Now, ...
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0answers
17 views

Action on Flag manifold

When $G$ is of type A,D,E and $B_4$ then the group of Dynkin diagram automorphisms is non-trivial. If $B$ is a Borel subgroup of $G$, then is there a nice action of the Dynkin diagram automorphism ...
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1answer
40 views

Dimension of a finite irreducible algebraic group

Let $G$ be an irreducible algebraic group over the field $K$ of characterstic 0. Let $A=K[x_1,...,x_n]/I(G)$ be the coordinate ring and $K(X)=Q(R)$ be the quotient field of $A$. (Since $G$ is ...
2
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1answer
55 views

Quotient $G \to G/N$ induces quotient $H \to H/N$ by restriction?

Let $G$ be a linear algebraic group over an algebraically closed field $k$. Consider closed subgroups $N \subseteq H \subseteq G$ such that $N$ is a normal subgroup of $G$. Then restricting the ...
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0answers
15 views

Borel-Weil theorem, confused about statement in special case of $SL_2(\mathbb{C})$

Suppose that $g = sl_2(C)$, and denote by $V(m)$ the weight $m$ irreducible representation of $g$. Let $B$ be the Borel subgroup of $G = SL_2(\mathbb{C})$ consiting of the upper triangular matrices. ...
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1answer
34 views

Show that $Y$ is constructible

I'm stuck on the following problem: Let $X$ be a Noetherian space, and let $Y \subseteq X$ have the property that for every irreducible closed set $Z \subseteq X$, $Y \cap Z$ contains an open dense ...
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0answers
28 views

One-One correspondence between elements of G (algebraic group) and maximal ideals of K[G]

We are given that $G$ is an algebraic group over $K$ and $K[G]= K[x_1,...,x_n]/I(G)$ where $I(G)$ is the ideal consisting of all polynomials of which elements of $G$ are the common zeroes. Now,if we ...
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0answers
17 views

Maximal torus of a compact algebraic group

An algebraic group $G$ is a group object in the category of algebraic varieties, i.e. it is an algebraic variety with Zariski topology and group structures. Example for linear algebraic groups are ...
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0answers
35 views

Why are none of the $\overline{Y_i}$ contained in one another?

Let $X$ be a topological space which is a union of finitely many irreducible closed sets $X_1, ... , X_n$. Lemma: if none of the $X_i$ are contained in one another, then $X_1, ... , X_n$ are the ...
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0answers
21 views

Finite groups as intersection of algebraic groups

Well known that any finite number of points can be seen as intersection of two algebraic curves. Is it true that any finite group $G$ can be seen as intersection of two (connected) one dimensional ...
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0answers
63 views

Nonstandard construction of sheafification

Let $F$ be a presheaf on a topological space $X$ of some category of "sets with structure." In Borel's Linear Algebraic Groups, he gives the following explanation for how to construct the associated ...
2
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1answer
37 views

Can the cohomology of the Grassmannian identified with the cohomology of a specific dense open subvariety?

Let $(\mathbb{C}^{2p},Q)$ be a $2p$-dimensional complex vector space equipped with a nondegenerate symmetric bilinear form $Q$ where $p\geq 3$. Let $l\leq p-2$. You may assume that $l$ is odd if this ...
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1answer
41 views

Lie Algebra of a connected simple linear algebraic group

Let $G$ be a linear algebraic group and $A=K[G]$ (K is a field of characterstic 0) be the coordinate ring of $G$. In Humphreys, the Lie algebra of $G$ is defined as the space of left invariant ...
2
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1answer
46 views

Linear Algebraic Group acting on the co-ordinate ring

Let $G$ be a linear algebraic group and let $C[G] = C[x_1,...,x_n] / I(G)$ denote the coordinate ring of $G$. (Note that $I(G)$ is the ideal containing all those polynomials in $C[x_1,...,x_n]$ of ...
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1answer
23 views

Automorphism of the coordinate ring of a linear algebraic group

Let me define a linear algebraic group first of all: A linear algebraic group is an affine variety (zeroes of a set of polynomials in $n$ variables with coefficients in a field $C$) such that the ...
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0answers
25 views

What is the category of models?

I'm taking a look at Introduction to Algebraic Geometry and Algebraic Groups by Michael Demazure and Peter Gabriel, and I'm confused about some terminology. Early in the book it says "A ...
2
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1answer
34 views

Equivalent definition of almost geometric quotient

I am trying to prove the following lemma - Lemma - Let $X$ be a variety and let $G$ be an algebraic group acting algebraically on $X$. Let $\pi:X\rightarrow X//G$ be a good categorical quotient. Then ...
11
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0answers
102 views

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 ...
2
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0answers
38 views

Why are $\mathfrak{pgl}_n\simeq\mathfrak{sl}_n$ when characteristic does not divide $n$?

Suppose $k$ is some algebraically closed field whose characteristic does not divide $n$. Why can we identify the lie algebras $\mathfrak{pgl}_n\simeq\mathfrak{sl_n}$ of the projective linear group and ...
1
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1answer
37 views

Is the inverse image of an irreducible variety under the natural projection irreducible (in the setting of homogeneous spaces)?

Let $p\colon Z\to X$ be a morphism between irreducible varieties (= reduced schemes of finite type over $\mathbb{C}$). Assume that every fiber of $p$ over a closed point of $X$ is also irreducible. ...
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1answer
25 views

Boundary is a union of orbits with strictly lower dimension

I'm stuck on the proof of the following propsition in Humphreys Linear Algebraic Groups: Let $G$ be an algebraic group acting morphically on a variety $X$. Then each orbit is a smooth, locally ...
0
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1answer
16 views

Linear algebraic group inside $GL_n$

Let $G$ be a linear algebraic group. Consider the closed embedding $G \hookrightarrow GL_n$. Let $K$ be any field. Let $x \in GL_n(K)$. Now suppose we know that $x^n \in G(K) $ for some positive ...
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0answers
39 views

About normalizer subgroup scheme

Let $S$ be a scheme and let $G$ be a group scheme over $S$. Let $X$ be an $S$-subscheme of $G$; we can define the controvariant functor: $$ \mathbf{N}_G(X):\mathbf{Sch}_{S}\to\mathbf{Group}\\ \forall ...
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0answers
59 views

Action on algebraic variety and adjoint bundles

Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $Spec\mathbb{C}$, and the underlying set of $G$ is a ...
0
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0answers
41 views

equivalent definition of a good categorical quotient?

Working solely with varieties, as you may know, a pair ($Y$, $\pi$) is a "good categorical quotient" for the $G$-variety $X$ if: 1) $\pi$ is surjective and constant along orbits 2) for any open $U ...
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0answers
39 views

Finite Strong Completeness Theorem for BL logic

I need to prove the finite strong completeness theorem for Basic Logic, where $BL$ is: $\vdash_{BL} =\vDash_{FLew}$ + linearity + division. The strong completeness is: $$\varphi_1,...,\varphi_n ...
2
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0answers
50 views

Why does $\textrm{SL}_n(R)$ (with a very abstract definition) coincide with the usual definition?

I'm reading J.S. Milne's notes on algebraic groups (http://www.jmilne.org/math/CourseNotes/iAG200.pdf). Here $k$ is a field, and an algebraic scheme over $k$ is a locally ringed space $(X, \mathcal ...
3
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3answers
73 views

Is a group object in the category of algebraic schemes over $k$ an honest group?

Let $G$ be an algebraic scheme over a field $k$ (in the sense of these notes http://www.jmilne.org/math/CourseNotes/iAG200.pdf, I can try to explain more if anyone needs), and $m: G \times G ...
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0answers
26 views

What does it mean for a subfunctor to be “defined by an open subscheme?”

I'm trying to read J.S. Milne's notes on Algebraic Groups. $X$ is a functor from the category $\mathscr C$ of $k$-algebras of the form $k[X_1, ... , X_n]/\mathfrak a, \mathfrak a$ an ideal of $k[X_1, ...
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1answer
53 views

Typo in Milne's notes? What should this be?

In section A.26 page 496 here (http://www.jmilne.org/math/CourseNotes/iAG200.pdf), it is written: A pair $(A,a), a \in F(A)$ is said to represent $F$ if the natural transformation $$T_a: h^A ...
0
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1answer
29 views

Unique sheaf determining an affine $k$-scheme, equivalent definitions?

Let $k$ be a field, and $A$ a finitely generated $k$-algebra. Let $X$ be the space of maximal ideals of $A$ in the Zariski topology. For $D$ a basic open set in $X$, let $$S_D = \{f \in A : f ...
0
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1answer
23 views

Dimension of centralizer of unitary matrix

Let $G=U(n)$ be the unitary group. We know that any unitary matrix is diagonalizable. Let $x$ be a unitary matrix. Then I've read the statement that any matrix $b$ commuting with $x$ is block-diagonal ...
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0answers
27 views

How to prove that a certain type of isogeny of a reductive group is a Frobenius for some $\mathbb{F}_q$-structure

It is well known that a connected reductive linear algebraic group $G$ over $\mathbb{F} = \overline{\mathbb{F}_p}$ can be classified via its root datum $\Psi(G,T) = (X(T),\Phi, Y(T), \Phi^\vee)$. ...
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1answer
23 views

Closed orbits for reductive group actions

Let $G$ be a complex reductive group acting algebraically on a complex affine variety $X$. Is it true that an orbit $G.x$ is closed in $X$ if and only if the stabilizer $G(x)$ of $x$ is a reductive ...
3
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0answers
27 views

Definition for Shimura datum

The following definition for $\textbf{shimura datum}$ is due to wikipedia. Let $S=\mathrm{Res}_\mathbb{R}^\mathbb{C}G_m$ be the Weil restriction of the multiplicative group from complex field ...
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0answers
48 views

Where is it used that $R^G$ is finitely generated?

In Lemma 5.0.4 of Toric Varieties by Cox, Little and Shenck I don't understand which part of the proof uses that $R^G$ is finitely generated. Can someone please help me? Lemma : Let $G$ act on ...
0
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1answer
31 views

Gauss decomposition of an algebraic group.

Let $K$ be any field. Consider $GL_n(\mathbb{K})$ as an algebraic group. I know that it has a Gauss decomposition, i.e $GL_n(K)=I^- D I^+$, where $I^-$ and $I^+$ are the lower unipotent matrix and ...
3
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0answers
48 views

The Picard group of an Elliptic Curve

Let $(E,O)$ be an elliptic curve. Let $\operatorname{Pic}^0(E)$ stand for the divisors that have degree $0$ where : $$D = \sum_{p\in E}n_p(P) \text{ and } \deg D = \sum_{p\in E}n_p.$$ I understand ...
2
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0answers
25 views

complex reductive Lie group

I am reading A. L. Oniscik's paper Decompositions of Reductive Lie Groups, and the author cited a proposition that a complex reductive Lie group $G=ZS$ is locally isomorphic to the reductive ...
2
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1answer
72 views

Action of $\text{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ on $X(T)$ permutes root system

I got these questions while reading Chapter 3 of "Representations of Finite Groups of Lie Type" by Digne-Michel. Let $T$ be a torus defined over $\mathbb{F}_q$. Then ...