Articles of lie algebras

3-dim simple complex Lie algebra

If we know that a 3-dimensional Lie algebra $L$ with $[L,L]=L$ is simple. How to prove that the only (up to isomorphism) 3-dimensional complex Lie algebra $L$ with $L=[L,L]$ is $sl_2(\mathbb C)$?

Definitions of length function on a Weyl group

Let $\Phi$ be an irreducible root system and $W$ the Weyl group of $\Phi$. Denote by $\Delta=\{\alpha_1, \alpha_2,\ldots,\alpha_l\}$ the corresponding base. Can anyone give me the standard definition of length function of an element of $W$? Furthermore, are there any upper-bound or lower-bound of lengths? For instance, let $w=r_{\beta_1}\ldots r_{\beta_k}$, $\beta_i \in \Phi$ be any […]

Is the exponential map ever not injective?

Let $G\subseteq GL_n(\mathbb R)$ and let $\mathfrak g$ denote its Lie algebra. Let $e: \mathfrak g \to G$ be the map $X \mapsto e^X$. Does there exist an example of $G$ and $\mathfrak g$ such that $e$ is not injective? Of course I think the answer is no, there is no such example because $e: […]

Where does the ambiguity in choosing a basis for a Lie algebra come from?

This is a follow-up to this question. For matrix Lie algebras, we can define the Lie algebra $g$ of a group $G$ as the set $T_a \in g$ that yield an element of $G$ when put into the exponential map: $$ e^{\sum_a \alpha_a T_a} \in G $$ for some numbers $\alpha_a$ For the $SO(n)$ groups, […]

Computing the differential of the map $\phi: M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$

Let $\phi: M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ be the map $$(B_1,B_2)\mapsto [B_1,B_2]$$ which takes two $2\times 2$ matrices to its Lie bracket. Then why does $d\phi_{(B_1,B_2)}:M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ send $$(D_1,D_2)\mapsto [B_1,D_2]+[D_1,B_2]?$$ $$ $$ Taking $B_1=(g_{ij})$ and $B_2=(h_{ij})$, I do not think taking the partials of the Lie bracket $[B_1,B_2]=$ $$ \left[ \begin{array}{cc} g_{12} h_{21} – g_{21} h_{12} […]

Fundamental weights of $A_n$

I have the following problem: Let $\mathfrak{g}$ be the Lie algebra of type $A_n$. We choose $e_i^*-e_{i+1}^*$ as simple roots. Is there a closed formula for the fundamental weights? Thank you!

$Hom_G(\pi,\sigma)$ = $Hom_{\mathfrak{g}}(d\pi,d\sigma)$?

Let $G$ be a Lie group. Let $\mathfrak{g}$ be the corresponding Lie algebra. Let $(\pi,V)$ and $(\sigma, W)$ be representations of $G$, with corresponding differentials $d\pi$ and $d\sigma$, which are $\mathfrak{g}$ representations. I want to know about the relationship between $Hom_G(\pi,\sigma)$ and $Hom_{\mathfrak{g}}(d\pi,d\sigma)$. Are they equal? Any help or reference is very much appreciated.

How to calculate the derivative of $F=\frac{e^{W}-1}{W}$ where $W = _\times$ in $\omega$?

How to calculate the derivative of $F=\frac{e^{W}-1}{W}$ in $\omega$ where $W = [\omega]_\times$ and $\omega\in\mathbb{R}^3$? I am aware of a solution (solution 1) and briefly show below, but I am not happy with it for its complexity and am looking for a possibly better solution (solution 2). Inspired by hans’ brilliant answer to a related […]

Tangent bundle of the quotient of matrix Lie groups

Let $G:=GL^+(n)$ (all invertible $n \times n$-matrices with positive determinant) and $K:=SO(n)$. Let $\mathfrak{g}, \mathfrak{k}$ denote their Lie algebras and $E:=\mathfrak{g}/\mathfrak{k}$. I would like to understand Why the (total space of) tangent space $T(G/K)$ of $G/K$ can be identified with $(G \times E)/K$. Some facts which are familiar for me: 0. $K$ acts on $E$ […]

Exterior product generates the infinitesimal rotations — what is the geometric significance?

Question: What is the geometric significance of the fact that the exterior products of the unit basis vectors in $\mathbb{R}^3$ generate a basis for the Lie algebra of the pure rotation group $SO(3)$? Also, if applicable, what is the algebraic significance or the physical significance? This is one of the first things I found to […]