# Questions tagged [linear-algebraic-groups]

The linear-algebraic-groups tag has no usage guidance, but it has a tag wiki.

**7**

votes

**1**answer

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### “Almost-ideals” in the (simple) Lie algebra of an algebraic group?

Let $G<\mathrm{GL_n}$ be a simple linear algebraic group defined over a finite field $K$. Let $\mathfrak{g}$ be its Lie algebra. Assume $\mathfrak{g}$ is simple.
Is it necessarily the case that ...

**3**

votes

**1**answer

201 views

### Definition of simple linear algebraic group

Why is it that many sources define simple (or almost-simple) linear algebraic group $G/k$ to be a connected, semisimple linear algebraic group such that every proper connected normal subgroup is ...

**4**

votes

**2**answers

225 views

### How does multiplication affect degrees?

Let $M(n) \sim \mathbb{A}^{n^2}$ be the space of $n$-by-$n$ matrices, seen as an affine space over a field $K$, and endowed with the usual matrix multiplication. Let $V$ and $W$ be subvarieties of $M(...

**7**

votes

**1**answer

167 views

### Simple Lie algebras: making subspaces 'very transversal'

Let $G$ be a Lie group or group of Lie type whose Lie algebra $\mathfrak{g}$ is simple. Because the Lie algebra is simple, for any proper subspace $V\subset \mathfrak{g}$,
there is a $g\in G$ such ...

**5**

votes

**0**answers

93 views

### $\mathbb{F}_q$-rational elements in unipotent classes of a finite group of Lie-type

I've tried posting this question on MSE, but didn't manage to get an answer there, so I'm trying again here. Sorry in advance if this question is trivial or trivially false. I haven't managed to find ...

**1**

vote

**1**answer

98 views

### Connectedness of centralizers of semisimple Lie-algebra elements under the action of a semisimple algebraic group

Let $\newcommand{\GG}{\mathbf{G}}\newcommand{\g}{\mathfrak{g}}\GG$ be a connected semisimple algebraic group over the algebraically closed field $k=\overline{\mathbb{F}_q}$, and let $\g$ be its Lie-...

**3**

votes

**1**answer

116 views

### Is the subgroup generated by a conjugacy class of semisimple elements Zariski closed?

Let $k$ be an arbitrary field with $\operatorname{char}(k) \neq 2$. Let $G$ be a linear algebraic group over $k$. Let $X$ be the conjugacy class of a semisimple element $s \in G(k)$ of order 2 (or a ...

**1**

vote

**0**answers

118 views

### Integral smooth model of unramified reductive groups

My question is motivated by the following observations. Let $\mathrm{T}$ be a torus defined over a $p$-adic field $K$, then by theories of tori, we have it is uniquely determined by a free $\mathbb{Z}$...

**4**

votes

**0**answers

172 views

### A basic question on a base change of a homogeneous space of a linear algebraic group

I asked this basic question in MSE and got a comment "This belongs to Mathoverflow", so I ask my question here.
Let $G$ be a linear algebraic group over a field $k$, and $H\subset G$ be a $k$-...

**0**

votes

**1**answer

69 views

### Haar measure of algebraic orbits

Let $G$ be a simple linear algebraic group acting on a projective variety $X$ through rational maps. Let $x_0\in X$ with stabilizer group $H$ and assume that $G/H$ in not compact and carries a $G$-...