Questions tagged [lie-theory]
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13 questions
7
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1
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285
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What are the intermediate semisimple groups of type A?
Background: The first examples one sees of reductive groups over a field $k$ are $\text{GL}_n$, $\text{SL}_n$, and $\text{PGL}_n$. We all know the definitions of $\text{GL}_n$ and $\text{SL}_n$, and ...
- 815
0
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0
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71
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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 ...
4
votes
0
answers
147
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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?
- 201
2
votes
1
answer
117
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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 ...
- 1,253
3
votes
1
answer
519
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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\...
- 428
0
votes
0
answers
67
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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.
8
votes
1
answer
447
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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 ...
- 263
2
votes
1
answer
171
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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
172
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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&...
- 3,020
2
votes
1
answer
150
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 ...
- 21
7
votes
2
answers
1k
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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 \...
- 7,320
4
votes
3
answers
676
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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'...
- 5,562
8
votes
2
answers
2k
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Does there exist a complex Lie group G such that ...
... every Riemann surface of genus $1$ appears as a complex one-parameter subgroup of $G$?
- 952