Questions tagged [lie-theory]

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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 ...
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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 $G$?