1
vote
1answer
33 views

For an element $x$ in an algebraic group $G$, why do we have $\mathscr{L}(C_G(x))\subset\mathfrak{c}_{\mathfrak{g}}(x)$?

I'm reading Humphreys' Linear Algebraic Groups, trying to understand the following argument found on the top of pg. 76. Let $G$ be an algebraic group over some field $k$, with $x\in G$. Let ...
3
votes
1answer
25 views

Does the exponential map respect module actions?

Setup: Let $k$ be a field and $G \subseteq \mathrm{GL}_n(k)$ an algebraic group, reductive if that makes a difference. Let $\mathfrak g \subseteq \mathfrak{gl}_n(k)$ be the Lie algebra of $G$ with ...
1
vote
0answers
38 views

Questions about affine Weyl group and extended affine Weyl group for SL2.

Let $G=SL_2$. Then the Weyl group is generated by $s_1$. On page 3 of the lecture notes, it is said that the affine Weyl group is generated by $s_0, s_1$. (1) The element $s_0s_1$ can be identified ...
0
votes
0answers
28 views

References request about exponentials in Lie algebras.

I saw two formulas about Lie algebras. Let $G$ be an algebraic group over $k$ and $\mathfrak{g}$ its Lie algebra. For any $x \in \mathfrak{g}$, $a \in k$ and $g \in G$, we have $$ g \exp(ax) g^{-1} = ...
0
votes
0answers
34 views

Lie algebra of the unipotent radical of a standard parabolic subgroup in $GL_n$

Let $k$ be a field, and consider the algebraic group $G=GL_n(k)$. For any partition $n_1+n_2+\ldots+n_m=n$, we have a parabolic subgroup of the form ...
1
vote
0answers
43 views

Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
3
votes
0answers
32 views

Correspondence between unipotent and nilpotent elements

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. Let $\mathcal{U}(G)$ be the closed subvariety of unipotent elements of $G$, i.e., all elements whose ...
3
votes
0answers
47 views

Computing the fundamental groups of simple algebraic groups of type $A$

I'm interested in seeing the computation for the fundamental groups of the simple algebraic groups of type $A$. Below is the definition of the fundamental group for a simple algebraic group $G$. Let ...
2
votes
0answers
39 views

Relationship between invariants of a simple algebraic group

Let $G$ be a simple algebraic group over an algebraically closed field $k$. I believe all of the following invariants are well-defined. Besides the coxeter number, I haven't read about the others, ...
5
votes
0answers
84 views

Motivation?: Lie algebra and algebraic group Cohomology

This is just an a-priori question to get a motivational heuristic idea: If an algebraic group G (more generally, G an affine group scheme), is connected over an algebraically closed base-field k. ...
1
vote
0answers
30 views

If Lie(H) preserves a subspace, must H also preserve that subspace?

Assume $H \subset G$ is a closed connected subgroup of a linear algebraic group over an arbitrary field (both assumed to be smooth). Assume $G$ acts linearly on the (finite dimensional) vector space ...
0
votes
1answer
79 views

A question about Lie algebras corresponding to Lie groups and algebraic groups

Lie groups and algebraic groups both correspond with Lie algebras, which are by definition the left invariant vector field. But the topology of Lie groups and algebraic groups are different. Are their ...
2
votes
0answers
128 views

When is the adjoint representation self-dual?

Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question ...
11
votes
2answers
676 views

References on Linear Algebraic Groups/Lie Theory

I am currently doing a course on Lie groups, Lie Algebras and Representation theory based on Brian Hall's book of the same name. We should cover upto chapter 4/5 in this book by the end of the ...
5
votes
1answer
213 views

When do all derivations come from automorphisms?

Let $k$ be a field (algebraically closed for simplicity) and let $A$ be an $n$-dimensional algebra over $k$ (not necessarily commutative or even associative). The group $G=\mbox{Aut}(A)$ is an ...
2
votes
1answer
147 views

Highest or positive weights (or roots)

Let $T= (\mathbb{C}^*)^2$ be embedded in $GL_2$ along its diagonal entries, and suppose $T$ acts on $M_2(\mathbb{C})$ via conjugation. Denote $\chi_i(g)=z_i$ where $$ g = \left( \begin{array}{cc} z_1 ...
1
vote
0answers
37 views

The nonexistence of nontrivial solvable series in $M_n(k)$

I am a bit confused about semisimple Lie algebras. For the sake of simplicity, let's take $\mathfrak{g}=M_n(k)$ where $k=\bar{k}$. According to Wiki, $M_n(k)$ is solvable if the radical of $M_n(k)$ ...
0
votes
1answer
146 views

bilinear forms and linear algebraic groups

Let $G$ be a linear algebraic group and $\phi$ a finite dimensional regular representation of $G$ into $GL(V)$ I would like to know about bilinear forms on $V$ and when they are $G$-invariant. ...
8
votes
0answers
106 views

Why are parabolic subgroups called “parabolic” subgroups?

I used to think that things called "parabolic" must have something to do with parabolas or their defining quadratic equations. In fact, terms like parabolic coordinate, parabolic partial differential ...
7
votes
1answer
128 views

Are Lie algebras of non-isomorphic central simple algebras non-isomorphic?

Comments to my answer to this MO question, which is isomorphic to this MSE question, point out that I was tacitly assuming that the associated Lie algebras to non-isomorphic quaternion algebras over ...
1
vote
0answers
56 views

The form of a subgroup of $GL(n,K)$ when the derived group is of certain form

The famous Lie-Kolchin theorem in the theory of algebraic groups states: Let $G$ be a connected solvable subgroup of $GL(V)$, $0 \neq V$ finite dimensional. Then $G$ has a common eigenvector in ...
1
vote
0answers
21 views

When does $C_G(s) \times Cl_G(s)s^{-1}$ equal $G$

I have read on James E. Humphreys' Linear Algberaic Groups If $G$ is an algebraic subgroup contained in $GL(n,K)$, and $s$ is a semisimple element of $G$, then $\mathfrak{g}$ has the ...
0
votes
1answer
70 views

If $U$ is a closed subgroup of $GL(V)$ consisting of unipotent elements, show that $\log x$ belongs to $\mathfrak{u}$

If $U$ is a closed subgroup of $GL(V)$ consisting of unipotent elements, show that $\log x$ $(x \in U)$ belongs to $\mathfrak{u}$. Here, for an unipotent element $x \in GL(V)$, $\log x = ...
3
votes
0answers
52 views

Proving $\mathscr L(C_G(H)) \subseteq \mathfrak c_{\mathfrak g}(\mathfrak h) = \{ \mathrm x \in \mathfrak g \mid [\mathrm x, \mathfrak h] = 0\}$

Let $H$ be a closed subgroup of the algebraic group $G$, $C = C_G(H)$. Prove that $\mathfrak{c} = \mathscr{L}(C_G(H)) \subseteq \mathfrak{c}_{\mathfrak{g}}(\mathfrak{h}) = \{ \mathrm{x} \in ...
0
votes
1answer
125 views

Is there any connection between fundamental group and diagram automorphism (for algebraic groups and root systems)

There is a conclusion on my textbook: All semisimple algebraic groups of type $F_4$ are isomorphic. I was confused because of the fundamental groups. Is it true that two semisimple algebraic ...