# Tagged Questions

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 (group-...

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### Is $(\mathbb R^3\setminus \{0\})/\mathbb R^*$ a smooth manifold?

Let $G=\mathbb R^*$ act on $X=\mathbb R^3\setminus\{0\}$ by pointwise multiplication. That is for any $t\in\mathbb G$ and $(x_1,x_2,x_3)\in X$ we have $$t\cdot(x_1,x_2,x_3)=(tx_1,tx_2,tx_3)$$ Is ...
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### References for the representation theory of $SU(2, 1)$

I couldn't find any reference with the representation theory of this specific case. I found some general stuff but never explicit computations or realizations. The only thing I found on $SU(2, 1)$ ...
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### lie group jacobian function deduce

hi guys I have read the paper at http://www.ethaneade.org/lie.pdf , and regarding the equation (87) I have coded it for proven but found not correct , the code is like this ...
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### Irreducible root system decomposition

I am looking for the name of and a good reference on the following theorem Theorem: let $G$ be a connected, compact and semisimple Lie group, and $T \subset G$ a maximal torus of $G$, there exists a ...
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### Determining whether a Lie group contains more than one conjugacy class of subgroups of a particular isomorphism type

Suppose I have a Lie group $G$. How can one determine whether there is more than one conjugacy class in $G$ of subgroups isomorphic to a given Lie subgroup $H$? Put another way: Fix a Lie ...
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### Compact Lie subgroup of $GL_n(\mathbb{R})$

Let $K\leq GL_n(\mathbb{R})$ be a compact Lie subgroup. I need to prove that $K$ is a conjugate of a subgroup of $O(n)$. The hint is to use the Haar measure, but I really don't see how to do this.
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### Determining the compact roots of the Cartan subalgebras of $\mathfrak sp(2,\mathbb R)$

I want to understand the notions of real vs imaginary roots and compact vs noncompact roots (among the imaginary ones) in the theory of Cartan subalgebras (CSA's) of real semisimple Lie algebras. I ...
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### Decomposition of semi-simple Lie algebras

Background: Let $G$ be a finite-dimensional Lie group, with Lie algebra $L(G)$. A subspace $I\subset L(G)$ which satisfies $[L(G),I]\subset I$ is called an ideal of $L(G)$. A non-abelian Lie algebra ...
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### maximal torus by dimension count?

Suppose $T$ is a maximal torus of $G$ with dimension = $n$. If there is another torus $H \subset G$ of the same dimension, could I then conclude that $H$ is also a maximal torus? In other words once ...
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### Why is the tangent space to the orbit through $p\in\mu^{-1}(0)$ an isotropic subspace of $T_pM$?

I'm reading symplectic geometry notes by Ana Cannas da Silva. The set up is a Hamiltonian action $G\curvearrowright(M,\omega)$ of a Lie group $G$ on the symplectic manifold $(M,\omega)$, with moment ...
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### Lagrangian densities, Lie Groups and Lie Algebras

I'm quite new to Physics and I was having a look for the first time to the Standard model. I'm not sure if the mechanism that I'm describing is directly from Weyl or from others but what I found quite ...
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### Killing form is negative definite

Let $G$ be a compact connected semisimple Lie group and $\frak g$ its Lie algebra. It is known that the Killing form of $\frak g$ is negative definite. What about the Killing form $B$ of the complex ...
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### Determining if two given matrices in the symplectic Lie group $Sp(2)$?

Define the following quaternionic matrices $1=\pmatrix{1&0\\0&1}, i=\pmatrix{0&-1\\1&0}, j=\pmatrix{0&-i\\-i&0}, k=\pmatrix{i&0\\0&-i}$ I am given that the symplectic ...
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### Decomposing representations

The problem I am trying to do is the following: Show that vector representation 5 and adjoint representation 10 in SO(5) decompose respectively into representations of SO(4) as: 5 →4⊕1 10→6⊕4 I ...
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### How many examples exist of Lie groups that are 2-dimesional surfaces?

It is relatively easy to show that $\mathbb{R}^2$ or $\mathbb{T}^2$ are 2-dimensional surfaces with a structure of Lie groups. I can not find other surface which are also a Lie group, there are more ...
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### The orbit of a compact Lie group action

Let $G$ be a compact Lie group acting on a manifold $M$. For each $p\in M$, we define the orbit of $p$ as $G\cdot p:=\{g\cdot p: g\in G\}$. The isotropy group of $p$ is $G_{p}=\{g\in G:g\cdot p = p\}$....
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### Restriction of an isogeny is still an isogeny?

Given that $E \times F \twoheadrightarrow G$ is an isogeny where $E,F$ are both subgroups of $G$, is its restriction to a subgroup $H$ of $G$, $E \times (F \cap H) \twoheadrightarrow (G \cap H) = H$, ...
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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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### Finding the generators of $SU(3)$ different from the Gell-Mann matrices?

I want to find a set of generators of SU(3) different from the Gell-Mann matrices. How should I go about it? Can I construct it in such a way that at least three of the 8 generators when squared gives ...
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### Endomorphism vector bundle isomorphic to the adjoint bundle of its frame bundle?

Could somebody help me to prove the following isomorphism (in particular what is the isomorphism)? End(\xi) \cong ad(E_{\xi}) = E_{\xi} \times_{GL(n,\mathbb{R})} \text{Mat}_n(\...
Let $G \subset \mathrm{GL}_n(k)$ be a connected affine algebraic group over a field $k$ with the following property: for any two distinct elements $g,h \in G$ there exists a vector $x \in k^n, x\neq 0$...
### Is the BGG category $\mathcal{O}$ a Serre subcategory of $\mathfrak{g}$-mod? [duplicate]
Let $\mathcal{O}$ be the BGG category for a be a finite-dimensional, semi-simple complex Lie algebra $\mathfrak{g}$. Let $\mathfrak{g}$-mod be the category of all $\mathfrak{g}$-modules. Is the BGG ...