Questions tagged [lie-theory]
The lie-theory tag has no usage guidance.
13
questions
0
votes
0
answers
68
views
Integrating homomorphisms of Borel subalgebras
Let $G$ be a connected simple complex Lie group and $\mathfrak{g}$ be its Lie algebra. Let us fix a root decomposition, let $\mathfrak{b}_\pm$, $\mathfrak{n}_+$ and $\mathfrak{h}$ be the corresponding ...
0
votes
0
answers
36
views
Simple Lie Moufang loops
Are there any simple Lie Moufang loops besides simple Lie groups and (modulo the center) special linear, orthogonal, and unitary loops over the octonions, split-octonions, and bioctonions?
4
votes
0
answers
138
views
Is the homogeneous coordinate ring of a flag variety a UFD?
I was wondering if $G$ is a semisimple complex algebraic group, then is the homogeneous coordinate ring of a flag variety a UFD or not?
2
votes
1
answer
115
views
Special cases of Lie II for groupoids using elementary techniques
I asked a similar question on math.stackexchange but did not get any responses, so I thought I'd kick it up to mathoverflow.
In Crainic and Fernandes's "Integrability of Lie Brackets" (and ...
3
votes
1
answer
477
views
Centraliser of regular semisimple element in $G^F$, for a connected reductive algebraic group $G$
Let $G$ be an connected reductive algebraic group over $k=\bar{\mathbb{F}_p}$. Suppose $G$ is defined over $\mathbb{F}_q$. Let $G^{F}$ be the corresponding finite group associated to $G$. Suppose $s\...
8
votes
1
answer
402
views
Shortest vectors in a root lattice
Let $R$ be a simply-laced root system in a Euclidean vector space $E$, with inner product normalized so that every root has length $\sqrt{2}$. Let $L \subseteq E$ be the lattice spanned by $R$. Is ...
0
votes
0
answers
59
views
Classifications of the indefinite generalized Cartan matrix
I want to know that the present results about classifications of generalized indefinite Cartan matrices.
I only have known that the classifications of hyperbolic matrces.
2
votes
1
answer
162
views
Injection of the Universal enveloping algebra
Let L1 and L2 be two Lie algebras.If U(L1)is isomorphic to U(L2)as associative algebra,then L1 is isomorphic to L2 ?
8
votes
0
answers
162
views
Generalizing a theorem of Kostant to arbitrary parabolics
Let $\mathfrak{g}$ be a simple complex Lie algebra and let $\Delta$ be a system of positive roots relative a choice of Cartan subalgebra and $\mathfrak{b}$ the corresponding Borel subalgebra. Let $B&...
2
votes
1
answer
144
views
When does an irreducible G-module admit an invariant quadratic form of signature (n,n+1)
Let $G$ be a connected real reductive Lie group and $V$ be a finite dimensional real irreducible $G$-module. When does $V$ admit an invariant non-degenerate quadratic form of signature $(n,n+1)$? I ...
7
votes
2
answers
1k
views
Malcev Lie algebra and associated graded Lie algebra
Suppose $L$ is a nilpotent finite-dimensional Lie algebra over $\mathbb{Q}$ of class $c$. We can define an associated graded Lie algebra to $L$ that, as a vector space, is:
$$\bigoplus_{i=1}^c \...
4
votes
3
answers
634
views
About $G$-modules versus $Lie(G)$-modules for algebraic groups
Hello,
I would like to know clear references about the following facts:
Let $G$ be a connected algebraic group (over alg. closed field in char. 0), $Lie(G)$ its Lie algebra, $M$ a $G$-module. I don'...
8
votes
2
answers
1k
views
Does there exist a complex Lie group G such that ...
... every Riemann surface of genus $1$ appears as a complex one-parameter subgroup of $G$?