A Lie group is a group (in the sense of abstract algebra) that is also a differentiable manifold, such that the group operations (addition and inversion) are smooth, and so we can study them with differential calculus. They are a special type of topological group. Consider using with the ...

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At what position do we insert the new coefficient in the weights for extended Dynkin Diagrams?

Given a set of weights of a representation and the corresponding extended Dynkin diagram for some Lie algebra, we can delete a node, which yields the maximal subalgebra. I know how to draw the ...
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Right invariance of Casimir (Laplacian)

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. The Casimir element $\Delta\in \mathfrak{zu}(\mathfrak{g})$, considered as an operator on $C^{\infty}(G)$ is right invariant, that is, $\Delta ...
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$5$ dimensional space over $\mathbb{R}$

When coming up with a double cover of $SO(5)$, I used conjugation by matrices of the form $$\begin{pmatrix} r & q\\ \overline{q} & r \end{pmatrix}$$ where $r\in\mathbb{R}$ and $q$ is a ...
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Explicit homotopy equivalence of homogeneous spaces $O(2n)/U(n)$ and $GL(2n,\mathbb{R})/GL(n,\mathbb{C})$

Exercise 2.25 of symplectic topology by McDuff and Salamon asks me to prove that $O(2n)/U(n)$ is homotopy equivalent to $GL(2n,\mathbb{R})/GL(n,\mathbb{C})$. They suggest to use the polar ...