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Results tagged with lie-algebras
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user 430
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.
10
votes
Three-dimensional simple Lie algebras over the rationals
I don't know if you can find the complete list written down anywhere (and I don't know how "nice" such a list would turn out to be), but let me point out that there is a lowbrow approach to this probl …
- 10.3k
34
votes
Accepted
Lie algebra semisimple if and only if perfect?
No. A Lie algebra satisfying that property is called perfect. For an example of a perfect Lie algebra that isn't semisimple, take a semisimple $L$ and a nontrivial irreducible representation $V$ of $L …
- 10.3k
22
votes
What is significant about the half-sum of positive roots?
This is actually a fairly deep question. Your suspicion that there may be multiple answers is correct, but there might be some surprising connections between seemingly unrelated answers. Let me give o …
- 10.3k
3
votes
Kostant's theorem on principal 3-dimensional subalgebras
Is it true that the centralizer $Z_{\frak{g}}(\frak{a})=\{\xi\in\frak{g}:[\xi,\eta]= $0$ \text{ }\forall\eta\in\frak{a}\}$ is trivial (or equivalently, that the trivial one-dimensional representati …
- 10.3k
3
votes
Accepted
Moving Between Weight Spaces in Highest-Weight Representations
They're the ones perpendicular to $\lambda$.
To see this, note, first off, that $\mathfrak g_\alpha v_\lambda \in V_{\lambda+\alpha}$ so if $g_\alpha v_\lambda \neq 0$ then $\lambda+\alpha$ must be a …
- 10.3k
3
votes
Accepted
Geometric structure of flag manifolds, Borel -Weil-Bott theorem
Correct. You can be fairly explicit here. For each root $\alpha$, let $\omega_\alpha \in \mathfrak g^\ast$ be a left-invariant form on $G$ that is dual to $\mathfrak g_\alpha$. Then for $\lambda \in …
- 10.3k
1
vote
Reconstructing a Lie group Banach representation from the Lie algebra rep. on analytic vectors
In general you can't expect to get an action of $G$ on $V^\omega$. Instead, what one has is that if $U$ is a $U(\mathfrak g_{\mathbb C})$-invariant subspace of $V^\omega$ then its closure $\overline{U …
- 10.3k
14
votes
Why study Lie algebras?
Although the title is about Lie algebras, the question body mentions Lie groups, and my answer will deal more with these. As mentioned in other answers, Lie groups show up frequently in geometry as gr …
- 10.3k