Articles of root systems

Lie algebra-like structure corresponding to noncrystallographic root systems

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 […]

How to understand Weyl chambers?

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 […]

— Cartan matrix for an exotic type of Lie algebra —

(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 […]

Relation between root systems and representations of complex semisimple Lie algebras

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 […]

How to prove that $B^\vee$ is a base for coroots?

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 $\alpha \in \Phi$ […]