1
vote
0answers
24 views

Why is the restricted nullcone a variety?

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $(\mathfrak{g},[\cdot,\cdot],(\cdot)^{[p]})$ be a finite-dimensional restricted Lie algebra. Define the restricted ...
5
votes
1answer
96 views

Lie-brackets and solution space of PDE

I have a linear, first-order homogeneous PDE system with polynomial coefficients $$L_j\, f =0,\text{ for } j=1,..,J\quad \text{ where } L_j \text{ is a first order, diff. operator with polynomial ...
0
votes
0answers
25 views

parabolic subalgebra

Let $G$ be a semisimple lie group, a parabolic subgroup of $P$ is a connected subgroup that contains a conjugate of $B$, (which $B$ is Borel subgroup of $G$) then I can not see why lie algebra of $P$ ...
2
votes
0answers
49 views

Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let G be an affine group scheme, and Dist(G) its hyperalgebra. I am wondering what is the relationship between Dist(G) and G interms of Cohomology? Is there a cohomology theory for Dist(G), if so ...
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. ...
6
votes
1answer
113 views

Ideal defining the nilpotent cone of $\mathfrak{gl}_n(k)$

Let $k$ be an algebraically closed field, and let $\mathfrak{g}=\mathfrak{gl}_n(k)$. Let $\mathcal{N}\subset\mathfrak{g}$ be the nilpotent cone, that is: $$\mathcal{N}=\{A\in\mathfrak{g}\mid ...
7
votes
1answer
124 views

Trivial summand of a representation's symmetric power

The following comes from Exercise 13.17 of Fulton and Harris's book, Representation Theory: A First Course. Let $V$ denote the standard representation of $\mathfrak{sl}_3\mathbb{C}$, with weights ...
0
votes
0answers
46 views

Map algebras between scheme and Lie algebra

Kindly asking for any hints about the following questions: Suppose, $X$ be an arbitrary scheme over an algebraically closed field $k$. 1- In general, what is the structure of $A= ...
0
votes
0answers
55 views

Lie algebra homomorphism as an scheme morphism

Kindly asking for any hints about the following questions: Assume $g$ is a finite-dimensional Lie algebra. We denote the group of Lie algebra automorphisms of $g$ by $\rm Aut_k g$. Any Lie algebra ...
2
votes
0answers
58 views

Finite-dimensional Lie algebra as a scheme

Kindly asking for any hints about the following questions: Suppose $k$ is an algebraically closed field of characteristic zero and $g$ is a finite-dimensional Lie algebra over $k$. Then $g$ is ...
2
votes
1answer
182 views

$GL_n(k)$ (General linear group over a algebraically closed field) as a affine variety?

In the context of linar algebraic groups, I read in my notes from the lecture that's already some while ago that $GL_n(k)$ is an algebraic variety because $GL_n=D(\det)$, $ \det \in k [ (X_{ij})_{i,j} ...