# Tagged Questions

For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

20 views

33 views

### Is every element contained in a Borel subalgebra?

Let $\frak g$ be a complex semisimple Lie algebra. Is every $X\in\frak g$ contained in some Borel subalgebra $\frak b$? Attempt: I know that a Borel subalgebra is by definition a maximal solvable ...
10 views

### How many cyclic conjugate class can $su(2)$ has

In the lie algebra of $su(2)$, we can easily find out three linear independent elements $t_1,t_2,t_3\in su(2)$ such that $[t_1,t_2]=t_3,[t_3,t_1]=t_2,[t_2,t_3]=t_1$ and this relation is preserved ...
27 views

### Compute the associated induced Lie algebra action $\text{d}\pi$

Let $G=\mathrm{SL}_2(\mathbb{C})$ and consider the action of $G$ on the space of smooth functions on column vectors $\mathbb{C^2}$ given by $\big(\pi(g)\phi\big)(v)=\phi\left({g^\top}\,v\right)$ for ...
37 views

### Let $G$ be a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogeneous space $\frac{G}{H}$ are connected, then $G$ is connected.

Let $G$ ge a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogenuous space $\frac{G}{H}$ are connected, then $G$ is connected. Remark: For this proof I use the following ...
42 views

### Why can we find a basis for the elements of the Lie algebra?

I am a physicist and we do Lie algebras pretty informally, so I hope my question makes any sense to a mathematician. There is one thing that I don't quite understand, which is why we can find a basis ...
30 views

### Finite dimensional, irreducible representations of the Lie superalgebra gl(1|1)

I am wondering how the finite dimensional, irreducible representations of the Lie superalgebra gl(1|1) are parametrized. I understand that they are all highest weight, and that the only non-trivial ...
10 views

19 views

### Calculating the Killing form of the classical algebras

I was reading in the book Parabolic Geometries (p.170-172). But I didn't get how he obtained the Killing form from the roots of each simple Lie algebra. For example: In case $sl(n,\mathbb C)$, given ...
30 views

### Lie algebra associated to Leibniz algebra

We know that for any Leibniz algebra $L$ we can associated its Lie algebra denoted by $L_{Lie}$. for example the ideal generated by $\{[x,x] | x\in L\}$ determines the non-Lie character of $L$. Is it ...
38 views

17 views

37 views

### Compute $ad_X$, $ad_Y$, and $ad_Z$ relative to a basis

For a lie algebra $\mathbb{g}$ we can define the adjoint representation as: $ad: \mathbb{g} \rightarrow End(\mathbb{g})$ as the map such that $ad_x(y)=[x, y]$ for all $\in \mathbb{g}$ I am ...
22 views

### Lie algebra of $SL_2(\mathbb{R})$ and show $\exp(X)=I+X$ where $I \in SL_2(\mathbb{R})$ and $X \in sl_2(\mathbb{R})$

I am doing an undergraduate course on Representation Theory and am trying to solve these consecutive questions. The first two I am ok with (I just included them for context), but I could do with some ...
21 views

### Lie subalgebra generated by a subset of a basis of root system

Let $L$ be a semisimple Lie algebra, ad let $\Phi$ be a root system. Fix a fundamental root system $\Delta$ of $\Phi$ with corresponding to $\Phi^+$. I would like to understand the subalgebra ...
63 views

21 views

47 views

### Why does $\frac{d}{dt}e^{X+tY} |_{t=0}$ depend linearly on $Y$ with $X$ fixed?

I'm studying the proof of Baker-Campbell-Hausdorff formula from Brian Hall's book Lie Groups, Lie Algebras and Representations. I am stuck at this part: I don't get why continuity of exp implies ...
65 views

### Direct calculation of the tangent space of $SO(3)$

Let $SO(3)$={$RR^T=I$, $det(R)=1$}, I need to show that a base of the tangent space in the identity is given by: $$E_i=\frac{d}{dt}\exp(tL_i)|_{t=0}$$ where L_1= \left(\begin{matrix} 0& 1& 0\...
45 views

### finding high weight vector in Verma module

Let $\frak{g}$ be a (semi-)simple lie algebra. Let $\lambda$ be a dominant integral weight. Denote $L(\lambda)$ to be the irreducible representation of highest weight $\lambda$. From BGG resolution, ...
19 views

### How to prove this lemma about Weyl group?

Let $\mathscr{W}$ be the Weyl group of a root system $\Phi$ with basis $\Delta$. If $\sigma\in \mathscr{W}$, $\sigma = \sigma_{\alpha_1} .. \sigma_{\alpha_t}$ where $\alpha_1, ...,\alpha_t \in \Delta$...
41 views

Let $G_1 \subset G$ be Lie groups and $\mathfrak{g}_1, \ \mathfrak{g}$ the corresponding Lie algebras. Assume that there is a non-degenerate bilinear form $\langle \cdot, \cdot \rangle$ on $\mathfrak{... 0answers 16 views ### why$ t\oplus {\sqrt{-1}t}$is a Cartan subalgebra of$\mathfrak{g}$? Let$G$be the complexification of a connected, simply-connected compact Lie group$K$and$\mathfrak{g}$be a Lie algebra of$G$. If$t$is a maximal abelian subalgebra of the Lie algebra$\mathfrak{...
Let $L$ be a Lie algebra and $I$ be an ideal in $L$. If $M,N$ are $L$-module with $N\subset M$. If $M/N$ is an $L/I$-module and $K/N$ is am $L/I$-submodule of $M/N$ with $(L/I)\cdot (K/N)=0$. Does ...
### Does a $k$-derivation of $k[G]$ into $k$ induce an element of $k[G]$?
Let $G$ be a linear algebraic group over an algebraically closed field $k$, and let $e$ be the identity of $G$. Then $k$ is a $k[G]$-module via the action $f \cdot a = f(e)a$. The tangent space of \$...