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Results tagged with lie-algebras
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user 13972
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
15
votes
Accepted
Why, conceptually, does the torus normalizer in $G_2$ split?
Here's a description that doesn't use octonions; instead, it uses the definition of $\mathrm{G}_2$ as the stabilizer of a $3$-form on $\mathbb{R}^7$. For simplicity, I'll do this for the split-form, …
- 108k
3
votes
Heat kernel of left-invariant metric on 3-sphere
I don't know a formula for $d(e^x,e^y)^2$, and I suspect that there is no simple formula, but the answer to Q2' is 'no'. The right hand side of (3) is linear in $Q^{-1}$, but it is not hard to see th …
- 108k
7
votes
Accepted
Is it true $\left\|\log(RS)\right\|≤\left\|\log(R)+\log(S)\right\|$ for all $R,S \in \mathrm...
Now that I have written out the completely elementary proof above for the quaternions and for $\mathrm{SO}(3)$, I feel that I should point out that the statement $|\log(ab)|\le |\log(a) + \log(b)|$, s …
- 108k
7
votes
Is it true $\left\|\log(RS)\right\|≤\left\|\log(R)+\log(S)\right\|$ for all $R,S \in \mathrm...
I realized a problem with my 'counterexample', so I no longer claim that the desired inequality does not hold on $\mathrm{SO}(3)$. The proof that it does hold on the quaternions is still OK. I'll po …
- 108k
5
votes
Accepted
Does the isometry group determine the Riemannian metric?
I think that you are missing some hypotheses on the action of $G$, otherwise there are trivial counterexamples. For example, if $G\subset\mathrm{Iso}(M,g)$ is the trivial group, then one clearly cann …
- 108k
6
votes
Accepted
Submanifolds of Lie groups with abelian normal bundle
One class of examples in a compact, connected Lie group $G$ of rank $r$ are the conjugacy classes of codimension $r$. Taking an element $a\in G$ whose centralizer is a maximal torus (a generic condit …
- 108k
10
votes
Groups associated with infinite dimensional Lie algebras
Here is an informative example that illustrates the difficulties: Consider the Lie algebra ${\frak{g}} = \mathrm{Vect}(\mathbb{S})$ of smooth vector fields on the circle $\mathbb{S}$. The flow of an …
- 12.9k
7
votes
Accepted
Viewing exceptional Lie algebras via the classical ones
Élie Cartan himself, recognized and used the following description of $\mathfrak{e}_6$: Let $V$ be a vector space of dimension $6$ and let $W$ be a vector space of dimension $2$. Then there is a vect …
- 108k
10
votes
Accepted
Is there a unique "natural" action of $\mathsf{SL}_{n+1}$ on $\mathbb{R}^n$?
There are two aspects to this question, the global question and the local question. Also, the case $n=1$ is different from $n>1$. Basically, the answer is 'essentially yes, but with some caveats'.
H …
- 108k
9
votes
Accepted
Are invariant forms on homogeneous spaces necessarily closed?
Note that the answer depends on the pair $(G,K)$.
For example, if $K=\{e\}$, then one is asking whether the ring of left-invariant forms on $G$ consists only of closed forms. This only happens when $ …
- 108k
8
votes
Accepted
3-dimensional Riemannian manifolds with 4-dimensional isometry group
There is a uniform way to describe these Riemannian $3$-folds using the geometry of the Lie group of isometries, as YCor mentioned in his comment. The following description is essentially drawn from …
- 108k
8
votes
When is the Lie algebra of automorphisms of a geometrical structure finite-dimensional?
The general procedure for deciding the 'local' version of this question, at least in the real-analytic connected case, was certainly known to Élie Cartan and, probably known to Lie in some form. The …
- 108k
2
votes
Accepted
Right Invariant Randers metrics
You are asking about a particular case of the general right invariant Lagrangian for curves on a Lie group. This is a well-known story, but I can summarize it here:
Let $G$ be a Lie group with Lie a …
- 108k
23
votes
First Explicit Irreducible Representations
If all you really want to know are the irreducible representations of ${\frak{so}}(7)$ and ${\frak{so}}(8)$ up to dimension $30$, I can save you the trouble of looking up these tables:
For ${\frak{so …
- 108k
10
votes
Constructing $E_8$ from its branching to $A_8$
Of course, the original description of this branching is due to Élie Cartan, himself. For example, see Chapitre IX of his 1914 paper Les groupes réels simples, finis et continu (Annales scientifiques …
- 108k