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Let $G$ be a compact Lie group assumed to be a subgroup of $U(n)$. Also, let $T$ be a maximal torus of G. Then there exists a basis $\{v_1, \ldots ,v_d\}$ of the Lie algebra of $G$, $\mathfrak{g}$, for which $${\rm Ad}_{g_0^r}(v_{2k-1}) = \cos(r\theta_k)v_{2k-1} + \sin(r\theta_k)v_{2k}$$ $${\rm Ad}_{g_0^r}(v_{2k}) = -\sin(r\theta_k)v_{2k-1} + \cos(r\theta_k)v_{2k}$$ where $\theta_i\in [0,2\pi)$, $i=1\ldots m$, $d=2m$ is the dimension of $G$ and $g_0$ is a topological generator for $T$.

So for $k=1\ldots m$ there are continuous homomorphisms $\tilde{\gamma}_k: T\to SO(2) \simeq \mathbb{T}$ with $\tilde{y}_k(g_{0^r}) = R(r\theta_k)$ where

$$R(\theta) = \begin{bmatrix}\cos(\theta)&-\sin(\theta)\\ \sin(\theta)&\cos(\theta) \end{bmatrix}.$$

Now for each $\tilde{\gamma}$ we take its derivative and by identifying the Lie algebra of $\mathbb{T}$ with $\mathbb{R}$ we get an $\mathbb{R}$-linear transformation $\gamma : \mathfrak{t}\to\mathbb{R}$. We will call such a guy for a root.

Now assume that $\gamma$ is non-trivial (i.e. $\gamma \neq 0$) we define $\mathfrak{g}(\pm\gamma) \subseteq \mathfrak{g}$ to be the maximal subspace for which there is a basis $\{u_1, \ldots, u_{2k}\}$ such that for $x\in\mathfrak{t}$ and $j=1,\ldots, k,$ then

$${\rm Ad}_{\exp(x)}(u_{2j-1}) = \cos\gamma(x)u_{2j-1} + \sin(\gamma(x))u_{2j}$$ $${\rm Ad}_{\exp(x)}(u_{2j}) = -\sin\gamma(x)u_{2j-1} + \cos(\gamma(x))u_{2j}.$$ This is called the root space associated to the root pair $\pm\gamma$ and by definition it is non-trivial.

Questions

  1. Why is the root space non-trivial?
  2. Assume that I calculate ${\rm Ad}_{\exp(x)}(u_{2j-1})$ and ${\rm Ad}_{\exp(x)}(u_{2j})$ as above and find that they are on the given form for some function $\gamma(x)$: how is that sufficient to say that $\gamma(x)$ actually is the derivative of some $\tilde{\gamma}$? E.g: with a proper choice of basis $U, V, W$ for $\mathfrak{so}(3)$ with $U \in \mathfrak{t}$ then $$\exp(tU)V\exp(tU)^{-1} = \cos t V + \sin tW$$ and $$\exp(tU)W\exp(tU)^{-1} = -\sin tV + \cos tW.$$ How is this sufficient to deduce that there are two non-trivial roots $\pm\gamma$ where $\gamma(tU) = t$ for $t\in\mathbb{R}$?

Any help is greatly appreciated.

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