All Questions
Tagged with lie-groups lie-algebras
816 questions
2
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answers
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Reductive Lie algebra of a Lie group
In the answer of my question:
On the full reducibility of representations of reductive Lie algebras
James E. Humphreys replied to me saying that:"the notion of "reductive" for a Lie algebra in ...
- 671
4
votes
1
answer
446
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Dolbeault Operators for $CP^1$ as $\mathfrak{su}(2)$ Actions.
This question is related to a previous question of mine. More specifically, it results from my attempts to understand the simplest incarnation of a phenomenon mentioned therein.
Put a grading on the ...
- 3,399
10
votes
3
answers
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subgroup of SU(N) with maximal manifold dimension
Given the group SU(N) of NxN unitary matrices, does there exist a subgroup S
with a manifold dimension larger than the SU(N-1) manifold dimension and
smaller than the SU(N) one? S should not ...
- 1,207
19
votes
5
answers
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Matrix representation for $F_4$
Has anyone ever bothered to write down the 26-dimensional fundamental representation of $F_4$? I wouldn't mind looking at it. Is it in $\mathfrak{so}(26)$?
I'm familiar with the construction of the ...
- 2,123
6
votes
2
answers
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Why can the Dolbeault Operators be Realised as Lie Algebra Actions
I've been looking at an example in the non-commutative geometry literature and I'm having trouble figuring out what the classical motivation is. I'll just describe the classical case here: Recall that ...
- 3,399
19
votes
5
answers
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Understanding moment maps and Lie brackets
I'm trying to learn about moment maps in symplectic topology (suppose our Lie group is $G$ with Lie algebra $\mathfrak g$, acting on the symplectic manifold $(M,\omega)$ by symplectomorphisms). I'm ...
- 1,129
2
votes
1
answer
197
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Polytopes related to the conjugation action of a Lie group on multiple copies of itself?
Let G be a finite dimensional real Lie group. As I understand it, the quotient space of G acting on itself by conjugation is a well studied polytope which can be identified with the fundamental alcove ...
- 27.5k
26
votes
2
answers
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Cohomology of Lie groups and Lie algebras
The length of this question has got a little bit out of hand. I apologize.
Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe ...
- 23.5k
9
votes
3
answers
4k
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A terminology issue with the Killing form
I understand the definition of the Killing form $B$ as $$B(X,Y)=\mathrm{Tr}(\mathrm{ad}(X)\mathrm{ad}(Y)).$$
When the Lie group is semisimple the negative of the Killing form can serve as a Riemannian ...
- 3,541
4
votes
0
answers
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Why are Lie Algebras/ Lie Groups so much like crossed modules, and not?
A crossed module consists of a pair of groups $G$ and $H$ with a group homomorphism, $t:H \rightarrow G$, and $\alpha: G \times H \rightarrow H$ that defines an action of $G$ on $H$, $\tilde{\alpha}$: ...
- 5,264
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votes
7
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Why is Lie's Third Theorem difficult?
Recall the following classical theorem of Cartan (!):
Theorem (Lie III): Any finite-dimensional Lie algebra over $\mathbb R$ is the Lie algebra of some analytic Lie group.
Similarly, one can propose ...
- 54.6k
21
votes
2
answers
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Is every finite-dimensional Lie algebra the Lie algebra of a closed linear Lie group?
This question is closely related to this one.
Ado's theorem states that given a finite-dimensional Lie algebra $\mathfrak g$, there exists a faithful representation $\rho\colon\mathfrak g \to \...
- 1,050
5
votes
2
answers
1k
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Lie Groups and Lie Algebras
What is the exact relationship between Lie groups and Lie algebras? I know it's not bijective because all commutative Lie groups have isomorphic Lie algebras.
- 3,399
77
votes
7
answers
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What is the symbol of a differential operator?
I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion....
- 54.6k
42
votes
9
answers
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Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?
Harold Williams, Pablo Solis, and I were chatting and the following question came up.
In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra g, you can ...
38
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18
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Learning about Lie groups
Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.