# Tagged Questions

For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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### Proving a Jacobi-like identity with four elements in a Lie algebra

Let ${\frak g}$ be a Lie algebra. Then $$[[[X,Y],Z],W] +[[[Y,X],W],Z]+[[[Z,W],X],Y]+[[[W,Z],Y],X]=0$$for all $X,Y,Z,W \in {\frak g}$. I started using the Jacobi identity four times to get: \...
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### Lie Algebra: Dimension of a one-parameter group (dimension of orbit)?

I am reading a book about Applications of Lie Groups to Differential Equations by Peter Olver. Lets say we have a PDE with $p$ independent variables and $q$ dependent variables In Chapter 3.1 (page ...
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### Finding lie algebra of a group by using exp map and tangent space [duplicate]

I'm studying Lie groups and I am in trouble with finding lie algebras of the classical groups. How can I calculate $\mathfrak{sp}(n,\mathbb{C})$ or $\mathfrak{so}(n,\mathbb{R})$ using exp map and ...
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### What does the set of dominant integral elements in a Cartan sub algebra look like?

I'm reading about the theorem of the highest weight: Irreducible finite dimensional representations of a complex semisimple Lie algebra (with a fixed Cartan sub algebra, $\frak{h}$ and choice of ...
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### Functoriality of the adjoint representation

Just a simply question. I came across the following statement which is used for deriving Weyl's integral formula: ''$\text{Ad}_G(h)|_{\mathfrak{h}} = \text{Ad}_H(h)$ due to functoriality in the Lie ...
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### A semisimple Lie group has no character; Am I right?

Let $G$ be a compact connected Lie group with semisimple Lie algebra ${\frak g}$. With the following reasoning, I show that there is no non-trivial Lie group homomorphism $$\chi:G\to S^1.$$ Is that ...
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Consider the Lie algebra $u(N)$. If I exponentiate an $x \in u(N)$, I will obtain an element of the $U(N)$ group. My understanding is that $exp(u(N))= \{exp(x)| x \in u(N)\}$ is, in fact, the group $U(... 1answer 29 views ### Algebraic groups and restricted Lie algebras If$G$is an algebraic group with coordinate algebra$A=\mathcal O(G)$, say over a field$k$of characteristic$p$, then its Lie algebra$\mathfrak g$can be endowed with the structure of a restricted ... 0answers 29 views ### Topologies of partially exponentiated lie algebras, especially in regard to$SU(2)$Consider the fundamental respresentation of$\mathfrak{su}(2)$given in terms of the Pauli matrices as$\mathfrak{su}(2) = \langle \frac{i\sigma_1}{2},\frac{i\sigma_2}{2},\frac{i\sigma_2}{2}\rangle_{\...
Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...