In the classification of Coxeter groups, or equivalently root systems: $$A_n, B_n/C_n, D_n, E_6, E_7, E_8, F_4, G_2, H_2, H_3, H_4, I_2(p)$$ with $p \geq 7$, the last four fail to generate any simple finite dimensional Lie algebras over fields of characteristic zero because of the crystallographic restriction theorem. However, I know that there exist […]

Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane that is perpendicular to $\alpha$ the elements of root system $\Phi$ in $V$. I want to show that: The Weyl chambers are open, convex and […]

(1) Is there a notion of Cartan matrix for non-semisimple Lie algebra? For example, consider this Lie algebra: $$ [X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [X_i,Y^j] = – f_{ik}{}^j Y^k \qquad\qquad [Y^i,Y^j] = 0 $$ Here, for example, consider 3 generators $X_1,X_2,X_3$ generate a compact semi-simple SU(2) Lie algebra with $f_{ij}{}^k$ given by $f_{12}{}^3=1$ and […]

I’m trying to understand the machinery of root systems for the purpose of classifying complex semisimple Lie algebras. During this process i lost the overview, espacially when it came to highest weight representations and all that, so i’m looking for a short summary: Do isomorphic root systems imply isomorphic Lie algebras (to good to be […]

Let $\Phi$ be a root system in a real inner product space $E$. Define $\alpha^\vee = \frac{2\alpha}{(\alpha, \alpha)}$. Then the set $\Phi^\vee = \{\alpha^\vee: \alpha \in \Phi \}$ is also a root system. Let $B$ be a base for the root system $\Phi$, ie. $B$ is a basis for $E$ and each $\alpha \in \Phi$ […]

Intereting Posts

Show that a graph with 9 vertices has at least five vertices of degree 6, or at least six vertices of degree 5.
Maximum number of linearly independent anti commuting matrices
Find an abelian infinite group such that every proper subgroup is finite
Finding a point along a line a certain distance away from another point!
graph vertex chromatic number in a union of 2 sub-graphs
What is the least value of $k$ for which $B^k = I$?
The reflexivity of the product $L^p(I)\times L^p(I)$
Why must such a group be dihedral?
How to prove that a bounded linear operator is compact?
What is $\limsup\limits_{n\to\infty} \cos (n)$, when $n$ is a natural number?
When is the derived category abelian?
Find $\sum\limits_{k=1}^{12}\tan \frac{k\pi}{13}\cdot \tan \frac{3k\pi}{13}$
Computing diagonal Length of a Square
Line joining the orthocenter to the circumcenter of a triangle ABC is inclined to BC at an angle $\tan^{-1}(\frac{3-\tan B\tan C}{\tan B-\tan C})$
Poincare Duality Reference