# Tagged Questions

Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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### Integral cohomology of $G/N(T)$

Let $G$ be a compact connected simple Lie group, $T$ a maximal torus, $N(T)$ the normalizer of $T$, and $W=N(T)/T$ the Weyl group. It is well-known that $H^*(G/T,\mathbb{Q})$ is the regular ...
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### Maximal subgroups of $\mathrm{SL}(n,\mathbb{R})$

I would like to find a list (or at least a description) of the maximal closed connected subgroups of $\mathrm{SL}(n, \mathbb{R})$ , and also of $\mathrm{SU}(p,q)$. In the following MO discussion is ...
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### Try to understand the relation between Poisson Lie groups and Lie bialgebras

I am trying to understand the relation between Poisson Lie groups and Lie bialgebras. In the book Lectures on quantum groups, Page 20, Theorem 22 says that given a Poisson Lie group, we can construct ...
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### intuitive connection between The KdV equations and the Virasoro bott group

I posted this on stack exchange but had no joy, perhaps someone here can answer : The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group....
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### A careful roadtrip from locally symmetric spaces to algebra

I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning completely)...
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### Harmonicity on semisimple groups

I asked this on Math.SE and got no answer, so I'll try my luck here. Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in ...
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### Order of metaplectic operator

I have a weak background on this subject. Suppose $S$ be a $2m \times 2m$ symplectic matrix of order $n$. Suppose $W_S$ be the corresponding metaplectic operator on $\mathcal{S}(\mathbb{R}^m)$, the ...
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### Why the sub-Laplacian $\Delta_{sub}$ on the Heisenberg group $H^3$ is sub-elliptic but not elliptic?

I want to know why the sub-Laplacian $\Delta_{sub}= X^2 + Y^2$ on the Heisenberg group $H^3 = \mathbb C \times \mathbb R$ is sub-elliptic but not elliptic, where $X$ and $Y$ are the left-invariant ...
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### Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant

The Lie group $GL(n)$ being a manifold is locally path-connected. Consider its connected component of the identity $C\subseteq\mathbb{R}^{n^2}$. What is a good way of showing that $C$ is a tame ...
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### multiplicity-free action on $SO(n+1)/SO(n-1)$

I'm trying to show that the Lie group $G=SO(n+1) \times SO(2)$ acts multiplicity-free on the cotangentbundle $T^* (SO(n+1)/SO(n-1))$. That means: 1) There exists an $\operatorname{Ad}^*_G$-...
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### What is this Lie algebra?

Consider two matrices $A,B \in \mathfrak{su}(N)$ which are both diagonal in the standard basis and non-zero. If we consider the new matrix $\tilde{B} := FBF^{\dagger}$ where $F$ is the `quantum' ...
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### Expressing $SO_8$ element as product of $L_u$ and $R_u$ for unit octonions $u$

Welcome octonions friends ! Long time ago when I travelled through octonion land, I conjectured that every $SO_8$ element can be expressed as product $L_a L_b R_c R_d$ for unit octonions $a$, $b$, $c$...
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### When this Ad-invariant function on a Lie algebra is zero?

Let $G$ be a compact Lie group with Haar measure $dg$ and (finite-dimensional real) Lie algebra $\frak g$. Endow $\frak g$ with an $\hbox{Ad}$-invariant norm $\|\cdot\|_{\frak g}$ so that $\frak g$ ...
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### Automorphism group of real orthogonal Lie groups

I would like to know what is the "outer-automorphism group" $Out$ of $SO(p,q)$ and $O(p,q)$, where $p+q >0$. My working definition of $Out$ is as follows: Let us denote by $Aut(G)$ the ...
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### Vector fields, diffeomorphism subgroups and lie group actions

Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization: Let $\{X_j\} \in Vect(M)$ be a ...
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### Does there exist finite dimensional irreducible representation of Euclidean or Poincare group in which translation and rotation both act nontrivially?

Does there exist any finite dimensional irreducible rep. of Euclidean or Poincare group in which translation and rotation both act nontrivially? Let me firstly clarify my question. For example, we ...
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### Projections of orbifolds

A while back I came across orbifolds, in particular the quotients $SU(2)/U(1)\cong S^2$, $SU(3)/(SU(2)\times U(1)\cong \mathbb{C}P^2$ and $SU(3)/(U(1)\times U(1))$. The way I needed them, was as an ...
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### Orientability of orbit type strata of Lie group actions

Let $G$ be a compact Lie group that acts on a smooth, finite dimensional, oriented manifold $M$, and suppose that such action preserves orientation, i.e., for each $g\in G$, the diffeomorphism $\mu_g$ ...
### Connected subgroups of $SL(2,C)$
Where can I find a list of all connected real Lie groups inside the 6-dimensional real Lie group $SL(2,C)$, up to conjugacy? How can one verify that a partial list is complete? I found on wikipedia a ...