Identifying the simple finite-dimensional Lie algebras over $\mathbb{C}$ by means of simple sequences
Kai Neergård
TL;DR
The paper addresses identifying Dynkin diagrams that correspond to simple finite-dimensional Lie algebras over $\mathbb{C}$ by introducing a universal criterion: a Cartan matrix $A$ is realizable via a positive-definite quadratic form if and only if its symmetrised matrix $B$ is positive definite, which yields a finite root system under Weyl reflections. It relies on the symmetrisability condition $B_{ij} = c_i A_{ij} c_j^{-1}$ with positive $c_i$, and uses Sylvester's criterion applied to a tridiagonal representation to efficiently decide positive definiteness through a sequence $p_i$ defined by $p_0=1$, $p_1=2$, $p_i=2p_{i-1}-(T_{i,i-1})^2 p_{i-2}$. The main results classify the connected positive-definite Coxeter diagrams as $A_l$, $B_l/C_l$, $D_l$, $E_l$ ($6\le l\le8$), $F_4$, and $G_2$, from which the Dynkin diagrams (via orientation of multiple lines) are obtained, including the distinct $B_l$ and $C_l$ cases for $l\ge3$. This yields a unified, constructive route to the classical Dynkin diagram list and eliminates the need for case-by-case root-system constructions in many instances.
Abstract
A novel method of determining which Dynkin diagrams represent simple finite-dimensional Lie algebras over $\mathbb{C}$ is presented. It is based on a condition that is both necessary and sufficient for a suitably defined Cartan matrix to be expressible by scalar products in a Euclidean vector space. The sufficiency of this condition makes unnecessary subsequent verification of the existence of a Lie algebra or root system corresponding to each Dynkin diagram by explicit construction. The Dynkin diagrams are selected by examination of an easily calculated sequence of minors of a symmetrised Cartan matrix. These minors are mostly integers.