Articles of lie groups

Proofs that: $\text{Sp}(2n,\mathbb{C})$ is Lie Group and $\text{sp}(2n,\mathbb{C})$ is Lie Algebra

Consider following Lie Group: $$ \text{Sp}(2n,\mathbb{C})=\{g\in\text{Mat}_{2n}(\mathbb{C})\mid J=g^TJg\}\quad\ where\quad J=\begin{pmatrix} 0 & 1_n \\ -1_n & 0 \end{pmatrix} $$ And the corresponding Lie Algebra: $$ \text{sp}(2n,\mathbb{C})=\{g\in\text{Mat}_{2n}(\mathbb{C})\mid g^TJ+Jg=0\} $$ Are there any basic proofs that $\text{Sp}(2n,\mathbb{C})$ is a Lie Group and that $\text{sp}(2n,\mathbb{C})$ is the corresponding Lie Algebra without using submersions (seen here: Why is $Sp(2m)$ as […]

What are the length of the longest element in a Coexter group for every type?

What are the length of the longest element in a Coxeter group for every type? Thank you very much.

$SL(3,\mathbb{C})$ acting on Complex Polynomials of $3$ variables of degree $2$

So I’m given the following definition: $h(g)p(z)=p(g^{-1}z)$ where g is an element of $SL(3,\mathbb{C})$, $p$ is in the vector space of homogenous complex polynomials of $3$ variables and $z$ is in $\mathbb{C}^3$. What I’m having trouble showing is that mapping $g$ to $h(g)$ is a group homomorphism. Namely, I know that $h(ab)p(z)=p(b^{-1}a^{-1}z)$, but I can’t […]

The complex structure of a complex torus

A complex torus $X$ of complex dimension $1$ is just a quotient of $\mathbb{C}$ by a lattice $\Lambda=\mathbb{Z}\oplus \tau\mathbb{Z}$ where $\tau=a+ib$ is a complex number with $b>0$. The complex structure is determined by this $\tau$. I want to undestand this complex structure as an endomorphism $J:TX \to TX$ such that $J^2=-1$ (an almost complex structure). […]

Connections of Geometric Group Theory with other areas of mathematics.

I’m a master’s student in the Turin University. At the end of my studies, I have to write a master thesis. My main interest is geometric group theory, but it is not a research area of the Turin’s mathematical department. My professors in algebra and geometry are principally interested in algebraic geometry, commutative algebra or […]

Weyl group, permutation group

Let $U(n)$ be the unitary group and $T$ its maximal torus (group of diagonal matrix) and $N(T)$ the normalizer of $T$ in $G$. Why $N(T)/T$ is the permutation group $S_{n}$?

How To Formalize the Fact that $(g, h)\mapsto dL_g|_h$ is smooth where $g, h\in G$ a Lie Group

Let $G$ be a Lie group. I am wondering if there is a way to say that the map $(g, h)\mapsto dL_g|_h$ defined on $G\times G$ is a smooth map (Here $L_g$ is the left translation map from $G$ to $G$ and by $dL_g|_h$ I mean the differential of $L_g$ at $h$). The challenge here […]

Finding the basis of $\mathfrak{so}(2,2)$ (Lie-Algebra of $SO(2,2)$)

I want to calculate the Basis of the Lie-Algebra $\mathfrak{so}(2,2)$. My idea was, to use a similar Argument as in this Question. The $SO(2,2)$ is defined by: $$ SO(2,2) := \left\{ X \in Mat_4(\mathbb R): X^t\eta X = \eta,\; \det(X) = 1 \right\} $$ (With $\eta = diag(1,1,-1,-1)$) With the argument from the link, i […]

The injectivity of torus in the category of abelian Lie groups

HI: I have the following question: Definition: A Lie group $T$ is called a torus if $T\cong \prod_{1\leq i\leq k} \mathbb{R}/\mathbb{Z}$ for some $k\in \mathbb{N}$. ${\bf Question}$: Is it true that a torus is an injective object in the category of abelian Lie groups? Thanks very much!

List of connected Lie subgroups of $\mathrm{SL}(2,\mathbb{C})$.

I am not familiar with the theory of Lie groups, so I am having a hard time finding all the connected closed real Lie subgroups of $\mathrm{SL}(2, \mathbb{C})$ up to conjugation. One can find the real and complex parabolic, elliptic, hyperbolic, subgroups, $\mathrm{SU}(2)$, $\mathrm{SU}(1,1)$ and $\mathrm{SL}(2,\mathbb{R})$ (the last two ones are isomorphic though), the subgroup […]