All Questions
39
questions
6
votes
0
answers
267
views
Tits construction of algebraic groups of type D₆ and E₇ via C₃
As shown in the Freudenthal magic square, the Tits construction of $D_6$ takes as input
an quaternion algebra and the Jordan algebra of a quaternion algebra (see The Book of Involutions § 41). In ...
- 1,429
1
vote
1
answer
203
views
Lie algebras and pulled back group schemes
Suppose I have an extension of fields $L/K$, a group scheme $G_K$ over $\operatorname {Spec} K$. Let $G_L$ denote the pullback of $G_K$ to $\operatorname{Spec} L$. Then, under what conditions on the ...
- 13
3
votes
0
answers
202
views
A quantity computed from weights of representations -- Have you seen it?
The following quantity has come up in some work my collaborators and I are doing on equivariant D-modules, and in that particular context it seems to be very significant (i.e. it's the only "...
- 3,039
1
vote
0
answers
149
views
Why do such a birational map exists? And why it is unique?
Let $G$ be a complex linear algebraic group which is connected and reductive and let $\mathfrak g$ be its Lie algebra.
Suppose that $H \subset G$ is a 1-dimensional torus such that
the action of $H$ ...
- 381
5
votes
1
answer
400
views
Lifting $\mathfrak{sl}_2$-triples
Let
$k$ be an algebraically closed field,
$G$ a (smooth, connected) reductive algebraic group over $k$,
$H$ a (smooth, connected) reductive group of semisimple rank 1, and
$T$ a maximal torus in $H$.
...
- 12k
2
votes
0
answers
142
views
Commutators and brackets in nilpotent Lie algebras
Let $\mathfrak{g}$ be a finite-dimensional nilpotent Lie algebra over an algebraically closed field $k$ of characteristic zero. Throughout, let $x,y$ and $z$ be elements of $\mathfrak{g}$. The Baker-...
- 533
5
votes
0
answers
123
views
Conjugacy classes of plane k-jet group
Define $G(n, k)$ as a subgroup of $\rm{Aut}(\Bbb C[[x_1, \dots, x_n]]/\mathfrak m^{k+1})$ with identity linear part (so, group of $k$-jets of selfmaps of $\Bbb C^n$). I'm interested in the map from $G(...
- 4,531
7
votes
1
answer
272
views
Is a 8-dimensional quadratic form recognized by its Lie algebra, modulo equivalence and scalar multiplication?
Question. Let $K$ be a field of characteristic zero (large characteristic should be fine too). Let $q,q'$ be two non-degenerate quadratic forms on $K^n$ with $n=8$. Suppose that the Lie algebras $\...
- 62.3k
4
votes
1
answer
191
views
The Lie algebra of the subgroup of $GL(n)$ preserving a given variety
Let $V=k^n$ for an algebraically closed field $k$ of characteristic 0, and let $W \subseteq V$ a subspace. Let $G_W\subseteq GL(V)$ be the set of invertible linear maps that preserve $W$, i.e.
$$
G_W=\...
- 970
2
votes
1
answer
171
views
Do rational points in a split reductive group act transitively on the orbits of the Cartan subalgebra (w.r.t. automorphism group of Lie algebra)?
Let $(G,T,M)$ be a split reductive group (over say, the integers), with Lie algebra $(\mathfrak{g}, \mathfrak{t})$, and let $R$ be a commutative ring. When $R$ is an algebraically closed field, it is ...
- 45.3k
7
votes
1
answer
520
views
Lie Algebra of Automorphism Group of $\mathbb{P}_k^1$
Let $X$ be a scheme over an algebraically closed field $k$ and let $\operatorname{Aut}(X)$ denote the functor sending a $k$-scheme $T$ to the group $\operatorname{Aut}_T(X \times_k T)$ of ...
- 729
2
votes
1
answer
107
views
Nice Form of Vector Field
Let $G$ be a reductive algebraic group (maybe reductive is not necessary) over an algebraically closed field $k$ of characteristic zero. Let $X$ be a homogeneous affine $G$-variety, i.e. $X=G/K$ for ...
- 1,077
5
votes
0
answers
185
views
Homeomorphisms of Springer fibers
Let $V$ be a complex $n$-dimensional vector space and denote by ${\cal F}$ its space of complete flags. Let $g \in Gl(V)$ be unipotent and consider the Springer fiber ${\cal F}_g$ of its fixed points ...
- 1,123
8
votes
0
answers
145
views
Semisimple Lie groups admitting a free algebra of invariants
Assume we work over an algebraically closed field of characteristic zero.
I know that for a connected semisimple algebraic group there is an upper bound for the number of isomorphism classes of ...
- 189
10
votes
2
answers
701
views
Square root in complex reductive groups
Let $G$ be a connected complex reductive linear algebraic group. Does every $g\in G$ have a square root? (That is, some $a\in G$ such that $a^2=g$.)
- 109