# Tag Info

Accepted

### The sum of two roots when their pairing is zero

In a root system $R$ of type $B_2$, there are various pairs of short roots $\alpha, \beta$ with $(\alpha,\beta)=0$ but both $\alpha \pm \beta \in R$.
• 25.3k
Accepted

• 20k

### Finding the lie algebra of the symplectic lie group

A much simpler proof exists using standard properties of the matrix exponential. We wish to show that if $X^TJ + JX = 0$ holds, then $\exp(tX)^TJ\exp(tX) = J$ for all $t$. To that end, we note that ...
• 222k

### Geometry of Semi-Simple Lie Groups

The sentence "so it can be considered merely as a vector field in dimension one" makes no sense as a manifold is not the same thing as a vector field. There is no single vector field ...
• 76.3k
Accepted

### A compact connected Lie group with trivial center is a direct product of simple groups

This is true: a compact connected Lie group with trivial center is a product of simple compact connected Lie groups with trivial center. Suppose $G$ is a compact connected Lie group. Then $G$ has a ...
Accepted

### Prove all Lie group homomorphisms of the circle have certain form

I think you should just impose more. If $w\in \mathrm{ker}(\phi)$ such that $w^m=1$ then, passing to a power of $w$ if necessary, we can assume that $gcd(m,N)=1$, with $w^m=1$. The goal is to now show ...
• 2,662
Accepted

### Explicit Lie group embedding of $O(n)$ into $SO(n+1)$

I’ll just write up an answer with a few ways of attacking the problem, and let you fill in some of the details (which can easily be found if you search the site or any decent book). First, let me make ...
• 51k
Accepted

### Differential of conjugation map is smooth

This is basically by definition of smoothness: smooth means infinitely differentiable, so the derivative is still smooth. More precisely, suppose $M$ and $N$ are smooth manifolds and $f:M\to N$ is ...
• 326k
Accepted

### Geometric Intuition for Hamiltonian Actions

Per my comment, this answer is largely cribbed from an excellent text by Peter Woit called Quantum Theory, Groups and Representations: An Introduction. In my edition this discussion occurs in Chapter ...
• 5,565

### Trivial principal bundles and curvature.

In the non-abelian case, the curvature forms $F_{\mathcal{M}}^{A}$ and $F_{s}$ are generally not equal. This is because the connection form $A$ and the section $s$ transform differently under gauge ...
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• 1,076
Accepted

• 151k
Accepted

### Natural rep of $SL(2,\mathbb{Z})$ and decomposing tensor powers into irreps

It's just the same as for $SL_2(\mathbb{C})$. Restricting an algebraic representation of $SL_2(\mathbb{C})$ to $SL_2(\mathbb{Z})$ preserves irreducibility. The key fact here is that $SL_2(\mathbb{Z})$ ...
• 11.1k

### Can every Lie group be realized as conformal group of smooth manifolds

First of all, one cannot talk about conformal transformations of a smooth manifold: For the notion of conformality to be defined you need an extra structure besides a smooth atlas. I will work with ...
• 93.8k

### What does $5 \bigotimes 5 = 10 \bigoplus 5$ mean for SU(5) Lie group?

Let $V=\Bbb C^5$ be the "natural" representation of $SU(5)$, with $\bar V$ the conjugate representation. Then $V\otimes V$ splits into $V\odot V$ and $V\wedge V$, symmetric and skew-...
• 124k
Accepted

### Calculating the differential of the map $A \mapsto A A^T$, $A \in M(n\times n, \mathbb{R})$

Regarding your 2nd question: the tangent space of the space of matrices $M(n\times n, \mathbb R)$ at a point $A$ is, as you said, identified with $M(n\times n, \mathbb R)$ itself. Regarding your 1st ...
• 418